The Geometry of Geodesics (Pure and Applied Mathematics Volume 6)

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THE GEOMETRY OF GEODESICS

PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks EDITED BY

PAUL A. SMITH and SAMUEL EILENBERG Columbia University, New York, N. Y.

I : A. SOMMERFELD, Partial Differential Equations in Physics, 1949 (Lectures on Theoretical Physics, Volume VI) 11: R BAER, Linear Algebra and Projective Geometry, 1952 111: H. BUSEMANN AND P. J. KELLY, Projective Geometry and Projective Metrics, 1953 IV: S. BEROMAN AND M. SCHIFFER, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, 1953 V: R.P. BOAB, JR., Entire Functions, 1954 VI: H. BUSEMANN, The Geometry of Geodesics, 1955 VII: C. CHEVALLEY, Fundamental Concepts of Algebra, in preparation

ACADEMIC PRESS INC., P U B L I S H E R S

THE GEOMETRY OF

GEODESICS HERBERT BUSEMANN Department of Mathematics University of Southern California

ACADEMIC PRESS INC., Publishers NEW YORK, N. Y. 1955

ACADEMIC PRESS INC. 125 East 23rd Street New York 10, N . Y .

All Rights Reserved

NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT

WRITTEN PERMISSION FROM THE PUBLISHERS

Libvary of Corigrcss Catalog Card Number: 55 - 11291

PREFACE CONTENTA N D METHODS

“A geometric approach to qualitative problems in intrinsic differential geometry” would be a brief description of the present book, which extends the methods developed in the author’s “Metric Methods in Finsler Spaces and in the Foundations of Geometry,” Princeton, 1942. In the earlier book the emphasis was on spaces in which the geodesic through two points is unique, whereas it is now on the general case in which the geodesics have only local uniqueness properties. Consequently, the relations to the foundations of geometry are no longer very relevant and Finsler spaces are the principal subject. For two reasons the latter do not appear in the title: Firstly, “Geometry of Geodesics” expresses clearly the method of attack and the intrinsic aspect. Secondly, the term Finsler Space means to many not only a type of space but also a definite approach: The space is considered as a set of line elements to which euclidean metrics are attached. The main problems are connected with parallelism. In spite of the great success of Finder’s thesis, the later development of this aspect lacks simple geometric facts to the extent that their existence in non-Riemannian geometry has been doubted. Here we take the attitude that Finsler spaces are point spaces of an essentially new character which we must approach directly without being prejudiced by euclidean or Riemannian methods. Of course, it is impossible to live up to this program. The euclidean tradition is much too strong to be shaken off readily, and it may take several generations of mathematicians to gain real freedom. However, the direct approach has produced many new results and has materially generalized many known phenomena,l thereby revealing two startling facts: much of Riemannian geometry is not truly Riemannian and much of differential geometry requires no derivatives. See page 413. All notes to the text appear on pages 413-415.

vi

PREFACE

Actually only the first fact is startling; the second is to a certain extent a consequence of the first. If all theorems were omitted from the present volume which have no bearing on, or become trivial for, Riemann spaces, and the remaining facts were applied only to the latter, there would still remain an interesting book, because it contains - besides many facts which are new also for Riemannian spaces - numerous well-known results of Hadamard, Cartan, Cohn-Vossen, Nielsen and others. Thus the non-existence of concrete geometric facts in Finsler spaces as compared to Riemann spaces proves to be a myth and there emerges the highly important problem of gaining a clear understanding of the true realm of Riemannian geometry, i. e., to recognize the character of the theorems for which it is essential that the local unit spheres be ellipsoids rather than arbitrary convex surfaces with center. At present we have unfortunately not the slightest idea how such a characterization might look. I t is now clear why differentiability hypotheses are often unnecessary : our geometric axioms for the existence of geodesics (see the definition of G-space on page 37) extract from Finsler spaces an essential part of their qualitative geometric properties, but not from Riemann spaces. The difference between the two types of spaces is of an essentially infinitesimal nature, which explains why there are hardly &y Riemannian theorems proved without differentiability.a The present methods also offer the esthetical and methodological advantage of a geometric approach to geometric problems and they extend to differential geometry the trend of replacing calculations by syllogisms which has so deeply permeated modem mathematics. Some differential geometers find this tendency unnatural. We need not take them seriously: the term “differential geometry” carries no obligation to use derivatives, just as names like ‘‘calculus of variations” or “algebra” do not confine us to operating within their original meanings. Nor is there any reason why reduction of hypotheses (and their exact formulation!) should be undesirable in differential geometry, but meritorious in all other parts of mathematics.a Moreover, once G-spaces have been introduced, our basic assumptions remain the same throughout the book. These form, therefore, an adequate basis for many geometric theories, and it does not seem wholly unreasonable to expect that the concept of G-space may eventually be generally accepted in the same way as, for example, the concept of field in algebra.

PREFACE

vii

PREREQUISITES Large parts of the book can be understood with a very moderate mathematical background. The first three chapters use only the elements of point set topology, a little dimension theory and a few facts on convex bodies. The remaining part, except for the last four sections, is accessible to anyone who knows the idea of covering space and its relations t 3 the first fundamental group. The last four sections use topological and Lie groups. However, in order to grasp the significance of the problems, some familiarity with noneuclidean geometry and classical differential geometry is necessary. The book “Projective Geometry and Projective Metrics” by the author and I-’.J. Kelly (1953), which appeared in this series, Will prove very useful as preparation for the present work. ACKNOWLEDGMENTS It had been the author’s intention to “sometime” write a comprehensive introduction to his geometric approach to Finder spaces. Professor H. G. Forder persuaded him to do so at the present time, read those parts which were written in New Zealand, and made many suggestions. Sincere thanks go to him as well as to the Fulbright authorities, who made the association with Professor Forder possible. For assistance in proofreading the author is greatly indebted t o Dr. Flemming P. Pedersen and to Mr. Eugene Zaustinsky.

H. B.

This Page Intentionally Left Blank

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTERI . THE BASICCONCEPTS . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Compact and finitely compact metric spaces . . . . . . . . . . . . . 3 . Convergence of point sets . . . . . . . . . . . . . . . . . . . . . . 4 . Motion and isometry . . . . . . . . . . . . . . . . . . . . . . . . 5. Curves and their lengths . . . . . . . . . . . . . . . . . . . . . .

Preface

& S eg m en t s . . . . . . . . . . . . . . . . . . . 7. Geodesics . . . . . . . . . . . . . . . . . . . 8 . G-spaces . . . . . . . . . . . . . . . . . . . . 9 . Multiplicity . Geodesics without multiple points . 10. Two-dimensional G-spaces . . . . . . . . . . . . 11. Plane metrics without conjugate points . . . .

. . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . . . .......... . . . . . . . . . . .

v 1 1 3 10 14 19 27 30 36 44 49 56

.................... 65 12. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 13. Planes with the Desargues property . . . . . . . . . . . . . . . . . 66 14. Spaces which contain planes . . . . . . . . . . . . . . . . . . . . 76 15. Riemann and Finder spaces. Beltrami's theorem . . . . . . . . . . . 82 16. Convex sets in affine space . . . . . . . . . . . . . . . . . . . . . 87 17. Minkowskian geometry . . . . . . . . . . . . . . . . . . . . . . . 94 105 18. Hilbert's geometry . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTERI1. DESARGUESIAN SPACES

CHAPTER111. PERPENDICULARS A N D PARALLELS . . .

. . . . . . . . . . . . 115 19. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 20. Convexity of spheres and perpendicularity . . . . . . . . . . . . . . 117 21. Characterization of the higher-dimensional elliptic geometry . . . . . . 124 22 . Limit spheres and co-rays in G-spaces . . . . . . . . . . . . . . . . 130 23 . Asymptotes and parallels in straight spaces . . . . . . . . . . . . . 137 24. Characterizations of the higher-dimensional Minkowskian geometry . . . 144 25 . Characterization of the Minkowski plane . . . . . . . . . . . . . . . 153

...................... 26 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 . Locally isometric spaces . . . . . . . . . . . . . . . . . . . . . .

CHAPTERI V . COVERING SPACES

.

28 The universal covering space 29 . Fundamental sets . . . . .

.................... ....................

165 165 167 174 181

TABLE OF CONTENTS

X

. Locally Minkowskian. hyperbolic.

30 31 32 33 34

or spherical spaces

.........

188

. Spaces in which two points determine a geodesic . . . . . . . . . . . 199

. Free homotopy and closed geodesics . . . . . . . . . . . . . . . . . 204 . Metrics without conjugate points on the torus . . . . . . . . . . . . $16 . Transitive geodesics on surfaces of higher genus . . . . . . . . . . . 223

.

CHAPTER v THE INFLUENCE 36. 36. 37 38 39 40 41 42. 43 . 44 .

. . . . .

OF THE

SIGN OF

THE

CURVATURE ON

THE

.

GEODESICS 235

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . Non-positive curvature in the theory of parallels Straightness of the universal covering space r. . . . . . . . . . . The fundamental groups of spaces with convex capsules . . . . . . Geodesics in spaces with negative curvature Relation to non-positive curvature in standard sense . . . . . . . Angular measure . . . . . . . . . . . . . . . . . . . . . . . . . . Excesa and characteristic Simple monogons. total excess. surfaces with positive excess

236

236

. . . . . . . . . . . 248 . . 264 . . 258 . . . . . . . . . . . . . 262 . . 267 273 . . . . . . . . . . . . . . . . . . . . . . 282 . . . . . . 292 SPACES . . . . . . . . . . . . . . . . . . . . 307 CHAPTER VI . HOMOGENEOUS 46. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 . . . . . . . . . . . . . . . . . . . . 309 46. Spaces with flat bisectors I 47 . Spaces with flat bisectors I1 . . . . . . . . . . . . . . . . . . . . 320 48. Applications of the bisector theorem . The Helmholtz-Lie Problem . . 333 49. Involutoric motions . . . . . . . . . . . . . . . . . . . . . . . . 343 60 . New characterizations of the Minkowskian spaces . . . . . . . . . . . 360 61. Translations along two lines . . . . . . . . . . . . . . . . . . . . 359 52 . Surfaces with transitive groups of motions . . . . . . . . . . . . . . 366 63. The hermitian elliptic and similar spaces . . . . . . . . . . . . . . . 374 54. Compact spaces with pairwise transitive groups of motions . . . . . . 386 I

G5. Odd-dimensional spaces with pairwise transitive groups of motions

...

394

. . . . . . . . . . . . . . . . . . . .403 NOTESTO THE TEXT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 APPENDIX. Problems and Theorems

CHAPTER I

THE BASIC CONCEPTS 1. Introduction

The spaces occurring in this book are metric and share two essential pro perties with the spaces usually considered in differential geometry : the firsi guarantees that the space behaves in many respects like a finite-dimensionaJ space which cannot be extended without increasing the dimension ; in differential geometry it is frequently called completeness or normality. I t is equivalent to the Bolzano-Weierstrass Theorem : a boundecl infinite set has an accumulation point. We call a metric space with this property /initely c o m ~ a c t ;the term seems to go back to M. Morse. The second property is the arrangement of the points into geodesics, and is obtained here by imposing conditions on the existence and the uniqueness of solutions of the triangle equality. In Sections 2-5 we study the implications of finite compactness alone. For the benefit of those who approach the subject from a background of classical differential geometry we show first that certain standard theorems in coordinate spaces about uniform continuity, extrema, and selection of subsequences follow from finite compactness. For others it will suffice to simply check on the terminology which varies slightly with different authors. In Section 3 we define convergence of sets once for ever in terms of Hausdorff's closed limit. Otherwise convergence would have to be defined in each case as it arises, for instance, for the hyperplanes in E". Nowadays we are not satisfied with the mere definition of a limit, we wish to ascertain the nature of the topology induced by this limit. We show that the closed subsets of a finitely compact space may be metrized such that they become again a finitely compact space and the resulting 1imit.k equivalent to the closed limit. Hausdorff had proved the corresponding result for compact spaces. Again, in order to obviate special agreements in individual cases the next Section 4 derives a similar result for motions, i. e., distance preserving mappings of the space on itself. It is shown that the motions of a finitely compact space form with a suitable right-invariant metric themselves a finitely compact space, in which convergence is equivalent to pointwise convergence. 1

2

GEOMETRY OF GEODESICS

1

We then discuss length of curves. The definition and properties of length are very much the same as in elementary spaces. The length of any curve is at least the distance of its endpoints. If the two numbers are equal we call the curve a segment, because it is then isometric to a segment in ordinary space. In the intrinsic differential geometry of surfaces a segment is a geodesic arc which is a shortest connection of its endpoints. A consistent use of segments would shorten many arguments in differential geometry, but as the literature shows, the concept is not indispensable. In the present setup it is basic. It is the subject of Section ti, where Menger's convexity is introduced as a sufficient condition for the existence of segments. A geodesic (Section 7 ) is a curve which extends indefinitely in both directions and behaves locally like a segment. This definition merely abstracts the process of continuing indefinitely a solution of the differential equations for a geodesic. To guarantee the existence of geodesics it must be postulated that prolongation is locally possible. So far questions of uniqueness have been entirely left aside. In differential geometry the assumptions are always such that a line element determines a geodesic, hence continuation of a solution can be effected in one way only. Our final postulate is therefore uniqueness of prolongation. The spaces which satisfy all five postulates, viz., that they are metric, finitely compact, and convex in Menger's sense and that prolongation is locally possible and unique, will be c d e d C-spaces. They are the primary subject of this book. Section 8 discusses the most elementary properties of these spaces. We then (Section 9) define the nbultiplicity for a point of a geodesic and treat the simplest types of geodesics. In Section 10 we show that a two-dimensional G-space is always a manifold. Whether this is true for higher dimensions is an interesting, but at present quite inaccessible, problem. Particiilarly simple and important are those G-spaces where prolongation is possible in the large. We call such spaces straight, because each geodesic is isometric to an ordinary straight line and is uniquely determined by any two of its points. In the language of the calculus of variations they are the simply connected spaces without conjugate points. A two-dimensional straight space is homeomorphic to the plane. Each geodesic is an open Jordan curve which tends to infinity if traversed in either direction, and there is exactly one curve through two given points. In Section 11 we prove: if a system of Jordan curves in the plane with these two properties is given, then the plane can be metrized as a straight space with the given curves as geodesics. For plane metrics r~4houtconjugate points we thus solve the inverse problem of the calculus

2

THE BASIC CONCEPTS

3

of variations in the large. This is a first example where the present methods not only reestablish a classical result without differentiability hypotheses, but go far beyond it by providing a solution in the large. The result is used for the construction of non-Desarguesian spaces which illustrate Chapter 11.

2. Compact and liriitcly compact met,ric spaces The basic concept underlying all the investigations of this book is that of the distance between two points of a space. The properties of distance will depend to some extent on the space considered, but we shall always assume that distance satisfies the following conditions: (1) the distance of a point from itself is zero,

(2) the distance of two distinct points is positive, and the distance of the first point from the second is the same as that of the second from the first, (3) distance is shortest distance; if we interpolate a point between two given points, the distance between the given points is less than the sum of their distances to the interpolated point.

These assumptions can be stated a little more exactly by formulae. We consider a set R of elements a, b, . ., y , z; we call the elements, “points”, and the set R, a “space”.

.

Our assumptions are:

A real valued function x y is defined on all pairs x , y of points in R which has the properties: x x = 0,

(1)

xy

(2)

(3)

xy

= yx

+ yz 2 xz

> O , if

x i . y,

(the triangle inequality).

A space with these properties is called “metric”. As a minor point, we may mention that these conditions can be replaced by x y = 0 if and only if x = y ,

xy+zy2xy.

4

8.1


COMMENTS ON

THE AXIOMS

These properties are too general to serve as a basis for an interesting geometric theory; for they are satisfied by the following trivial space R,: Take any set of elements and define x y = 0 when x = y, xy = 1, when x # y. Nevertheless the conditions exclude many interesting spaces, for example, spaces which occur in the calculus of variations frequently do not satisfy x y = y x ; for a practical illustration, if the distance between two points x , y on the surface of the earth be measured by the time it takes to walk from one to the other, then x y and y x will differ if one direction is uphill, and the other downhill. Much of the early work in this book has been extended to non-symmetric distance’, but the most interesting later results are often meaningless or have not been carried over. Thus the extension is, at least at present, not justified by the results. The condition x y > O for x # y fails in the spaces considered in the theory of relativity. Finally, if we omit the triangle inequality, distance loses its meaning of shortest distance, and so much of its geometric interest. Investigations based on (l),(2) only have therefore a different aim and trend. Partly they concern spaces in which (3) holds, but is not assumed*, because it implies the following very strong continuity property of distance : IPq -P‘4’1
0, contains all but a finite number of the points of {p,}, we say that {p,,} “converges to P I ’ , and that p is its “ h i t point”, we write, p = lim p,, or p , --c p . This is equivalent to

a,

..

1)

2.6


0. A set M is “bounded” if there is a number a such that we have < a for any pair of points x , y of M.Then, if p is any point of R, there is a number /3 such that M C S(p, p). Conversely, if there is a point p, and a number p such that M C S(p, p), then M is bounded. A sequence {p,} of points is “bounded”, if the point set which it traverses is bounded; or if p p, < p for some p, p and all 1’.

p,p

-*

xy

(2.6) A converging sequence is bounded and has just one accumulation point. I n a finitely compact space (see below) the converse holds.

COMPACTAND

FINITELY COMPACT SETS

A set C is “compact” if every infinite subset of C has an accumulation point, and this lies in C. Thus in our terminology, a compact set is always closed. A compact set C satisfies the Heine-Bore1 Theorem: (2.7) If C is contained in the union of a (not necessarily denumerable*) collection of open sets, then it is contained in the union of a finite subcollection. Apply this to the set of all spheres S(p, 7) with T,I fixed, centers p in C and we have (2.8) A compact set C contains, for each 7 > 0, a finite set of points C, such that C c S(C,,, 7). We make an important deduction : Since each S(p,7) is bounded, so is S(C,, y), hence C is bounded. Take 7 = 1, 1/2, 1/3, . ; altogether we get a countable set of points which is dense in C. In the usual terminology C is “separable”.

..

(2.9) A compact set (in a metric space) i s bounded, closed, and separable. For our geometric investigations it would not suffice to consider only compact spaces, for this would exclude even Euclidean space. But the BolzanoWeierstrass Theorem holds for that space and for all those which we will consider, namely : (2.10) A hounded infinite set has at least one accumulation point. If (2.10) holds in a set, with the accumulation point in the set, we call the set “finitely compact”[finitely, because (2.10) makes a statement only for bounded sets!. (2.11) A closed subset of a finitely compact set is finitely compact. A bounded closed subset of a finitely compact set is compact. (2.12) A finite1;y compact set F is closed and separable.

2.14

7

THE BASIC CONCEPTS

To show separability, let S, = F n S ( p , v ) then by (2.11), g,.is compact, and hence by (2.8), it contains a finite subset T ,with S(T,, v - l ) 3 9,3 F n S ( p ,v ) . m

Now the set I=U1 T , is countable and it is dense in F , since u S ( p , v ) 3 F . If R, R1 be two metric spaces, and M C R, we say M is “mapped i n R’” if there is a correspondence, or function f ( p ) such that if p E M , then f ( p ) E R1. If N C M then f ( N ) denotes the set of points f ( p ) in R1 obtained when p traverses N , and we say f “maps N on f ( N ) ” . If M‘ c R then /-‘(MI) is the “original” of M’, consisting of the points p in M for which f ( P ) E M’. This set may be empty. If the mapping p’ = f ( p ) is one-to-one, then f - l ( P ’ ) is a single point for each p ’ ~ f ( M )and , p = f-’(p’) is a mapping of M’ on M . The mapping p’ = f ( p ) of M in R’ is “continuous”, if f(p,) f ( p ) whenever p,, p E M and Pv p . Thus if p E M and E > 0, then a number 6, depending on E , p exists such that

-

-+

Pq < b(E,PI

implies

/(PI f(4) < E.

(2.13) A continuous mapping of a compact set C i s uniformly continuous. This means that, given E > 0, a positive number 6 exists such that f ( x ) f ( y ) < E whenever x , y E C and x y :. 6; this follows from the HeineBore1 Theorem. For with the above meaning of S(E, p ) , there is a finite number

of spheres S (pv,J! 6 ( ~ / 29,)) , which cover C . Let 6 be the least of 6 ( ~ / 2p,, ) / 2 and let x y < 6. Then if x E S (p,, $ B ( E / ~ ,9,)), we have

Y P,

< y x + x P, < 6 + 4 w 2 , P,) < w 2 , P,) ;

and by the definition of 6 ( ~ / 2p,), ,

f(Y)

< f ( x ) f(P,) + t(P,lf(?t) -=: E/2 + 4 2 =

E.

(2.14) If the mapping f ( p ) = p’ of a compact set C i s continuous, then f ( C ) is com?rict. I f , further, f ( p ) is one-to-one, then f-’(p’) i s continuous. For, if (4,’) = (f(q,)) is any sequence of points in f ( C ) ,it has an accumulation point q in C ; hence (4,) has a subsequence (4,) such that qfi q. As the mapping is continuous, f(q,) -+ f ( q ) E f(C). If I(#) is one-to-one and 4’, = f(q,) q’ = f(q) then no subsequence (4,) of (4,) can converge to a point p # q, because then wouldf(q,) f ( P ) # f(q)=q‘. But as (4,) is bounded, it has an accumulation point; hence by (2.6) q, -+ q. A one-to-one mapping p’ = f ( p ) of a subset M of R on a subset M ’of R’ for which both f ( p ) and I-’($) are continuous is called “topological”, and two sets M , M’ for which such a mapping exists are “homeomorphic”. Thus a one-to-one continuous mapping of a compact set is topological.

-

+

+

8

2.16


SELECTION THEOREMS The reader will have noticed the analogy between (2.13) and an elementary theorem in analysis. Certain standard selection theorems in analysis also hold for finitely compact spaces, and this is one reason for their importance. We first apply Cantor’s diagonal process.

f,(p)

.. .)

of the denumerable subset N of R in the finitely compact metric space R’ have the property that {f,(p)} (v = 1,2, . ..) is bounded for each p E N . Then a suitable subsequence of {f,(p)} converges for all p of N. For let pl, pz, . .. be the points of N. Since {f,(pl)} is bounded and R’ is finitely compact, some subsequence {fv,l(pl)}converges. Similarly the sequence {I,,, (Pa)} has a convergent subsequence { f , , ~( P a ) } , and of course { f , , a ( P J } converges. So continuing, we find a sequence {/,,,(#)} converging for p = P I , pz, . , ., p,, and we can take x = 1, 2, ... indefinitely. The diagonal sequence {f,, ( p ) } converges for all p, E N, since for Y 2 x it is a subsequence of {L,, (PI}. Various forms of Ascoli’s Theorem are based on this process: A sequence of mappings {f,(p)} of M in R’ is “equicontinuous” if, when E > 0 is given, there is some 6 > 0 such that (2.16) Let the mappings

(2.16)

f,(p) f,(q) < E

(Y

= 1,2,

for all Y, and for any p , q in M with pq

< 6.

This is sometimes called “uniform equicontinuity”.

{f,(p)} is bounded for each p E M , and M contains a countable dense set N , and if the f,(P) are equicontinuous in M , then some subsequence {f,(p)} of {f,(p)} tends to a uniformly continuous mapping (2.17) If R’ is finitely compact and

of M in R’. Proof: By (2.16) there is a subsequence { p } of {Y} such that {/,(p)} converges for all p in N. Let E be any given positive number and choose 6 so that, if 9. q E M and pq < 6, then f,(p) f,,(q) < e/3. Now if p is any point of M choose q in N with pq < 6, then (2.18)

8. Hence f,,(p) f,,(p) < E. As R’ is finitely compact {f,(p)} converges. But as 3/ depends on q and thus on p , the convergence need not be uniform in general. Finally, f ( P ) = lim f,(p) is uniformly continuous, since, if pq < 6, then f,(P) f,k) < E* and f ( P ) f(4) 8.

1/6; the set C,,, contains a point q with Pq < 6. Now fi exists such that f,,(q) f,,(q) < 4 3 for all q in C,,,, when pl,pa > 8. Thus fi is independent of p , and (2.18) gives f,,(p) f J P ) < e when pU1,p2 > 8. For ,Ua -+ 00, we find f,,(p) / ( p ) Q E , when p1 > fi. This shows uniforniity of the convergence.

REAL VALUED

FUNCTIONS

Two special cases of mappings of R in R’ which are of great importance occur when either R or R’ is the real axis T. We now take the second case. We shall always define “distance” on T in the natural way: the distance of the numbers r,, ra shall be Ir, - rll. If R’ = I’, then f ( p ) is a real valued function of p . From (2.14): (2.20) A real continuous function f ( p ) defined on a compact set C attains i t s maximum and minimum. For the image f ( C ) is a compact, hence closed bounded set on T and includes a greatest and a smallest number. An important example of a continuous function in R is the distance PA of a variable point p from a fixed set A (See 2.3). If there is a point f of A such that pf = PA then f is a “foot” of p on A. If A is compact then by (2.20) a foot always exists, but it need not be unique; e. g. in ordinary space, each point of a spherical surface is a foot for its center. (2.21)

If H is finitely compact (and not empty), then each point p has a foot on H .

-’a, where

For if a = p H = inf fix, let p x ,

{ x ” } is a sequence of points

X P H

on H . Then { x , } is bounded and, since H is finitely compact, contains a subsequence {x,} which converges to a point f of H. By (2.1), p x , f i f = a. Iff, is a foot of x, on A and x, For if g is any point on A , then

(2.22)

xg - x f 2 x”g - xx,

hence

XR

2 xf.

- (xx,

-+

x, f ,

-

+

f E A , then f is a foot of x on A.

+ +f”f) 2 X”f,

-

2 x x , - f,f

+

0

10

2.23


(2.23) If H is finitely compact and each point x of M has a unique foot f ( x ) on H then f ( x ) depends continuously on x ; thus f ( x ) is a continuous map of M in H. For let x, x , where x,, X E M. By (2.3) {f(x,)} is a bounded sequencc; xf(x,) xx, 4- x,f(x,) = xx, x,H 2 xx, x H . By (2.22) each accumulation point of {f(x,)} is a foot of x, and hence coincides with f ( x ) ; an accumulation point exists on H,since H is finitely compact (compare (2.6)).

-.

0 contains points of all but a finite number of the M,. Obviously lim inf M, c lim sup M,. If these limits coincide we say {M,,} converges: lirn M , = M means M = lirn inf M , = lim sup M,.

CONSEQUENCESOF

THE DEFINITIONS

{a,}

a,,

The upper and the lower limits of are those of {M,}. For M , C and a sphere which contains a point of contains points of M,. Further, the upper and lower limits of { M , } are closed sets. For if p is the limit of points p’ in lim sup M,, there is a subscript p, > Y , and a point

w,

3.4

THE BASIC CONCEPTS

11

P I ” E M,,,such that p’p;, < l / v ; hence p E lim sup M,.Similarly forthe lower limit. Example : Consider the sequence of straight lines 6= Y , v = 0, f1, f2,. in the (6,q) plane. According to the definition given, this converges to the empty set; but if to each line we adjoin a fixed point p to form a new set, these sets have p as limit. The lines, as it were, “disappear to infinity”. The same may be said of the semicircles, 6 2 Y in (3.1); the semicircles 6 < v “tend” in an intuitive fashion to 6 = 0. Those for 2 Y count less and less, so to speak, as we increase v.

..

These remarks on convergence of sets suffice for reading this book. The remainder of this section serves to elucidate the notion of convergence, and to make precise the intuitive statements above. Our aim is to define a distance 6 ( M ,N) between sets M ,N in a metric space R such that when R is finitely compact 6 ( M , , M ) 0, if and only if M, Hausdorff considered this problem for the bounded case only.

-.

- a.

DISTANCE OF SETS Guided by our example, we select a point the (non-empty) sets M , N as follows:

p

and define the “distance” of

( p x is, of course, the distance between p and x ) . Thus

6 p ( M N ) 2 IPM-

P”.

This definition diminishes the importance of points x at a large distance from p . The weighting factor e-px is chosen for convenience; many other functions which decrease fast enough would do. We note for reference

The number (3.2)is always finite, since, by (2.3) IxM-xNI<xM+xN 0 is given, there is a point a such that dp(L, N ) - E < IzL - zNIGp' 0, y% = y M = 0 ; and hence

We have shown (3.8) With OUY distance (3.2) the non-empty closed sets of a metric space themselves form a metric space.

We next show that the exceptional role of the point p is really immaterial. We introduce a notion of general importance. If S,(p, q) and d,(p, q) be two metrics defined for the same set (of points or of any elements), they are "topologically equiodent" if each of the relations d,(p,, q) 0, d,(p,, q) 0 implies the other. Thus any notions which involve convergence only coincide for the two metrics.

-.

-.

We easily show that the metrics d,, dq are topologically equivalent. Using (3.41,

Similarly S,(M, N)

< d,(M, N )

89.

3.13

LIMITAND (3.9)

13

THE BASIC CONCEPTS

I f dp(M,,M )

Proof: Let x

+

DISTANCE, FOR SETS

0, as v -,oc), then lim M ,

EM, then, noting

=a.

(3.7), xM = XZ= 0,

dp(M,, M ) 2 IxM, - xMI

= xM,

e-PX

- e+~.

Hence xM,+ 0; in M , thereis a point y , such that xy,+O. ThusM C lim inf M,. Next let y E lim sup M,. Then there is a subsequence { p } of { v } such that there are points y,, E M,, with y,, y and +

epy 6p(M,, M ) 2 l y g - yM,I

2 yM- y,,M,, - yy,,

=y a -yyp.

Hence yz = 0, lim sup M , C M. These two conclusions give the result. Note: The converse of (3.9) is not true for general metric spaces. Consider as an example a space consisting of a denumerable number of distinct points p,, Pa, . . . with the trivial distance p , p,, = 1 for Y # y. If M , = R - p , then lim M, = R, because any S(p,,, E ) contains for arbitrary positive E the point p , of M,, if v # v,,. But dp,(Mv,R) = p,M, e-ptpv = e-I for v > 1. However for finitely compact ,spaces, this converse is true.

-

(3.10)

If the space R is finitely compact, and lim M , = M Bp(M,, M ) - 0

as

v

.-.

=

a # 0 then

00.

Proof: If not, there is a subsequence { p } of { v } for which Bp(M,,M) 2 4 a >O. Thus there is a sequence { x,,} of points such that, for large p, l x , , ~ , ,- x , ~ e-pr, (

(3.11)

If

ZG

> 3 a.

M, then M,, contains a point z,, with z,, IxpMp - x&I

3 a.

Hence {px,,} is bounded. By (3.11) there is either a subsequence { A } of {y} with (3.12)

x l M ~ - x l M> 3 a

or a subsequence (0) of { p } with (3.13)

xeM - xeM0 > 3 a.

14

3.14


These caseswe treat separately. In the first, choose y , in M with xA y,< x,M + a. Thus {y,} is bounded and has an accumulation point y. Since M is closed we have EM. But

yiMr

+ a 2 %,Ma- xayi + a > xiMa- XAM> 3a.

Hence for large 1, y M , > y, M A- a > a which contradicts y E lim inf Ma. In the second case (3.13), choose Ug E M Bwith Ug xg < xo Me a. Then since ( x e } , { Z g } are bounded SO is {Ug}. By (3.13),

+

tJeM + a

xgM-#clxo

+ z > x ~ MxgMo > 2a.

Thus there would be an accumulation-point u of { M a } which was not in M , contrary to u E lim sup M , = M. This theorem, together with a result of Hausdorff, leads to the following statement which, although not explicitly, underlies many of our arguments. (3.14) I f R is finitely compact and {M,} a sequence of non-empty sets for which dP(M,,M,,) is bounded then {M,}contains a subsequence { M A }which converges to a non-empty set M . For as R is separable, Hausdorff’s result [2, p. 1471 assures us that {M,} contains a subsequence {M,} whose limit is a set M in R. By (3.2)

Hence {PM,} isbounded, and M,contains a point z, such that {pzA} is bounded. (2,) has an accumulation point z E lim sup M , = M. Thus M is not empty and (3.14) follows from (3.10). We now can state more than (3.8). By (3. 9, 10, 14): (3.15) Theormrc The non-empty closed sets of a finitely compact metric space R form with the metric (3.2) again a finitely compact metric space. If the space R is bounded the factor e4’ may be omitted in (3.2) and distance then coincides with Hausdorff‘s distance.’

4. Motion and Isometry The concept of motion as a transformation of points which preserves distances is of paraniount importance in geometry and particularly applicable to metric spaces. A mapping p’ = f ( P ) of the set M in the space R on a set M’ of R’ is “isometric”, or an “isometry”, if it preserves distances :

4.6

16

THE BASIC CONCEPTS

The mapping is topological, and it has an inverse which is an isometry, since, if pq > 0 then f ( p )f(q) > 0 , and so f ( P ) # f(q). Two sets M, M‘ of points are “congruent” if they can be related by an isometry. A “motion” of a metric space R may be defined as an isometry of R on itself. Notice that f ( P ) must traverse all points of R when p does so. Thus the mapping t’ = f ( t )= t 4-1 of the non negative real axis t 0 is not a motion of t 0, since the point f ( t )does not traverse the part 0 t < 1. In this section we treat properties of motions which hold for general, or at any rate for finitely compact spaces. Motions will be denoted by capital Greek letters, usually 0, Y l with or without subscripts. When @ is the motion 9‘ = f(p) we write p’ = p 0, and f ( M ) = M @ for sets. If Y is the motion p’ = g ( p ) , then the product @ Y is defined by 9’ = p @ Y= g(f(#)). Thus products of motions are read from left to right. The point p is a “fixed point” of @ if p = p @.

>

(4.2)

The fixed points of a motion form a closed set.

.

For if P I , Pa, . . are fixed points, and p , p , then pf(p) = lim Pvf(p) = lim f(P,) f ( p ) = lim p,p = 0. The motion @ “leaves the set M invariant”, if M = M @, i. e. if @ maps M on itself; it “leaves M point-wise invariant”, or “fixed”, if p 0 = p for each point p of M . We have obviously: -+

(4.3) The totality of motions which a metric space can carry form a group. (4.4) Of this group, the totality of motions which leaves a set M invariant form a subgroup. So do those niotions which leave M point-wise invariant. (4.5) If a motion Y exists which maps M on M‘, then the group of motions which leaves M (pointwise) invariant is isomorphic to the group of motions leaving M’ (pointwise) invariad.

For if M @ = M then since M Y = M’, we have M’ Y-’ @ Y= M‘. Thus Y = Y-’ @ !P leaves M invariant. Conversely if 0’ is any motion leaving M‘ invariant then Y 4j’Y-l leaves M invariant. Thus the groups correspond by an isomorphism. Similarly if the sets are pointwise fixed.

16


(4.6)

4.6

Isometric spaces have isomorphic growps of motions.

For if R is an isometry of space R on R' and @ is a motion of R then @ R is a motion of R', and every motion of R' can be written in this form. The isomorphism desired is obtained by making @, @ R correspond. Isometric spaces have the same geometric properties. For example, if f is a foot of x on a set A in R. then /R is a foot of xR in the set AR in R'.

DISTANCE BETWEEN

MOTIONS

In geometric discussions, we often use terms which presuppose that notions of continuity or of limit have been defined for motions: e. g. if we speak of a continuous rotation about a point, or say that the group of covering motions of a space is discrete. In these cases the group is explicitly given and it would be easy to define limits within the group. It is not at all obvious, and probably not true, that a general definition of limit could be given which would always coincide with that used in the special cases; but it is feasible for the finitely compact spaces which we treat. We have a problem and a difficulty like that encountered when we considered the convergence of sets. If the space R is bounded, then a satisfactory definition for the distance between the motions @ and Y would be sup x @ x Y . But this would be infinite even for such simple motions as x ER

rotations in a plane. Guided by (3.2) we define for any metric space R, a distance dp(@,Y) between motions @, Y, by taking an arbitrary fixed point p and putting dp(@,

(4.7)

Now x @ XY

0 everywhere.

Apply this result to the property P ( x , y) in (7.4). Now let p ( p ) be the least upper bound of those p for which P ( x , y ) holds in S ( p , p). Then by (7.5): Either p(p) = 00 for all p or 0 a. By (6.8) there is a segment T ( p ,zl) ; this contains a point q' with p q' = a, and since (q p z1), ( p q' zl)we have (q p 4'). It should be noted that we have not said, neither can we prove, that q' is unique, compare R, in the last section.

EXISTENCE OF GEODESICS (7.0) Theowrv~= Let R be a finitely compact, M-convex metric space which satisfies the axiom of local prolongability, and let x ( t ) , a t P, a < P, represent a segment. Then a geodesic g exists which has a representation y(t) such that y ( t ) = x ( t ) when a t 8. Thus the geodesic contains the given segment.

<
. a*' V

with

= a,-

Z a(y (aJ) --+ - 00 and a function

x- 1

y(t)

for .- 00

< r ,< p

7.11

35

THE BASIC CONCEPTS

<
0. Let K be the locus of points r with zr=a. If y E V , then T(z,y ) contains just one point z' in K. Define the operation R by y' =: yR, and call y' the "projection" of yon K. Then R is clearly a continuous mapping of V in K ; (10.5) will be trivial if R is one-to-one. But this is not the case for general C-spaces, since planes need not exist. z does not lie on l J therefore , p' = $ 0 will not lie in W' = WR. Since W'is closed, we have (10.6)

P'W'=26>0.

We first show that V' = VR has dimension at least two a t p'. For this it is enough to prove that if F' is any closed set in V ' n S(p', 6 ) which separates b' = bR from p' then dim F' 2 1. (Cf. Hurewicz and Wallman. [l, p 341). Now the set F = F'R-' is closed and F n W = 0, since F ' n W' = 0 by (10.6). Moreover, F separates p from b in V . For, if not, then there would be a continuum C in V connecting p and b but not meeting F . But this would entail that C' = CR were a continuum connecting p' and b' not meeting F', and then F' would not separate p' from b'. Thus F separates p from b in I/ and, as V is homeomorphic to a circular disk, F contains a continuum F , which already separates p from b, see Hausdorff [ l , p. 343, \JI]. Then Fo must contain points PI, p , with (ap,p), (pp,c), for otherwise T ( p ,a ) u T ( a ,c ) or T ( p ,c ) u T(c, b) would connect p to b, but not meet F, since W n F = 0. The projection F,' = F,Q of F,, is a continuum, since it is a continuous map of F,, and contains the two distinct points pi' = piR. But F,' c F', dim F,' 2 1. Hence dim I;' 2 1. Thus dim V' 2 2. If u is the mid-point of T(x',z ) , where x' E V' then u T(x',u ) is the topological product of V' by a segment of length a/2. Hence XI€

V'

by a theorem of Hurewiez [ I ] dim Zi 2 dim u T(x',ZL) X'E

= dim I/'

-t 1 2 3.

V'

CONSEQUENCES (10.7)

If R is a two-dimensional G-space and the geodesic through any pair

o/ points i s unique, then R is homeomorphic to the projective plaiie or to a proper subset of it.

Proof. Choose a , b, c , p as before. An elliptic plane E with distance y') for which the common length I' of its lines is at least 2 p ( p ) contains a quadruple a', b', c', p' congruent to a, b, c, p . Map again T(a,b ) , T(b,c) isometrically on the (unique) segments T(a',b'), T(b',c') so that b corresponds to b' in both &(XI,

64


10.8

maps. This defines a topological mapping of U’ := T(a,6 ) u T(b,c) on W’ = T(a‘,b’) u T(b’,c’). Since each geodesic through p contains a segment of length p ( p ) with mid-point p and each point can be joined to p we have

We now map, for fixed y , the geodesic g(B, y) on g(P’, y’) as follows: (a) If Q(P, y ) is agreat circle of length A,, let f l y , p i be respectively conjugate to p on ~ ( py ,) and to fi‘ on a(#’, y’). Let TZ be the segment fromp to fi, which contains y E W where y # c , and T i be the other segment. Define T t l , Tt# similarly for y‘ E W’, y’ # c‘. Then map the point x of T’y on the point x’ of Ti,, for which F(P‘,

x’)

L

+

i‘ p x 2 A, - -- --* 2 1 px ly

+

Observe that the mapping is consistent for x = p , since the right-hand side equals 4 1’ so that x’ = pi,.

pp,

= Ay/2, and

(b) If g(P, y) is a straight line let T i be the ray with origin p which contains y and T;the opposite ray. Then map the point x of T i on x’ of T:, for which

A‘ -p.r e(p’,x‘) = -

21+fix

That this mapping is topological follows since A, is a continuous function of y and (2 A,)/A, --* 1 as 1, m, compare (9, 11).

-.

+

Further applications of (10.3) which will be used frequently are the following: call a set M in a G-space convex, if the segment connecting any two points of the closure of M is unique and x , y E M implies T(x, y) c M. Then :

a

(10.8) Every point of a two-dimensional G-space is interior point of a closed and an open convex set.

The set W = u T ( p , x ) , x E T(a,b) u T(b,c ) used above is convex. For a given point y a set W can be constructed that contains y as interior point. If the three segments bounding W are omitted we obtain an open set containing y.

Two distinct segments of length p ( p ) with each other at p . (10.9)

p

as common midpoint cross

10.11

THE BASIC CONCEPTS

55

More precisely: either segment decomposes S ( p , p ( p ) / 2 ) into two sets either of which contains one of the half open segments into which p decomposes the other segment. A corollary of (10.9) and (9.10) is: (10.10) If the geodesic through two points of a two-dimensional G-space

is unique, and the geodesics g,, ga intersect at to g, and ga' close to g, will also intersect.

p , then two geodesics gl' close

(10.11) I n a two-dimensional G-space let p be the midpoint of a and b where 0 < y = ab and a < p(P)/S, fi < p ( $ ) / 8 , there exists in S ( p ,p ( p ) / 2 ) ,on either

+

side of the diameter D carrying T(a, b ) ,exact1 y one poirtt c with ac = fi and bc = a.

Proof. The circle K,: ax = fi lies in S ( p , p ( p ) / 2 ) and is therefore homeomorphic to an ordinary circle, similarly for the circle K,: bx = a. The diameter D divides the circles K,, K , into semicircles K:, K:' and K i , K i p ,where we assume that K: and K; lie on the same side D' of D. The semicircle K: connects the points q and qo on D with (qaq,) and qa = aq, = fi, and K i connects the points Y , ro with (rbr,) and rb = by, = a. Because la - PI < y and a 4- fi > y the pair q, qo separates the pair r , ro on D . Hence K:and K ; intersect at a point c.

Fig. 3.

To establish the uniqueness of c in D' assume the existence of another point C in I>' with a7 = fi, bC = a. It is clear that C cannot lie on T ( a ,c) u T ( c , b), nor c on T(a,C ) u T(C,b). Therefore either C lies in the interior of the triangle abc, or c in the interior of a b c , or neither. In the first case (10.3) implies the esistence of a point u with (aCu) and (bzcc). Then a

+ p =-= ac + cb = uc $- cu + ub > au + u b = a 7 + Cu + u b > > a 7 + 7 b : a + fi.

Interchanging c and 7 in this argument settles the second case.

58

11.1


In the third case either T ( a ,C) and T ( b ,c) or T(a, c) and T ( b ,7 )intersect. Assume v = T ( a ,7 )n T ( b ,c). Then

or

a-cv+vC=bv+vC>bC=a

VC>CV

and p-VC

+ cv = uv + z*c > vc = /?

or

ct! > I ! ; .

This proves (10.11).

11. Plane metries without conjugatu points The proof of (10.7) shows: (1I . 1) A two-dimensional straight space i s homeomorphic to the plane.

Such a space may therefore be regarded as a remetrization of the euclidean plane P. One might then pose the problem of determining all the metrizations of P for which P becomes a straight space. The following considerations will show that the problem is, in this form, too general to be interesting. But asking for all curve systems which can serve as geodesics for such metrizations of P turns out to be fruitful, because two simple, obviously necessary conditions prove to be sufficient: (1I .2) Theorem : In the euclidean plane P, with a euclidean distance e(x , y ) , let a system C of curves be given with the following two properties:

I. Each curve in Z is representable in the form p ( t ) , - 00 for It1 P ( 4 ) # P(t,) for 4 # te and 4 P ( O ) , PP)) +

11. There is exactly one curve of

< t < 00, +

such that

a.

Z through two given distinct points

of P.

Then P may be metrized as a straight space such that the curves in C are the geodesics. The proof of the theorem is quite long, but this is justified by the importance of the result. Besides leading to many important examples, the theorem constitutes a solution in the large, of the inverse problem of the calculation of variations' for plane metrics without conjugate points, a problem which is quite inaccessible to the standard methods. In addition, the length of the proof is largely due to the weakness of our hypotheses. For instance, it must be established that a curve in C varies continuously with two of its points, a fact which is obvious under the usual differentiability hypotheses.

11.2

67

THE BASIC CONCEPTS

We denote the unique curve of C (called Z-crcrve or line) through the two distinct points x and y by g ( x , y), and the arc of g ( x , y ) with endpoints x and y by T ( x ,y ) , and put T ( x ,x ) = x . Moreover if x # y then [ x z y ] denotes that z is an interior point of T ( x ,y). The points on T ( x ,y ) , x # y, form together with the points z for which [ x y z ] the ray R(x, y). The proof will be decomposed into several steps. ‘rOPOLOGICAL PROPERTIES OF

(a)

If [apb] and p lies on a Ccurve b # g(a, b ) , then b separates a from b.

For if [rps] and Y , s ~ t ) then , D = T ( a ,r ) u T(r,s) u T(s,u) is a closed Jordan curve whose interior lies on the same side of b as a. If b were also on this side, then R(p, b) - p would lie on this side and intersect D because of the Jordan curve theorem and I. But R ( p , a ) intersects each of the lines g(r, s), g(r, a ) , g(s, a ) , hence g(a, b) = R ( p , a) u ( R ( p ,b) -$) would intersect at least one of these lines twice. This proof implies: (b) The system Z satisfies the Axiom of Pasch: if b is not on g(a, c) and [adc], then each Ccurve through d intersects T(a,b) u T(b,c ) . We call a set Z-convex if it contains with x and y the whole arc T ( x ,y). If a, b, c do not lie on one Z-curve then T(a,b) u T(b,c ) u T(c,a ) is the boundary of a closed and an open Cconvex domain. We denote the former by F(abc). If a, b, c lie on one Z-curve, F(abc) denotes the smallest Z-segment containing a , b, c. If xl, , ., x, are any points then

.

C ( X )= C ( X ~ , ., xa) = u F ( x i ~ p . ~ p ) -

6

a.p.v

is readily seen to be convex. I t isobviously the smallest Z-convex set containing the points xl, ., x,. Unless all these points lie on one Z-curve, C ( x ) is

..

I

bounded by a Z-polygon Q of the form Q

=

u T ( X Qxa,,-l),

x + + ~= X I ,

V = l

where no three xi, lie on the same Z-curve and C ( x ) is the intersection of the closed half planes bounded by the lines g(xAv,xi,+J and containing C ( x ) . Thus (c) Any finite set of points xl,

. . ., x,

which is not contained in a Z-curve I

has a boundedZ-convex closure C ( x ) ,whose boundary has the form u T(xI.”,xi,+1), v-1

11.2


68

x % + ~=-. x,,. A n y point outside of C ( x ) can beseparated from the interior of C ( x )

b y a suitable C-cwve @(xl,, x,,+J.

Next, convergence of Csegments and C-curves will be discussed. The underlying limit concept is here, as always, that of Hausdorff’s closed limit defined in Section 3. (d) If e(a,, a )

+

0 and c(b,, b)

+

0, then lim T(a,, b,) = T ( a ,b).

In the proof we assume a # b, the modifications necessary for n = b are obvious. If lim sup T(a,, b,) C T ( a ,b) were not correct then a subsequence { p } of { Y } and points c,, E T(a,,,b,,) would exist, such that c = lirn c,, 4 T(a,b ) . There is a C-curve b separating c from T(a,b). This is obvious for c E @(a,b). If c 4 g(u, b) then a @(a,,b,) with [aa,c], [bb,c] separates c from T(a,b) because of (a). For large p the curve 4 also separates u,, and b,, from c,,. But then T(a,,,c,,) and T(c,,, b,,) would both intersect 8, and @(a,,,b,,) would intersect 8 twice. Now we show that every point c with [mb]is limit of a sequence c, E T(a,,b,). Let Q be any C-curve through c distinct from @(a,b). Then b separates a, from b, for large Y , hence intersects T(a,, b,) in a point c,. The first part of this proof shows that every accumulation point of {c,} lies on T(u,b). On the other hand, c is the only common point of (I and T(a,b), hence lim c, = c. (e) If e(u,, a)

+

0, e(b,, b)

-

0 and a # b then lirn @(a,,b,) = @(a,b ) .

For the proof of lim sup g(a,, b,) C a(a, b) it suffices, because of (d), to consider a subsequence { p } of { t i } for which a point c,, with [a,, b,, c,,] exists that converges to a point c. If c did not lie on @(a,a), a suitable 2-curve 8 would separate b from a and c, therefore also b,, from a,, and c,, for large p. Again @(a,,,b,J would intersect b at least twice. Let an arbitrary point c with [abc] be given. Put y = max e ( x , a ) . x €T(..C)

By I there is a point PI on R(a,, I,) with e(uv,p,) = 27. If a neighborhood N : e(c, x ) < E of c and a subsequence { p } of { v } esisted such that N contains no point of g(a,,, b,,), take a subsequence {pi} of {p,,} which converges to a point p . The first part of this proof implies that p E @(a,b), and (d) shows lirn T(a,, p,) = T(a,p ) . Because of [aAbfir]we have [a@] and, since up = 2 y , also [acp]. But then (d) would show that N contains points of T ( a ,c) for all large 1. (f) For a given point z and a given p > 0 a 2’-convex polygon exists that contains the disk N, : e(z, x ) < 2 p i n i t s interior.

THE BASIC CONCEPTS

11.2

69

For a proof consider the circle K : e(z, x ) = 2 p. Because of (d) there is a 6 > 0 such that for x , ~ E and K e ( x , y) < 6 the diameter of T ( x ,y ) is less than p / 2 . Select points xl, . . ., xu, xaF1= x1 in this order on K such that a

e(x,, x , + ~ )< 6.

Then

u T(x,,x , + ~ ) is a (not necessarily simple) closed v =1

polygon that separates N , from all points y with e(z, y) points x I ,

. . ., xu construct

> 3 p. For these T

u

C ( x ) as under (d). The boundary

T(xAv, x~,+~)

v=l

of C(x) satisfies the assertion. We denote as simple family @ of C-curves a set of curves in .Z that covers the plane simply, i. e., every point of P lies on exactly one curve in @. If the Z-curves are the euclidean straight lines, then the families of parallel lines are the only simple families. This fact indicates how to construct simple families in the general case.

P2

t'

4

b

L

X

11

+

R(P,d

Fig. 4.

( g ) Every cicrve b in Z is coiztained in a simple family of Z-curves @(I). Prool. Denote by $+ an orientation of $. Let p 4 $ and choose q and b on $+ such that b follows q, and any point a such that [gpa]. As x traverses b+ from b on, in the positive direction, the intersection y of T ( p , x ) with T(a,b) moves monotonically from b toward a, but does not reach a (because of (a) and (e)). Therefore as x tends on $+ to infinity, the point y tends to a limit position yo with [byouj, so that the line g ( p , x ) tends by (e) to the line ~ ( pyo). ,

60

11.2


We call t = Q(#, yo) the asymptote to $+ through p . The line t cannot intersect Q(+, y) would by (a) intersect b twice for y on T(h, yo) close to y,,. To prove (g) it suffices to see that t does not depend on the choice of on t. First let p , be on t on the same side as yo. Since T(pl, x ) lies for every x in the closed strip S bounded by b and t, the ray R(#,, x ) tends to a ray R, in S. If R, did not lie on t, it would contain an interior point s1 of S. Then the , would not intersect b, which contradicts the definition of t. line ~ ( psl) Now let pa be a point of t on the other side of p from yo. Again T(p,, x ) lies in S, hence R, = lim T(p,, x ) lies in S. If R, did not lie on t, then the asymptote t’ through p to the oriented line which contains H, as positive subray cannot be t because t intersects R, at p2,hence would have to intersect T(y,,y) in an interior point u without intersecting b which again contradicts the definition of t.

8, for then

#J

CONSTRUCTION OF

THE METRIC

Because of (e) there is a countable set of curves in C (for instance, all curves containing two distinct rational points) which is dense in C in the sense that every C-curve is limit of a sequence of curves in the set. Therefore there 08,. , such that the lines in u Gi is a countable set of simple families 01, are dense in 2. With each 0, we associate an auxiliary function 6,(x, y) as follows. We denote the curve in Oi through the point u by Lf and select a point z once for ever. Denote the two sides of Lf by a: and a t and put

.

ti(L) =

1

L cat L cal. if L = L ~

e(z, L) if -e(z, L) if

o

If the line L in 0, separates the lines L’ and L“ in Oj then

-=

t,(L’)< t,(L) ti(,”) The function 6,(x, y) is defined by

or

t,(L’)> ti(L)> tj(Lf’).

11.2

61

T H E BASIC CONCEPTS

and the equality sign holds if and only if Licoincides with L : from L:. (1)

S,(a,, a)

-,0

if

e(a,, a )

--+

Liar L:,

orseparates

0.

For if e(a,, a) --* 0, then L:,-+ L i since otherwise L:, would intersect L: for infinitely many v , and e(L:,, z) --c e(L:, z) follows immediately from the definition of L;,+ L:. d,(x, 2) Q 4%2 )

(4

because B,(x, z) = It,(L:) -t,(L:)l Put now

=;

lt,(Li)I = e(L:, z)

< e ( x , z).

m

t=l

This series converges for every pair x , y because by (k) and (m) 6,(x, Y )

< d,(x, z) + d,(z, Y )< 4%) i-e(z, y). 2)

Clearly 6 ( x , y) 0. Since for x # y a C-curve f different from g ( x , y) through a point z with [ x z y ] separates by (a)x from y and the curves in u 0, are dense in C,there is a curve L'* in a suitable @,* which separates x from y. It follows from (i) that dJx, y) > 0, hence 6 ( x , y ) = d(y, x ) > 0. Also 6 ( x , y ) satisfies the triangle inequality by (k). Thus 6 ( x , y) satisfies the axioms for a metric space. If the symbol (xyz) has the usual meaning with respect t o S ( x , y) then

(4

( x y z ) if and only if [ x y z j .

For let ( x y z ) . If [ x y z ! were not true we could separate y from x and z by a Z-curve (compare the proof (d)), hence also by a curve L'm in some @,. Then the remark on the equality sign in (k) shows S,#(x,y) 1-S,,(y, z) > d,,(x, z ) , hence 6 ( x , y) d(y, Z) > 6 ( x , z ) contrary to the hypothesis. If [ x y z ] , then the curves Li, LE,, L: either coincide or Lf separates L: from L:, hence (k) yields b,(x, y) 6,(y, z ) = S , ( x , z ) for every i, and S ( x , y) d(y, 2 ) = 6 ( x , 2 ) follows.

+

+

(0)

d(a, a,,)

-.

+

0 if and only if

e(a, a,)

-.

0.

Let e(a, a,) 0. Then d,(a,a,) is uniformly bounded because of (k) and (m). Therefore, given E > 0, ,u may be chosen such that +

rg

z d , ( u , u,)/T I-&

1

< 4 2 for all

v.

62

11.2


Because of (1) we can find A such that 6,(a,a,)/2’

Then 6(a,a,)

<E

< &/2”+’ for v

for

Y

>A

i = 1, . . .,p.

and

> A.

Let 6(a, a,) -* 0 and assume for an indirect proof that e(a, a,) > E > 0 for a subsequence { p ) of { v ) . We may also assume that the rays R(a,a,,) converge. There is a curve L’* in some Qii, separating a,, from a for large p. If q is any point on L’*we have S,@,

(I,,)

> &,(a, q) > 0

for large p ,

hence 6(a, a,,) > S,,(a, q) 2-’@> 0. If the metric 6 ( x , y ) is finitely compact, then it solves our problem. However, this may in general not be the case. We therefore make the following additional construction. Let N,, denote the disk e(z, x ) < Y. For every positive Y

integer

Y

construct by (f) a Z-convex polygon

QY:

1=1

containing N, in its interior and put y(v) = max (1

u T(x,, x a t l ) , xytl = x1

+ e(z, x ) ) .

x CQv

For every point p with e(z, p ) > y(v) there is a line fi(x,, x,+J separates f from N,. Define y,(t) by

which

Let vl, be a simple family containing Q(x,, x , , ~ ) and define t,(L) with respect to this family exactly as t,(L) with respect to 0;. If L$ denotes the curve in !PAthrough x we put Y

Y) --

2 IP”ML1)1 -

Y Y

M L 3 1I*

A =1

Since yYY(r) is a monotone function of z the above arguments show that E,,(x, y ) -- E,(Y, x ) 0, E , ( x , y) 1- e Y ( y ,z) 2 E J X , z) and that [ x y z ] implies

>

11.2 E,(x,

63

THE BASIC CONCEPTS

It follows from the definition of y , ( t ) and

y ) -k ~ ~ (z )y=, E,,(x, z ) . r ( z , x ) that

t,(L,)

V(Y) > Y then at least one line M Ey(z,

x)

< v/2.

= g(x,, x A , 1) separates x

from N , ,

2 I Y J y ( T ~ ( MI > ) ) v,(Y)== P ( l ’ )>. 1’.

Therefore

m

x y == d(x, y )

4-

J’t”(X,

Y

y)

-1

is finite for all pairs x , y and satisfies the condition zx,-

oc)

for

&,

xy)

+

a.

This implies that the plane with the metric x y is finitely compact, so that x y solves our problem.

REMARKS If the system Z has differentiability properties (in the inverse problem as originally conceived it consists of the solutions of a second order differential equation) then our summations can be replaced by integrations and a distance function x y with adequate differentiability properties is obtained. The same problem for higher dimensions is quite difficult. I t is known that conditions I and I1 no longer suffice, i. e., geodesics have additional properties of a topological nature. For the classical three-dimensional problem in the small, necessary and sufficient conditions have been found by J. Douglas 113. The problem is entirely unsolved for higher dimensions than three. NON-DESARCIJESIAN SYSTEMS

I t is of importance for the nest chapter to remember from the foundations of geometry, that the curves in a system with properties I and I1 need not satisfy Desargues’ Theorem or its converse (compare the Desargues Property in Section 13) even if Z satisfies strong additional conditions. A reader not familiar with the well-known example of Hilbert I 1, pp. 66-71] need not look it up; the present book provides several other esamples of such systems: the two systems constructed to verify (23.5 a, b) furnish examples

64

11.2


of non-desarguesian E which go into themselves under all translations of the euclidean plane. However, there is no metrization of the plane with either of these systems as geodesics, which is invariant under all translations (this follows from Theorem (50.1)). A non-desarguesian system C which (in Cartesian coordinates t,q) goes into itself under the translations =5 Y, Y an integer, 17’ = q p, p arbitrary real, and a metrization of the plane as a straight space with the curves of C as geodesics which admits these translations as motions are found in Section 33. Like Hilbert’s example this system satisfies the parallel axiom.

+

+

The importance of these examples for the next chapter is this: we call a subset L of a G-space R “flat”, if it is with the metric of R itself a C-space. If L has dimension p we call it briefly a “p-flat”. Thus the straight lines and great circles of a C-space are its one-flats. The euclidean, hyperbolic, and elliptic spaces have the property that any p 1 points which do not lie in a u-flat with u < p lie in exactly one p-flat, in particular, three points, which do not lie on one geodesic, determine a two-flat. Every two-dimensional straight space has trivially this property for p = 1, 2. But higher dimensional straight spaces need not possess it. For it is well known, and will be proved again in (14.4), that every 2-flat in the space must satisfy Desargues’ Theorem, if any three points lie in a 2-flat. Therefore, to obtain examples of straight spaces of dimension greater than two, in which 3 points do in general not lie in a 2-flat, it suffices to take a non-desarguesian two-dimensional straight space R and form the product [R x E’],, defined in (8.15), of R with the v-dimensional euclidean space E”. Even if R satisfies Desargues’ Theorem, [R x E’], need not have the property that any three points lie in a two-flat. For if R is the hyperbolic plane, then [ R x E’], is a Riemann space in which the curvature is not constant, hence three given points will, by Beltrami’s Theorem, (see Section 15) in general not lie in a 2-flat.

+

CHAPTER I1

DESARQUESIAN SPACES 12. Iiitroduction

G-spaces in which the ordinary lines are the geodesics are the subject of this chapter. More precisely, a G-space R falls in this category if it can be mapped topologically, or imbedded, in a projective space P i n such a way that each geodesic is mapped in, or lies on, a line of the P". Such G-spaces will be called Desarguesian because of the fundamental role played by Desargues' Theorem. Necessary conditions for R to be Desarguesian are obviously (1) That the geodesic through two distinct points i s unique.

( 2 ) If R i s two dimensional that Desargues' Theorem and its converse hold, whenever the intersections, with which these theorems deal, exist.

(3) I f R is highcr diwensional, that any three points lie i n a plane, that is in a two dimensional subset of R which, with the metric of K, is a G-space. I t was pointed out a t the end of Chapter I that (2) does not hold in every two-dimensional G-space that satisfies (1). A simple Riemannian example is furnished by the paraboloid z =-- x y in ordinary space, with the length of the shortest join on the surface as distance l. Spaces of dimensions greater than two which show that (1) does not imply (3) in higher dimensions, are found on the preceding page. In Section 13 it will be proved that (1) and (2) are sufficient for a twodimensional G-space to be Desarguesian, and in Section 14 it will be seen that (1) and (3) characterize higher dimensional Desarguesian G-spaces. In addition we find in both cases that R, considered as imbedded in P" (where n is the dimension of R), either covers all of P", in which case all geodesics are great circles of the same length, or leaves out an entire hyperplane of P" and may therefore be regarded as an open convex set i n the affine space A". That the proofs of these theorems are quite long, will not surprise readers familiar with the foundations of geometry. In the Riemannian case the only Desarguesian G-spaces are the euclidean, hyperbolic, and elliptic spaces, so that the Desarguesian character implies 65

66


13

strong mobility properties. Although the methods are quite foreign to the rest of the book, we prove this fundamental Theorem of Beltrami in Section 16, because it is by far the most striking example of a Riemannian theorem without a simple analogue in more general spaces. In fact, it is one of Hilbert's famous problems [2] to characterize the Desarguesian spaces among all G-spaces with certain differentiability properties. Whereas Hamel [ 11 has given a method for constructing all Desarguesian, sufficiently differentiable G-spaces, no entirely satisfactory infinitesimal characterization of these spaces, in terms of analogues to curvature tensors say, has ever been given. The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great, that it may be doubted, whether there really exists a convincing characterization of all Desarguesian spaces. At any rate, without differentiability properties (1) and (2) or (1) and (3) are most likely the simplest conditions. However, there are important special Desarguesian non-Riemannian spaces, in particular the Miitkowskian spaces, which may be considered as th.e prototype for all non-Riemannian spaces : every non-Riemannian space behaves locally like a Minkowski space, in the same way as the local behaviour of a Riemannian case is euclidean (Section 15). It is therefore not surprising that we will meet in the course of this book many different characterizations of Minkowski spaces. Convex surfaces play an important part in the theory of Minkowski spaces and in other parts of the book. In Section 16 we compile all material on convex surfaces or bodies which is needed later. This will partly be applied in the study of Minkowski spaces which is taken up in Section 17. Only those properties will be discussed which will prove important later. The theory of Minkowski spaces has lately been carried rather far, but in a different direction. Finally we discuss another Desarguesian space which was discovered by Hilbert. It will furnish valuable examples for the properties of parallels and of the spaces with non-positive curvature. This metric is also used to construct G-spaces in a given, open convex subset of A" with the ordinary lines as geodesics. 13. Pl811013with the Desargues Property This section is closely related to the classical results of the foundations of geometry, and the methods of this field are partly used here. We outline briefly the analogies as well as the differences of that work with the present.

13

67

DESARCUESIAN SPACES

In building up plane projective or affine geometry the following points are essential : I) Incidence conditions,

0 ) Order relations,

C ) Continuity,

H ) The uniqueness of the fourth harmonic point, D ) Desargues’ Theorem,

P )Pappus’ Theorem.

The interdependence between these is as follows:

( I ,D) H, -+

( I ,P ) 4 D ,

( I , H , 0)-c D,

( I , H,0,C)

-+

P.

If the lines are the geodesics of a two-dimensional G-space in which the geodesic through two points is unique, then the order and continuity relations on a given line are either those of affine or those of projective geometry. Hamel proved, and therefore we do not want to assume, that they are either on all lines of the affine type, or on all lines of the projective type. The order relations for a plane are guaranteed, at least locally, by Pasch’s Axiom. Owing to the continuity properties of a G-space P will follow from D or H. Of the incidence conditions only the existence of a unique line through two points is always true. A priori, there may be lines which intersect every other line and also pairs of lines which do not intersect. We assume Desargues’ Theorem in a weak form in which it follows from the existence of planes in a higher dimensional space. It is tempting to try to introduce ideal elements following the classical work by Hjelmslev and others. We cannot simply refer to this work, because the order relations need not be on every line of the affine type, nor do we know from the outset that Pappus’ Theorem holds. We therefore proceed differently : We establish the existence and uniqueness of the fourth harmonic point in special situations, construct on the basis of this a Moebius net in a small domain Q, use continuity to pass from the net to an imbedding of Q in Pa, and extend the imbedding of Q to cover the whole given G-space. The following is the form of Desargues’ Theorem assumed here: THE DESARCUES PROPERTY (1) I/ the geodesics g(a,, u2), g ( b 1 , b2), g(c,, c2) hare a common point, and c2) exist, then, the iittersections p = g(a,, b,) n g(a,, b,), q = a(bl, cl) n Zf two of ihe three intersections a(#, q) n g(c,, al), g(#, q) n s(c,, ~ 2 1 , d c 1 , 4 n g (c,, a,) exist, they coincide.

68


13.1

(2) I f the intersections a(al, 61) n @(aa,4 , a(b,, cl) n a(ba, C a ) , a(c,, al) n a(ca, ap) exist and are collinear (i. e., lie on a geodesic) and if two of the three intersections da,, a,) n g(61, 4 . g ( 4 , 6%)n a(cl, CJB a(c,, CJ n g(a1, aa) exist then they coincide.

We assume both parts because two geodesics do not necessarily meet, and the duality principle which reduces (2) to (1) does not hold. The object of the present section is the proof of the following important theorem : (13.1) l l ~ o r e m Let : R be a two dimensional G-space in which the geodesic through two points i s unique and in which the Desargues Property holds. Then Either all geodesics are great circles of the same length and R can be mapped topologically on the projective plane Pa in such a way that each great circle in R goes into a line in Pa Or all geodssics are straight lines and R can be mapped topologicallv on an open convex subset C of the affine plane A8 in such a way that each straight line in R goes into the intersection of C with a line in Aa. a

Instead of mapping R on P or C we may then consider H as coinciding with (or imbedded in) Pa, or coinciding with C. We express this by saying that we identify R with Pa or C. We decompose the long proof of (13.1) into several steps

(a) Harmonic points

*

Let D be any open non-empty convex set, see (10.8). If a, 6,d E D and (a 6 d ) , choose any point u E D not on @(a,d ) , then u with (a u u). Because of Pasch’s Axiom T(u, d ) and T(u, 6) intersect at a point w, put t = T(a, w) n T(6, u). Then g(u, t ) intersects T ( a , 6) in a point c with (u t c ) . We call c the fourth harmonic point to a, b, d in this order. That c does not depend on the choice of u and u may be seen by the following modification of the standard proof: Let u’E D be not on a(a, d ) or a(a, u). If (a u’ u’) and u‘ is sufficiently close to u’, and w’, t‘, c’ are constructed as before, then the intersections (see Figure 6) p = g(u, u’) n o(w, w‘), a(t, t’) n g(u. u‘) and g(u, u’)n a ( u , u’) exist (since they fall in u’ if u’ = u’). The triangles u, u, I and u‘, u’, w’ satisfy the hypothesis of the second part of the Desargues Property, hence a(u, u’), g(u, u’), g(w, w’) concur a t p . By the same argument g(u, u’), g(w, w’), g(t, t’) concur at p . Now the first part of the Desargues Property applied to the triangles u, u, t and u‘, u’, t’

13.1

DESARGUESIAN SPACES

69

yields that the intersections c of g(a, 6) with Q(U,t ) and c’ of g(a, b) with ~ ( u ’t’) , coincide. If now uo is any point in D not on g(a, d) and (a uo uo) then u‘ may be chosen so close to u‘, that also the intersections p,, = g(uo, u‘) n g(wo, w ’ ) , o(to, t‘) n g(vo, u’) and Q(u,, u’)n g(v0,v’) exist. Then, if co is the point obtained, co = c‘ hence co =; c. We add a simple consequence: U

P Fig. 5 .

-

d ) n T ( u , b), t = T(v,6) n T ( w ,Z) then the If (u Z u) and 6 = T(Z, intersection z of a (u,7) with T (v, w ) is, by definition the fourth harmonic point to u, w , d. On the other hand, the Desargues Property applied to a, 6, t and Z, 6 , t shows that u, t , 7 are collinear. Thus the fourth harmonic point z to u, w , d , where (a v u) and w = T(b,u) n T(v,d) lies, for variable v , on a line. Since t lies on this line, it is the fourth harmonic point to the intersections of T(a,u) and T(6,u) with g(d, t) and d itself.


70

13.1

(b) The Moebius net If a , b , u , w , t , c , z are as before, we put a = (O,O), b = (l,O),u = (O,l), w = (1,1), t = ('la, I/*), c = (I/@01, z = (I/*, 1). Then t is harmonic to c, z,u, also, as just observed, t is harmonic to (O,l/.J = T(u,u) n g(t, d ) , (1,1/2) = T(b,u) n g ( t , d), and d. Similarly put ('/4,3/4)

= T[(O,' / 2 ) , z 3 n T(b,4,

(V4,'I4)= T[(O,

4 n W, 4,

(3/4,1/4)

=

T [c, (1, ' / a ) ] n T(b,4 (1, 112)1n T ( a ,w )

(%, s14) = T [z,

U

Fig. 6.

d are collinear and the line connecting them The points (114, intersects T(a,u), T(c,u), T(b, u) in points (0, 'I4), (1, 114) which u and c, t, u and 6, (1, u are the fourth harmonic points to u, (0,'I4), respectively. A similar statement holds for the line connecting 3/4) and (a/,,, 3/4): it passes through d, intersects T(a,u) in the point (0, 3/4) harmonic to (0, (0, 1) and u, etc. Also (114, 1/4), *14), u are collinear and their line intersects T (a, b) in the point (I/.,, 0) harmonic to a, c , d and SO forth. Continuing this process yields points ( p , a) for any numbers p, u of the form

< x < 2". <
, r - 1. We show that p = r - 1. Assume p > Y - 1 and take any point a,,,. Then a,, . . ., a,+' lie in no flat of dimension less than I , hence lie in an r-flat L,'. Since n > Y there is a point b not on L,'. Then al, . . ., a,, b again lie in no flat of dimension less than r, hence they lie in an r-flat L," # Z-,'. This contradicts (14.2).

<
0. If EI, E; be two such ellipsoids we must show they coincide. By an affinity which keeps volumes invariant, we can transform E ; into a sphere El. This affinity maps E; on an ellipsoid E, and F on a set F'. In proper coordinates El and Ea have equations of the form Zx: a9 and Zx:/bf 1 respectively. Now Ei, E; are both minimal, hence IE1'l == lE,'l. But lEll =; x, a", lEal = x, b, b, . . . b,, x, = xJa r1 (n/2 1). Since the transformed sphere and ellipsoid contain F'

< +

-flat through g and a point of L n C would not then be an ellipse. Consider A" as imbedded in Y" and let p' be the harmonic conjugate of p for ql, q,; choose the hyperplane through H l n H , and p' as the plane a t


92

16.13

infinity. In the new affine space, p is the midpoint of q,, qa and H,,Ha are parallel; let H be the plane through p parallel to these. First let n = 3. Then H n C is an ellipse E. Each La through g meets C in an ellipse E' with tangents Ti in Hi. Since T,and TI are parallel, p is the center of E'. Moreover, the plane of E' meets H in a line T parallel to T i , that is, in the diameter conjugate to a. The tangents to E' at the points E ' n E are therefore parallel to a. Moreover 9 is the mid-point of the chord carried by Ta hence the center of E. With a suitable euclidean metric, E is a circle, a is normal to H,and pq, = pqa equals the radius of E. Then E' appears as circle with diameter qlqa, hence C is a sphere. Use induction: if n = 4, then H n C is an ellipsoid, from the case vt = 3. With a suitable euclidean metric, E is a sphere with center p and now a will be normal to H and the distance qd will equal the radius of E. Hence C is a sphere. (16.13) If for every family of parallel chords of a closed convex surface C, the mid-points of the chords lie on a hyperplane, then C is an ellipsoid.

Let La be any &-flat which contains points inside C. Any family F of parallel chords of E = C n La belongs to a family of parallel chords of C. By hypothesis, their mid-points lie on a hyperplane H,hence the mid-points of the chords of F lie on the line H n La. If (16.13) has been shown for curves, then E is an ellipse, and hence by (16.12) C is an ellipsoid. Assume therefore that the mid-points of a family of parallel chords of the closed plane convex curve C are collinear. Let F , be any family ,of parallel chords of C and g1 the line carrying their mid-points, moreover 8, the line through the mid-points of the family Fa of chords parallel to 8,. Let Gj be the affinity which maps each point of gi on itself and interchanges the ends of one non-degenerate (and so of each) chord in Fi. G1 maps C on itself, and sends every chord of Fa into another chord in Fa, moreover the mid-point of the first into the mid-point of the second. Hence G1 maps the line containing the mid-points of chords of I;a on itself. A similar statement holds for @a. Take affine coordinates in which g1 and have equations x1 = 0 and Xg = 0; the affinity Gi is then the transformation x.' = - x {, x.' = x . (j # i ) . Since @,@a maps C on itself and is the affinity x,' = - x,, Xa' = - Xg it follows that a = a, n 0, is the center of C. If now fi and q are any two distinct points of C and lZ; is the family of chords parallel to L ( p , q ) , where L ( p , q) denotes generally the line in A2 through p and q, then the affinity a, defined above, maps C on itself, p on q and leaves z fixed. Hence by (18.11) C is an ellipse.

,

,

16.14

DESARGUESIAN SPACES

93

We call a line L a supporting line of the closed convex surface C, if it contains a point on C but no point inside C, and prove the following important theorem due to Blaschke [l]. (16.14) Ilheoresn,: If the closed convex surface C i n A" (n 2 3) has the property that the points common to C and a given family of parallel supporting lines of C lie i n a hyperplane, then C is an ellipsoid.

For, because of (16.12) it suffices to show that every 3-flat L, containing interior points of C intersects C in an ellipsoid. Let g be any line in L, and Z'(g) the set of all points of C which lie on the supporting lines of C parallel to g. A supporting line of C in L, is also a supporting line of K = C n L, and conversely. Hence Z(g) = Z'(g) n L, is the set of points on the supporting lines of K parallel to g. Since Z'(g) lies in a hyperplane H , the set Z(g) lies in H n L, that is, in a 2-flat. Therefore K satisfies the hypothesis of the theorem and it suffices to establish the theorem for a closed convex surface K in A S . We observe that K is strictly convex. For if the supporting line g of K intersected K in a proper segment, then the set Z(g) could not lie in a plane. Consider any two parallel supporting planes H , and H , of K touching K a t p , and p,, and let $ = L(p,,p,). For any line g in H , through p1 the curve Z(g) goes through p , and P, and lies, by hypothesis, in a plane H ( g ) . Any plane P parallel to H , , H , and lying between them intersects K in a convex curve A , and cuts Z(g) in two points q,, q,. Their join T(q,, 4), lies in the plane H ( g ) containing $, and hence meets Q in the point c = P n Q. At the points q, the curve A has supporting lines parallel to g and hence to each other. Varying g it follows that A has parallel supporting lines a t the ends of any chord through c ; hence c is the center of A , see (16.7). Changing the position of P between H, and H , yields that T(P,, p,) is the locus of the mid-points of a family of parallel chords of Z(g). If now Z(g) is kept fixed and p , is allowed to vary on Z(g) it is seen that every family of parallel chords of Z(g) has collinear mid-points. Hence by (16.13), Z(g) is an ellipse. We cannot immediately apply lemma (16.12), since the planes H ( g ) are not yet known to be concurrent. With the same notation as before consider the ellipse Z($). Each ellipse Z(g) (with g in H , through PI) intersects Z($) in two points rl, r2 at which the tangents toZ(g) are parallel to$; hence T ( Yr,) ~, is the diameter of Z(g) conjugate to $. Consequently the tangents to Z(g) at p , and p , are parallel to T(y1, 1,). Thus HI and H , are parallel to the plane of Z ( $ )and the situation is the same as in the proof (16.12).


94

.

17.1

17. Minkowskian Quometry

A metric x y defined in the whole space A" is called "Mirtkowskian" if it is equivalent, on each affine line, to the (natural) topology of A" and if the affine mid-point z of two points x , y is also the mid-point in the sense of the metric xy, that is, if xz = zy xy/2. 1

We have assumed as little as possible in this definition; in particular, we have not assumed that x y is topologically equivalent to A" as a whole, because this will follow. The equivalence relation cannot be omitted altogether for, as there are discontinuous solutions of the functional equation

the mid-point condition is not enough. In the present discussion it will prove useful to call a euclidean metric e ( x , y ) defined in A", "associated" with A" if it is equivalent to Anand the straight lines defined by e(x, y ) coincide with the affine lines. Such a euclidean metric is a particular Minkowski metric. (17.1) The metric x y defined in the (whole) A" is Minkowskian if and only if for some associated metric e ( x , y ) the distances x y and e ( x , y ) are proportional on each given line L. This will their be true for all associated metrics e(x, y ) .

This means there is a function q(L)> 0 such that (17.2)

xy

=q(L)- e ( x ,y )

for

x, Y E L .

The sufficiency is trivial. To show necessity, let xy be a Minkowski metric and L a given line in A". For a given associated euclidean metric e(x, y ) take on L two points a, and a, with e(ao,al) = 1. Then put a, a1 = q >O and it is to be shown that x y == q e(x, y ) for all x , y on L. On L choose the euclidean arclength a as parameter such that a,, al correspond to a = 0 and a = 1, and denote generally by a, the point corresponding to a. Since ai is the euclidean mid-point of a,-l and ai+l, it follows that a j ai+l = q if i is an integer, for a, al = a, a, = a_, a, = Since a, is the mid-point of a-i and a , it follows that (a-j a, ai)and then by (6.6)that (@k a, a,) if k < 1 < m are integers, and hence aha, = q Ik - 11. Since a,,% is the euclidean and hence the Minkowski mid-point of ak, ah+, the same argument shows that a, a,, = q~Ip - uI when p, a have the forni 4 2 , and similarly when of the form k/2", k integer.

. . ..

17.6

95

DESARCUESIAN SPACES

As we assume x y and e ( x , y) are equivalent on L the rest easily follows. (17.3) A Minkowski metric in A" is equivalent to the euclidean metrics associated with A".

This will follow from: (17.4) In a definite associated euclidean metric e ( x , y) the function v ( L ) defined in (17.2) depends continuously on L. We show this by induction with respect to I t ; in A' there is nothing to prove. Assume that the statement holds in A"-', and let L, be any sequence of lines in A" tending to the line L. On L take two distinct points a,, a2 and a hyperplane Hi through a; that does not contain I*. For large v the line L, intersects H i in a point a: and e(a:, a;) 0. The distance x y in A" induces in H i a distance with the same property (that e ( x , y ) and x y are proportional 0, on a line in H i ) . By the inductive assumption e(al, a;) 0 implies a! ai hence by (2.1)

-

-+

lq~(L,)e(a;, a;) -cp(L)e(a,, u,) I ==la; a; - a, a,] Since e(a;, a;) v(L,) -+ v w .

-

+

< a; a, + a; a,

-

0.

e(a,, a2) # 0, it follows that

Next we show that (17.6)

where L

p ( L ) =v(L')

11 L'

if

L

( 1 L'.

means that L and L' are parallel.

L'

An equivalent statement is: (17.5') a b = a'b' if a, b and a', b' are opposite sides of a parallelogram.

b'

b

U For if a, be L and a', b' E L' then e(a, b) = a' = e(a', b'), hence a b = a'b', if and only if Fig. 12. v(L) =dL'). To prove (17.5') let (a b fi,) and let the line L,' through #, and a' intersect T(b,b') in b,. When bfi, = g?(L)4 6 , fi,) 00, then b, b'; hence, by the preceding remark, a' b, = v(L,') e(a', 6,) g@') e(a', b') = a' b'. But

--

-


96

17.6

hence

aa' 1--=--

ap,-aa' apv

ap. Since a a'/a

p,

-*

a'#, = a'b < __ , a P,

ab

< -a ' a + a p .

'

ap,

a a' =I+--. a Pv

0 it follows that a' b' = lim a' b, = a b.

This implies : (17.6) A Minkowski metric in A" is invariant under the translations of A".

+

The central reflection of A" in p is the affinity x i = - xi 2 pi, which maps any point x # p on the point x' on the line through p and x , for which p is the mid-point of x and x'. Any two points a , b not on a line through fl go into points a', b' which form together with a, b a parallelogram, hence a b = a'b'. Conversely if a parallelogram is given then the central reflection in the intersection of the diagonals will interchange opposite sides. Therefore (17.6)yields: (17.7) A Minkowski metric in A" is invariant under the central reflections of the A". The similitude xi' = b xi

+ ai,

b # 0,

transforms any euclidean segment T ( x ,y ) into a segment T(x',y') which is parallel to T ( x ,y ) and has length e(x', y') = Ibl e(x, y). It follows from (17.5) that x'y' = Ibl x y . (17.8) If x y i s a Minkowski metric in A" with affinecoordinates xI, . . ., x , lhen under the similitude x i = bx, + a, all distances x y are multiplied by (bl; that is, x'y' = Ibl x y.

THE SPHERES We now formulate several relevant facts in one theorem. (17.9) A Minkowski metric In A" i s finitely compact and M-convex. Prolongalion is possible in the large, or p ( p ) =a. The spheres K(p,p ) (the loci p x = p > 0) are convex and homothetic surfaces in A", and p i s the affine center of K ( p ,p ) . The space i s a G-space and therefore straight, if and only if the spheres are strictly convex.

17.9

97

DESARCUESIAN SPACES

Because of (17.6) cp(L) takes (for a definite associated euclidean metric) all its values on the compact set of lines through a fixed point. Since cp(L) is continuous, (17.4), it has a finite maximum P and a positive minimum a,so that

This implies that x y is finitely compact. By (17.8) the similitude xi' = u (xi - p J / p qi maps K(p, p) on K(q,a), therefore the spheres are homothetic. To prove the convexity of K ( p ,p) let p a = p b = p. I t is to be shown that pc p if c = (1 - t)a -t- t b, 0 < t < 1. Since the statement is trivial when p lies on L(a, b ) , assume that L(a, p ) and L ( b , p ) are different, and let the parallel to L ( p , a ) through b intersect L ( p , c) in d . Then

+

pd implies p > pc, hence the strict convexity of K ( p ,p ) . Since segments are unique in a straight space, it follows that the K ( p , p ) are strictly convex, when x y defines a G-space. Conversely, when the spheres are strictly convex, then x y defines a G-space. It is to be shown that prolongation is unique. Since prolongation of T ( p ,b) is always possible along L ( p , b) it suffices to show that, if p b + b d = p d for d not on L ( p ,b ) , a sphere can be found which is not strictly convex. If we take a on the line a through p parallel to L ( b , d) (on the proper side d of 9 ) such that p a pb, then the preceding Fig. 13. argument shows pc = p a =; ph, so that K ( p , pa) is not strictly convex. In view of all the properties which are necessary for a metric to be Minkowskian it is quite surprising how few are sufficient according to the original definition. Different Minkowski metrics defined on the same model of the A" are not necessarily intrinsically different, they may be isometric. The following 3=

+

pT 7

98


17.10

theorem states that there are as many different Minkowski metrics as there are affinely not equivalent, closed convex surfaces with center.

Two Minkowski metrics xy and m(x, y ) in A" are isometric if and only i/ an affinity exists which maps the sphere K :p x = 1 on the sphere K' :m(p,x)= 1. Note: In view of (17.6) and (17.8) it would have amounted to the same to require that p x = p > 0 can be mapped by an affinity on q x = a > 0. Assume an affinity @ exists mapping K on K'. Since p is the affine center of both K and K' it stays fixed under #. If e ( x , y ) is any euclidean metric associated with A" denote the line functions corresponding to x y and m(x, y ) by q ( L ) and q'(L) respectively. Let two distinct points y, 2 and their images y' = y 0,z' = z@ be given. If L and L' are the parallels to L ( y , 2) and L(y', z') through p , then (17.10)

yz = q ( L )e(y,2);

y'z'

= q(L') e(y', 2').

Because # leaves p fixed, L' = L #. An intersection x of K and L goes under @ into an intersection x' of L' and K , and 1 = p x = q ( L )e(p, x ) = m(p, x') = q'(L') e(p, x').

As an affinity, # multiplies all euclidean distances on parallel lines by the same factor, hence

y z = e ( y , z ) :e(p, x ) = e(y', z') :e ( p , x') = m(y',2'). Conversely, assume the two metrics to be isometric, that is, a mapping #' of the A"on itself exists such that yz = m(@', zV).If #" is the translation sending p' = p 0 for x # 0, 2) F(,u x ) = JpIF ( x ) , for any real { I , 3) F ( x ) is convex. Conversely, i f F ( x ) is defined itr Anand has these three properties, then (17.18) i s a Minkowski metric. The metric defines a G-space if and only if F ( x ) is strongly convex. Proof. Let x y be a Minkowski metric. If z denotes the origin we define

F(x)=z x. Property 1) is then obvious, 2) follows from (17.8),and 3) from (17.9) and (17.16). Since the metric is invariant under translations, the distance of x and y equals the distance of x - .y to the origin z , which proves (17.18). Conversely, if a function F ( x ) with properties l),2),3) is given, and x y is defined by (17.18),then x y isinvariant undertranslations,and /%IF(&xIxI-')=xz because of 2). Therefore x y is on each line proportional to Ix - yJ,hence will be Minkowskian, if we can establish that it is a distance. x y > 0 for x # y follows from 1). The symmetry x y = y x is equivalent to F ( x - y ) = F ( y - x ) which is contained in 2). The triangle inequality reduces to

F(x-u) - F ( ( x - y )

+ ( ~ - u ) ) < F ( x - y +) F ( y - u )

which follows from the convexity of F ( x ) .

17.20

101

DESARGUESIAN SPACES

The statement regarding C-spaces is a consequence of (17.9) and (17.16). This theorem contains as a special case the general expression for the associated euclidean metrics e ( x , y ) in a given affine coordinate system: the sphere e(z, x ) = 1 is an ellipsoid, see (17.11), with center z and has therefore an equation of the form

where the form Cgikxi xk is positive definite, because every line xi ty,, y # 0 intersects K ( z ,1) for some t # 0, so that Zgikyi yk = 1/t2 > 0. Since e(z, x ) must satisfy 2) it follows that =5

MOTIONS Denote by E ( x , y ) the associated euclidean metric in which the unique ellipsoid E of minimal volume that contains K ( z , 1) and has centre z, is the unit sphere (compare (16.10)). A motion of the Minkowski space is according to the second part of the proof of (17.10) an affinity

Since translations are Minkowskian motions, x,' = Za,, xk is also a motion. I t therefore carries K ( Z ,1) and, as in the proof of (16.11), also E into itself. This means that it is a motion for the metric E ( x , y ) . Therefore the original affinity is a motion for E ( x , y ) . Thus we have proved: (17.19) For a given Minkozoski metric in A" there i s a euclidean metric associated to A" such that each motion of the Minkowski metric is also a motion of the euclidean metric.

This implies together with (17.6), (17.7), and (16.11): (17.20) The group of motions of a n n-dimensional Minkowski space i s isomorphic to a subgrotif of the motions of Enzoihich contains all translations and centrtzl reflections. If for given points p , x , y with p x = p y > 0 a motion exists which leaves p fixed and carries x into y , then the metric i s euclidean.

102


17.21

At this point we break off the systematic development of Minkowskian geometry and discuss some isolated properties which will provc of importance later on. Since this book is concerned with G-spaces, we restrict ourselves to this case, although all properties hold with slight modifications for the general case.

LIMITSPHERES We introduce for a general metric space the notation K ( a , b ) = K ( a , ab) for the locus of points x with ax = ub. If x(z) represents a straight line, it will be shown later that the sphere K ( x ( t ) ,x(0)) converges for t -+ a0 to a limit, which we call a limit sphere. In a Minkowskian G-space x ( t ) is an affine line, all the spheres K ( x ( t ) , x ( 0 ) ) are homothetic and have a t x(0)the same tangent cone T. I t follows immediately that lim K ( x ( t ) ,x(0)) = T. Hence, compare (16.6): 7+m

(17.21) In a Minkowskian G-space the tangent cones of the spheres are the limit spheres. The limit spheres are hyperplanes if and only if the spheres are differentiable.

DISTANCE OF

POINT FROM LINE.

PERPENDICULARS

If in a Minkowski G-space (axb)and ax : xb = (1 - t') :T' then x is, because of (17.2), the point (1 -t') a t ' b , hence (17.8) yields:

+

(17.22) If in a Minkowskian G-space (ga'a), (gb'b) and ga' : ga = gb' : gb = t

then

a'b' : ab = t.

(17.23) I f x ( t ) represents a geodesic i n a Minkowskian G-space which does not pass through 9, the% p x ( t ) is a strictly convex function of t .

Given a < p and 0 < t < 1, determine u' with (pa'x(a))and pa' : px(a) = 1 -t. If c = x((1 -t)a then by (17.22) a'c : fix(/?) = T, hence

+ .a), p c < pa' + a'c = (1 - t)px(a)+ t p x ( p ) .

Since p x ( t ) ---* a for It1 co,and p x ( t ) is strictly convex, it reaches its minimum a t exactly one point %(to), which is the foot of # on the geodesic 4 7 ) according to the definition in section 1. Of course, convexity means much more than that the minimum is reached a t one point, which is sufficient for the uniqueness of the foot. These questions will occupy us very much later on. I t will be seen that uniqueness of the foot of a given point p on a given line L +

17.25

DESARCUESIAN SPACES

103

is, in any straight space, equivalent to convexity of the spheres and implies that all points on the geodesic g(p, f ) have f as foot on L. This leads naturally to the definition: The straight line L in a G-space is called a “perpendicular” to the set M at f , if f E L n M and every point of L has f as foot on M. In Minkowski spaces the existence of perpendiculars can be verified immediately : (17.24) If p does not lie on the r-flat L, of a Minkowskian G-space, then p has exactly one foot f on L, and L ( p , f ) is a perpendicular to L,.

The existence of a t least one foot follows from (2.21). Then L, cannot contain an interior point K ( 9 ,f ) , and hence lies on a supporting hyperplane L, - of K ( p ,f ) at 1. This hyperplane, and hence L,, cannot contain a second point of K ( p ,f ) since K ( p ,f ) is strictly convex. The spheres K ( x , f ) with x on L ( p , f ) and on the same side of f as p are homothetic and therefore have also L, as supporting plane. The fact that spheres have parallel supporting planes at diametrically opposite points shows the same for the x on L ( p , f ) on the other side of f.

ASYMMETRY OF

PERPENDICULARITY

This theorem shows in particular that there is exactly one perpendicular G to a given line L through a given point p not on L . It is very important to realize that L need not be perpendicular to C , or, as we will briefly say, that perpendicularity between lines is in general not symmetric. In fact, according to a surprising theorem of Blaschke [l] it is a very exceptional occurrence: (17.25) 1f , in a Minkowskian C-space of dimension greater than two, perpendicularity between lines is symmetric, then the metric is euclidean.

Proof. Consider a supporting hyperplane H of a sphere K ( p , 1) at a point g. The line G = L ( p ,g) is perpendicular to all lines in H through g. If the hyperplane parallel to H through p intersects K in 2, then L ( p ,z) is for z E 2 parallel to some line L, in H . Since G is perpendicular to L, it is perpendicular to L(p,z). By hypothesis L(p,z) is also perpendicular to G, hence also to the parallel G, to G through z. This means that G, must be a supporting line of K ( p , 1) at z. Therefore 2 is the locus of the points in which the supporting lines of K ( p , 1) parallel to G touch K ( # , 1). Thus K(+,1) satisfies the hypothesis of Blaschke’s Theorem (16.14) and is therefore an ellipsoid. (17.11) shows that the metric is euclidean.

104

17.26


The assumption that the space has dimension greater than two is essential. This is not to say that perpendicularity is always symnietric in a Minkowskian plane, but that there ure curves in the plane other than the ellipses which lead to symmetric perpendicularity. In a Minkowskian C-plane the perpendicular C to a line L at a given point f of L is unique. For a second perpendicular a t f would be perpendicular to L after being translated. It would then intersect C,and the point of intersection would have two feet on L, contrary to (17.24). If perpendicularity is symmetric, then Id is perpendicular to C and therefore f ) with unique. On the other hand every supporting line at f to a circle K(#, p E C has C as perpendicular. Hence the circles are differentiable. If the line G through p intersects K ( p , 1) a t fl and f a , and L,, L, are the tangents of K ( p , 1) at these points, then the line L parallel to Li through p must intersect K ( p , 1) in the points z,, z, at which the tangents C,,C, are parallel to C. Thus K ( p , 1) shares with the ellipse certain properties of conjugate diameters. That this property cannot generally hold for differentiable circles is obvious : for should the tangent a t zi turn out to be parallel to C,the circle can obviously be modified in the vicinity of z1 and z2, so that the tangents are no longer parallel to G. However, as stated before, there are other curves than ellipses with the properties of conjugate diameters derived above'. In polar co-ordinates Y, g, take a differentiable strictly convex arc Y = f(g,), 0 Q g, Q n/2, with f(0) = 1 and r cosg, = 1 as tangent and f(n/2) = 1 and r sing, = 1 as tangent. Let r = g(g,) be the image of = f ( y ) under the polar reciprocity in Y L- 1. This arc is also strictly convex and differentiable, passes through (0, 1) and (n/2,1) and has the same tangents as r=f(y) at these points. Then the equations

r =f(y),

0 0. If f is a foot of p on Q,then Q and S ( p , p ) have no common point. Hence K ( p ,1) and Q have at f a common supporting hyperplane H,'. If a and b denote again the intersections of L(p, f ) with C, then C has at a and b supporting planes Ha, H , such that their pencil contains H,'. This is an obvious consequence of the preceding discussion if K(P, f ) is differentiable at f . Since the spheres K ( p , f ) are differentiable for smooth C, the assertion follows in the general case by approximation of C with smooth surfaces%. If HP) denotes the plane in this pencil through p , and x E H, = H,' n D, then a segment connecting x to a foot of z of x on Q contains a point y of H,'. It follows from (18.7) that =1

h ( x ,Q)= h ( x , 4,

4%) Y )2 W , f)

with inequality for x =#p and strictly convex D. Therefore H,' is a supporting plane of S(Q, p) at p , so that S(Q, p) is convex, and strictly so if D is strictly convex. A particular case of (18.9) is of special interest. (18.10) I f the intersection L, of an r-flat L,' in A" with D i s not empty, then the equidistant locus h ( x ,L,) = p > 0 lies on n coni~exsurface in A", which is strictly convex if D i s strictly convex.

We will see in Chapter V that (18.9) is in general spaces much stronger than convexity of spheres. On the other hand the convexity of S(Q, p) for any convex Q, follows from the convexity of S(T, p) for segments T , compare (36.21).

18.12

109

DESARCUESIAN SPACES

PEAKLESS FUNCTIONS The type of function which is needed to describe these facts analytically does not seem to have received a name in the literature. Since these functions will be met frequently, we venture to define: A function /(t) which is defined and continuous on a convex set of the real t-axis is called “peakless” if (18.11)

/(t2

/(t3))

< t2 < t3

for

and the equality sign implies /(tl) = / ( t 3 ) . A function is peakless if and only if it belongs to one of the following types: /(t) 1) is constant 2) is strictly increasing or decreasing 3) takes its minimum a t one point, decreases strictly to the left of the point and increases strictly to the right 4) takes its minimum at all points of an interval and decreases strictly to the left and increases strictly to the right of this interval. We call a peakless function “strictly peakless” if it is not constant in any interval, thereby excluding cases 1) and 4). Convex functions are peakless, but not conversely. Nevertheless peaklessness may be interpreted a s a degenerate form of convesity: If /(T) is convex and positive, then f “ ( tis) convex for every a > 1, hence the requirement that f“(t)be convex becomes weaker as a increases. For a - + a it becomes the condition (18.11). For if f’(t)is convex and positive, t2 = (1 - 0) t, 4-0 t3, 0 < 0 < I , and for example /(tl) /(t3)then

) f ( t 3 ) For . a and the equality sign holds only when /(t2) = f ( t l= becomes (18.11) with the condition for the equality sign. We may now formulate (18.10) as follows.

-

this

(18.12) I n a Hilbert geometry, if x(t) represents a geodesic G zehich lies on u n a//ine line3 then the distance /(t) X ( T ) Q is a peakless function /or a n y convex set Q i n D . I / the space is straight, then p x ( t ) is strictly peakless. I / D is strictly convex lhen all fzinctions /(t) are strictly peakless. For if /(tl) p = max (/(tl),/(t3)) and tl < t2< t3,then the convexity of the closure F of S(Q, p ) implies that /(ta) p. If /(t2)= /(tl),then the convexity of F implies that the line ~ ( tcontains ) for t 2 t2no point of S(Q, p ) , hence /(?a) 2 f ( t 2 ) = h l ) and f b 3 ) = f(t1). =1

=1

0, hence they cannot be convex. This implies the existence of two numbers

and of two numbers

t"

> t'

> tl

t2

0 such that

0 such that

If, for sufficiently large to,the arc of C from %(to) to y ( t o )which contains T is replaced by an arc through q and s which forms with the remaining arc of C a strictly convex curve, then none of the points in the cross ratios which must be calculated to find the six distances occurring in the last two inequalities will change. Therefore the functions x(t) y ( t ) and ~ ( tH) also fail to be convex for general strictly convex C, although now x(t)y(t) + 00 and x(t) H 00 for It1 + 00.

-.

METRICS I N

CONVEX SETS OF

A"

We now come to point 2) and prove: (18.14) Given a non-empty convex open set D i n A", then D can. be nietrized as a straight space i.n which the itttersections of D with the lines in A" are the straight lines. First let D be a set that does not contain any complete line in A". If C is the surface bounding D, then any line L ( p , q) connecting two points p , q in D intersects C a t least once, say at a. With b we denote the second intersection of L ( p , q) with C if it exists, otherwise the point of L ( p , q) a t infinity. Then

4 P 8 9) = 1% IR(P,qla, b)I defines a metric in D which is finitely compact and M-convex and with p ( p ) = 00. The proof is the same as for (18.5), in fact the present case can be reduced to (18.5) by taking as new plane at infinity a finite plane which does not intersect D u C. If C is strictly convex, then h(p, q) defines a straight space, otherwise L ( p , q) e ( p , q ) , where e(p, q) is any euclidean metric associated with A", will define a straight space in D with the intersections of the affine lines with D as geodesics. The proof is quite trivial and may be left to the reader.

+

112

18.14


If D does contain a whole straight line and is not the whole A", then a maximal r, 1 I n - 1 exists such that D contains an r-flat L,', An (H - r)-flat that intersects L,' in a point, intersects D in a set D,,+ which does not contain a whole line. In D, we introduce a Hilbert metric h,,(p', q') as in the first part of the proof. If now p and q are any two points of D, let p' and y' be the intersection of the r-flats parallel to L,' through p and q with I),,-, (these r-flats lie entirely in D). Then

<
1.

p, = ( x i , y,), i = 1, 2,

+-

~ i ~ z = & ! ( ~ i - x *( Y) ~~ - Y ~ ) ~ I ' I~(%i)-t?~(xa)IIv(Y~)

--p(~z)I

defines, for every E > 0, a metric for which the affine lines are the geodesics. The points a = (2,0), b = (1, l ) , c = (0, 2) lie in this order on a segment, but with z = (0, 0) za=zc=2e+1,

hence zb enough.

< 1 za = 1 zc

and

will not be correct for

As second example we consider the set D : x 1

( x , y)-plane. The function y(r) = -t

hence for

pi = (xi, y,), i = I , 2,

V'L-E+~, a given 1< 2

ah=

1

+l-t

if

E

is small

> 0, 0 c y < 1 of

an affine

is monotone in 0 < t < 1,

18.11

113

DESARGUESIAN SPACES

defines a straight space in which the intersections of the affine lines with D are the geodesics. In particular, if 0 < a < then y = ax, y == fix, 0 < y < 1 are geodesics G and H . Call G+ and Hf the orientations of G and H for which traversing Gf or Hf in the positive sense corresponds to decreasing x. Let p ( t ) and q ( t ) represent G and H such that ordinates of p(0) and q(0) equal 4 and increasing t corresponds to decreasing x. The lines G+ and H + have a common “positive end-point”, and would be parallels in the language of hyperbolic geometry and are asymptotes in OUY lairguape (Section I1 and next chapter), the opposite orientations are not asymptotes. Nevertheless p ( t ) H and q ( t ) C are bounded for t 0 and lim p ( t ) H = lim q ( t ) G = cm i/ t 00. For if 9, and q,, are the points p ( t ) and q ( t ) with ordinate y, then

a.

-

-

0.

0 and the second tends


CHAPTER I11

PERPENDICULARS AND PARALLELS 19. Introduction

Perpendicular and parallel lines are basic concepts of elementary geometry. One would think that the extension of these concepts to more general spaces must have attracted the attention of mathematicians, but this is not the case’, and so it happens that the present chapter is less related to the work of others than anything else in this book. Therefore this introduction is quite detailed. A perpendicular to a straight line H is a straight line L that intersects H a t a point f and such that every point x of L has f as its foot on H. In the discussion of Minkowskian geometry we saw that perpendicularity need not be symmetric. The line H touches the spheres K ( x , f ) with center x through f at f and otherwise lies outside K ( x , . f ) . We map say that H is a supporting line of l i ( x , f ) and expect that in straight spaces the existence of perpendiculars is related to convexity of spheres. In fact, these conditions are equivalent. In Section 20 we first discuss briefly local convexity of sets in a G-space, then, also locally, the relation of convexity of spheres to uniqueness of the foot of a point on a segment. Then we apply these results to straight spaces and find that other equivalent conditions to convexity of spheres are : the function p x ( t ) is peakless* when ~ ( trepresents ) a straight line; a sphere intersects a line a t most twice. The latter condition may be applied to spheres of radius less than 2 in a space of the elliptic type whose geodesics have length 21. It is proved in Section 21 that for spaces of dimension greater than two this condition alone makes a space of the elliptic type actually an elliptic space. For two dimensions the problem offers peculiar difficulties and is open. One of the difficulties is that the Desarguesian character is for a two-dimensional space a much niore complex notion than for a higher dimensional space. The spheres are convex in any simply connected Riemann space with negative curvatures. Such spaces are, by Beltrami’s Theorem, Desarguesian only when the curvature is constant. Therefore convexity of spheres is a comparatively weak condition, when the situation regarding parallels is as in hyperbolic space. We have just seen that it is extremely restrictive in spaces 115

116


19

of the elliptic type, and expect therefore that thz boundary case, where parallels behave in the euclidean way, will prove z w y interesting. This question presupposes a theory of parallels. In view of the great fascination which parallels exerted for a long time it is most surprising that hardly any further work was done, after hyperbolic geometry had been discovered. We develop a theory of parallels in Sections 22 and 23. First we deal with C-spaces. Every point of a non-compact C-space is origin of at least one ray, that is a set isometric to the nownegative real axis. If r is any ray, q any point, then the limit 0 of a convergent sequence of segments T(q,,z,) with q, q and z, tending on t to infinity, is called a co-ra y from q to r. The co-ray from q to r is in general not unique, but the co-ray to t from any point q,, # q of 0 is unique and is a subray of 5. The proof of this remarkable fact is based on another characterization of co-rays. The spheres K(z,, p ) , fi E r, converge when z, tends on r to infinity to a set K,,,(r, p ) , the limit sfihere through p with r as central ray. The intersection of a coray s to r with K,(t, p ) is a foot of any point of s on K,(r, p ) . Hence the co-rays to r are the orthogonal trajectories of the limit sphere K m ( r ,p). Applying this to straight spaces we find in Section 23 that if g' is an oriented straight line and q a point not on 8, then as z tends on a+ in the positive direction to infinity, the line gc(q, d, so oriented that I follows q converges to an oriented line a+, which we call the asymptote through q to g+. The asymptote to g+ through any other point of a coincides with a+. Strangely enough the asymptote relation is in general neither symmetric nor transitive, even in two-dimensional straight spaces. Hardly any of the statements which we might venture to make on the basis of our euclidean or hyperbolic experience is true in general. We saw, for instance, at the end of the last chapter that the distance from 0 of a point x traversing a+ in the positive direction may tend to infinity, even when g+ is also an asymptote to a'. A notable exception is that r and 0 must be co-rays to each other, when sequences x, E r and v, E 0 tending to infinity exist such that x, y, 0. The absence of properties in general spaces leads to the task of ascertaining for special classes of spaces whether, and which of, their properties yield additional facts on parallels. There will be many instances later on where this is carried out. Returning to our original problem we now formulate the fiarallel axiom in a straight space as follows: asymptotes have the symmetry property and, if a+ is an asymptote to gf, then the opposite orientation a- of a+ is an asymptote to the opposite orientation g- of o+. In the plane case this means -+

-

20

117

PERPENDICULARS AND PARALLELS

exactly the same as the usual requirement: given a line g and a point on g, there is exactly one line through fi that does not intersect 9.

p

not

We then prove in Section 24 that a straight space of dimension greater than two is Minkowskian with differentiable spheres, in the usual sense, if i t satisfies the parallel axiom and its sfiheres are convex and differentiable in a sense which we define. The differentiability condition, as formulated here, is natural enough, but is a blemish, inasmuch as it is not known whether the theorem holds without it. We deduce this theorem from the fact that a higher dimensional straight space with flat limit spheres is Minkowskian with differentiable spheres. A third property characteristic for these spaces is that a set on which every point has a unique foot must bz convex and closed. We also obtain a characterizution of the higher dimensioiial errclidenn geometry in terms of the parallel axiom and the property af symmetric perpendicularity. Here it is not necessary to assume differentiability of spheres. We know from our discussion of plane Minkowskian geometry that these properties do not characterize the euclidean plane among two-dimensional straight spaces. Whether the characterizations of Minkowskian spaces which we gave, carry over to the plane is not known. We face here the same difficulty as in the elliptic case. As already mentioned, the convexity of the circles nieans that the functions fix(t-) are peakless. The stronger condition, that for some fixed a 2 1 the functions fix(^)" are convex, implies in conjunction with the parallel axiom that the metric is Minkowskian. This is shown in Section 25. No differentiability hypothesis enters here, so that we have a characterization of all plane Minkowskian geometries. 20. Convexity of spheres and perpendicularity

Before discussing the implications of convexity of spheres we discuss briefly general convex sets in G-spaces.

CONVEX SETS

IN

G-SPACES

The set M in a G-space R is convex if the segment T ( x ,y ) is unique for any two points x , y in the closure of M and if x , y E M implies T ( x , y ) E M . If M is convex, then is convex because x, x, y, y , where x,, Y , ' E M implies T ( x , ,y,) -, T ( x ,y), since T ( x ,y ) is unique. The set M is strictly convex if x , y E M implies that any point z with ( x z y ) is an interior point of M , i. e., S ( z , p) c M for sufficiently small positive p. These definitions are the same as in affine space. We show that some of the elementary properties of convex sets in affine space extend to this general case:

a

-+

-

20.1


118

n

(20.1) Let M be convex. If x is an interior point of M and y is a n y point of then every poiitt z with ( x z y) is an interior point of a. For there is a positive p such that S ( x , p) E M . Then V u , T ( y , u) where u E S ( x , p), lies in M and V - y is an open set. Hence no point of V - y can lie on the boundary of M ,so that z E V - y C M. A corollary of (20.1) is: =;

(20.2) The interior points of a convex set form a convex set. If x , y lie on the boundary of the convex set M ,then either T ( x ,y) lies on the boundary of M or every point z with ( x z y ) is an interior point of M. For if T ( x ,y) does not lie entirely on the boundary of M ,it contains an interior point zo of M. Then T ( x ,y) = T ( x ,zo) u T(zo,y ) and every point of T ( x ,z,,) - x and T(zo,y) - y is by (20.1) an interior point of M. In general spaces we cannot introduce supporting planes because planes will not exist. However, with some precautions supporting lines may be defined and proved to exist in sufficient numbers. It follows from (8.6) that for any two points x , y in S ( p , p ( p ) / 4 )a segment U ( x ,y) exists that has x as midpoint, passes through y and has length p ( p ) . If p ( p ) = a,then U ( x , y) means the entire geodesic through x and y. If M is a convex set with interior points in S ( p , p ( p ) / 4 )and q a boundary point of M , then a supporting line of M at q is a line U(q,x ) through q that contains no interior point of M. Denote by V , the set of points formed by q and all points which lie on supporting lines of M at q. Let A , consist of the points x in S = S ( p , p ( p ) / 4 ) for which an interior point y of M with ( x y q ) exists. Finally denote by B, the set formed by the points x in S for which an interior point y of M with ( x q y) exists. I t follows from (20.1) and (20.2) and the definition of supporting lines that the sets A,, B, are disjoint and A , U B , u ( V , n S ) = S.

If there is at least one supporting line of M at p , the point q need of course not be mentioned explicitly as element of V,. There is certainly no supporting line, when the space is one-dimensional. That there are supporting lines in every other case, is contained in (20.3) V , separates A , from B, i n S ( p , p ( p ) / 4 ) .

We may assume that the space has dimension greater than one. Let a curve x ( T ) , a < t /I. in S be given with x ( a ) E A , and x(B) E B, with X ( T ) # q for all t. I t is to be shown that x ( t o ) E V , for a t least one to.Let tobe the

a. There can be no interior point y of M with (%(to) y q) because this would imply the existence of y , with ( x ( t ) y, q) for small t - to> 0. Also, no interior point y of M with (y q x (to)) can exist because then for small to- t > 0 there would be a point 7,with (yrq x ( t ) ) . But by the definition of tothere is a t1< toarbitrarily close to toand an interior point y,, of M with (q y,, x ( t l ) ) . But then (yr,q yr,) contradicts (20.1). This proof yields a little more than was stated in the theorem. The point %(to) is not an arbitrary point in S on an arbitrary supporting line of M at q, but would in euclidean space be a point of the supporting cone of M a t q. For a later application we formulate this concept rigorously in the general case. Denote as ray with origin q E S a segment with q as initial point and of length p(p)/2. If p($) = 00 we mean a half geodesic x ( t ) , T 2 0 with x(0) = q. If the ray Q with origin q lies on a supporting line of M at q and is limit of rays (with origin q) that contain interior points of M , then we call R a supporting ray of M at q. Our proof contains: (20.4) If M c S ( p , p ( p ) / 4 ) is a convex set with interior points of an a t least two-dimensional C-space, then the supporting rays of M a t a given boundary point q form a set that separates A, (which contains all interior points of M ) from B,.

CONVEXITYOF

SPHERES AND UNIQUENESS OF FEET

We apply these concepts to spheres: For convenience we 0 < p c pl(p), (strictly) convex if S(p, p) is (strictly) s(p, p ) = S ( p ,p) u K(#,p) is also (strictly) convex. We convexity of all spheres with a fixed radius implies strict spheres with a smaller radius.

also call K(P, p ) , convex. Then show first that convexity of all

(20.6) Let 11 = inf p ( p ) > 0, and let K ( p , uo) be convex for p E F and a P EF fixed positive uo 7. Then each sphere K(q, a) with 0 < u < uoandS(q,uo-u)cF is strictly convex. Proof. A not strictly convex K(q, a)would yield points x , y, z with ( x z y). x q =y q u and q z 2 u. Since x , y E S(q, u,,) and K(q, a,,)is convex, z q < 0 ., and p I = a , . Then Since a, ,< q p(q) there is a point p with ( p q I ) p q = p z - q z = u o - u ; hence pc F. Moreover$ x < p q q x Q p q u 0 exist. If ~ ( trepresents line through a and b then q x ( T ) is not peakless. --+

-+

-+

--+

-+

-+

-.

, n - 2. If K(pJ contains K , put K8= K2. Otherwise the same method yields a p i n t p,l in K , such that K, = K 2 n K(p,') = K 2 nK(p,), so that K , is by (21.3) homeomorphic to PusIn either case K , = K ( p l )n K @ , ) n K(fi,) and n, = dim K, 2 n - 3. Continuing in this manner we find

'.

dim K(p,)nK(p2)n . . . nK(p,) 2 0. Therefore this intersection contains at least one point q. But q E K(PJ implies p, E K(q) which proves the assertion. We now identify our space with P" and use projective and affine terminology. We show next: (21.6) I n the affine space with K ( p )as hyperplane at infinity a sphere K ( p , a), 0 < a < 1, is an ellipsoid whose affine center is p .

Proof. Let x E K ( p , a),and x' be the conjugate point to x on g (p, x ) . Since every point of K(x') has distance 1 from x', the point x is, in a trivial way, a foot of x' on K(x'). Because of (20.6) the point x is then the only foot of p on K(x'). Hence K(x') is, in affine language, a supporting plane of K ( p , a) at x . Therefore K ( p ,a) is a convex surface. Let z E K ( p ) .Then K ( z )contains p and intersects K ( p , a) in a hyper plane set 2. If x E 2 and x' is defined as before, then ~ ( z x,) C K(x') because x E K(r') and the relations p E K(z),x E K ( z )imply x' E a($, x ) c K(z)so that also z E K(x'). Therefore ~(z, x ) is a supporting line of K ( p , a). Since z E K($) the lines g(z, x ) , x E Z are all parallel. The supporting lines of K ( p , a) parallel to g(z, s) therefore touch K ( p , a) in the plane set Z. Blaschke's Theorem (16.17) shows that K ( p , a) is an ellipsoid. The affine center of this ellipsoid is p because the plane K ( z ) containing Z passes through p .

21.6


127

It should be noticed that if we had any differentiability hypotheses like those for Finsler spaces in Section 16, the theorem would now be an immediate consequence of Beltrami’s Theorem. For the finite spheres K ( p , a) being ellipsoids (actually, homothetic ellipsoids) implies that an “infinitesimal sphere” about p , is an ellipsoid, so that the metric is Riemannian.

Fig. 19.

In absence of differentiability assumptions we proceed as follows: For an arbitrary point q let p E K(q) and let a, b be the intersection of K(p,a) with a line through p that neitherpasses through qnorlies in K(q).The two-dimensional is the conjugate point plane P through q, a, b intersects K(q) in a line. If #J’ of p on this line, then the intersection E of K ( P , a) with P is by (21.6) an ellipse with affine center p in P with g(q, p’) as line at infinity. The proof of (21.6) contains the fact that the line connecting the points of contact of the tangents to E from p‘ passes through p and q. Therefore p , p’, q is a self Hence a’ = g(q, a ) n g(#J’, b) and polar triangle with respect to E . b‘ = g(q, b) n g(p’, a ) are points of E collinear with p . It follows that pa‘ = pa, ab = a’b’. If g(a, a‘) intersects g ( p , p‘) or K(q) in no, then q, ao, a , a’ is a harmonic quadruple. The mapping @ of P“ on itself, which maps q and every point of K ( q ) on itself and maps any point u # q and not on K(q) on the point a’ on g(q, a) for which the points q, uo = g(q, a) n K(q), a, a’ are harmonic, is a

128


21.6

projectivity (more particularly in the terminology of projective geometry, a harmonic homology)'. As a projectivity it maps ~ ( ap,) on o(p, a'). The relations up = a'p and ab = a'b' show that Q, preserves distance on o(a, p ) . Since p is arbitrary in K(q) the mapping 4j preserves distances in general, hence is a motion. This shows that R possesses a reflection in the arbitrary hyperplane K(q). It is well known, and will be proved later, that K must then be elliptic. This establishes (21.2) and (21.1). The two main ideas of this proof were (1) using the flatness of K ( p ) to establish the Desarguesian character of the space, (2) using Blaschke's Theorem to show that the spheres are ellipsoids. Neither of these methods works in two dimensions, and the problem of whether Theorem (20.1) holds for two-dimensional spaces is open. One would conjecture that the theorem does not hold without assuming the Theorem of Desargues, but will hold if this theorem. is assumed. The hypothesis that a sphere K ( p , 0 ) does not intersect a geodesic more than twice, implies that the space is Desarguesian and the metric elliptic. The Desarguesian character alone does not imply that the metric is elliptic. G. Hamel [l] has shown that compact Desarguesian spaces other than the elliptic spaces exist.

SPHERELIRE SPACES

A not necessarily Desarguesian metrization of P4as a C-space of the elliptic type leads by means of the standard 2 to 1 mapping of the sphere Sz on Pg to a metrization of S* (compare Section 28) in which all geodesics are great circles in the metric sense, and any geodesic passing through a point Z of S* also passes through the antipodal point is' to is. It is a wellknown conjecture of Blaschke that the only Riemannian metric on Sa with this property is the spherical. The examples of Hamel just mentioned show that there are non-Riemannian metrics on S2in which the ordinary great circles on S2are the geodesics. If Blaschke's conjecture is correct, the proof must therefore use emphatically the Riemannian character of the metric, and not so niuch the configurational properties of the geodesics. This connection with metrics on the sphere leads naturally to the question, whether our theorems (21.1) and (21.2) cannot be used to characterize the higher dimensional spherical metrics. To achieve this we define.

21.9

129


A G-space R is "spherelike" if all geodesics are great circles and each geodesic which passes through a given point a also passes through a second point a', the "antipodal" point to ii. There cannot be more than one antipodal point to Z, because two distinct great circles cannot intersect more than twice. Since two geodesics through Z intersect a second time, they have the same length, and Z, a' are conjugate to each other on both geodesics, see (9.12). Hence Z' is also antipodal to Z, and the geodesic through two not antipodal points is unique. A n y two geodesics have the same length, because there is a geodesic intersecting both. (21.7) The mapping of a spherdike space points is a motion.

on itself which mates aiitipodal

_-

--

We must show that a h = a' b' for any two points a , b in and their antipodal points Z', b'. This is obvious for a = b o r Z = b'. If neither is the case, then Z and are not antipodal, hence the geodesic 5 through 5 and '6 is unique and contains both Z' and 8'. Hut S is a great circle, that is, congruent to an ordinary circle, therefore Z b = Z' 5'. We can now establish the connection with spaces of the elliptic type which leads to a characterization of spherical spaces:

a

(21.8)

space

The non-ordered pairs a form with the metric

==

(a,Z')

of

antipodal points

of

a spherelike

ZP),

a b = niin ( Z T ,

(21.9)

a space R of the elliptic type in which the great circles have length A = Zb'. The mapping Q, of 3 on itself which mates antipodal points has period two and forms therefore together with the identity a discrete group of motions of I t follows from a general theorem on covering spaces that R is with the metric (21.9) a G-space, compare (29.1). In the present case a direct proof of this fact is simple: because of (21.7)

x.

ab=min(Z'T,iib') =min(bZ,BZ')=ba. Obviously a 6 = 0 if and only if a = b. The triangle inequality holds because (21.7) yields the inequalities

a

-

T +h T > i i ~ , ZT'+TC=Z'T

aJ

+ T C > Z a ' T= UC'

+ Jc' 2 act, j 6' + b ~ l = +

Z'c' = a T ,

which also show that (a b c) only if Z, 5 , 5 lie on a great circle in

3.

130


21.10

R is obviously finitely compact. The definition (21.9)applied to the points on one great circle in ’T amounts to identifying antipodal or conjugate points on and therefore yields in R a great circle 0 with half the length of 0, that is 1. It is now clear that R is a G-space in which all geodesics are #eat circles of length 1. The-great - circle through - - two distinct points a, b of R is unique, because if a = ( a , a’) and b = ( b , b‘) then and ‘8 are not antipodal and the great circle through; and ‘8 is unique. In the spheres K ( 5 ,a), 0 < a < 1/2,and K@‘, 1 - a) coincide. K ( 9 ,a) and - K@‘, a) correspond to the sphere K ( P , a) in R. Therefore, if the geodesic g in intersects the sphere K(F,a), 0 < a < l/2,at most twice, the same holds for the corresponding geodesic 0 in R and the sphere K ( p ,a). If this is true for any geodesic in and any sphere K ( p ,a), 0 < a < 1/2, and is at least three-dimensional, then the metric in R is by (20.1)elliptic. We conclude that the metric in is spherical. Thus we have the theorems:

s,

s

a

x

x

s x

(21.10) Let R be a spherelike space of dimension greater than two, whose geodesics have length 4l. I f a geodesic intersects a sphere of radius a < 1always in at most two points, then R is a spherical space. (21.11) A n at least three-dimensional spherelike space whose geodesics have length 41 is spherical if its spheres of radiats 1are flat. 22. Limit spheres and co-rays in Q-spaces.

For the reason outlined in the introduction to this chapter we now turn to the theory of parallels for rays and straight lines, which we develop first for general non-compact G-spaces and then apply to straight spaces. A “ray” with origin p is a half geodesic ~ ( tt )2 0 with x ( 0 ) = p such that z(tl) x ( t a ) = Itl - tal for ti >, 0. In other words, a ray is a set isometric to the ray t 2 0 of the real axis. In a compact space the distances are bounded and rays cannot exist. However:

(22.1) I n a non-compact G-space every point is origin of at least one ray. For if q is a given point, then points qv with q q, -c 03 exist. Let xv (t) represent a geodesic for which the subarc 0 t q qy of x, (t)represents a segment T(q,qv). Because of (8.14) zJt) converges for a suitable subsequence { p } of { v } to a geodesic ~ ( t Then ). x (0) = q, since x,, (0) = q. Moreover, if tl taare any non-negative numbers, then q q,, > t2from a certain subscript on. Then z,,(tl)x,,(ta)= ta- tl because z , ( t ) is a segment for 0 t qq,,. It follows that x ( t l ) x ( r s )= t2- rl, hence ~ ( tt) 2 , 0 represents a ray.

<
a(r, P ) the “exterior of Km(rtp)”.

<
t - tl- x z ( t )

> t2- tl

--

xz(t2).

Put z(ti) = zi. If x is in the interior of K(z2,zl) then tz - tl> xz2 hence a(r, zl) > a(r, x ) so that x lies in the interior of Km(r,zl). If x lies on K(z,, al) then t2- tl = xz2. Because of (22.2) the equality sign holds in (22.13) only if ( x z g ( t ) ) . But then uniqueness of prolongation yields x = zl. We can now prove the following addition to (22.11):

-

(22.14) Tileorem: If z ( ~ represents ) the ray r and r is a subray of a ray r, that intersects Ka(r, p ) at a point q then for aity sequence t, m

lim W t , ) ,q) = K , (r, p ) = K , (r, a).

Such a ray ro always exists when r lies on a straight line.

+

> +

Let r be the subray z ( t ) = zo(t t o ) to , 0 of the ray to represented by zo(t). By hypothesis there is a 7 2 0 such that z o ( t ) = q. Because of (22.10) it suffices to show with z, = z(t,) = zo(t, to)that lim W,, q) = K,(ro, 4). (22.11) reduces this equation to

K,(ro, q) climinf K(z,,q).

+ +

+

If x E Km(r0,q) then (22.12) yields xz, 2 t, r0 - 7,when t, to> 7. A segment T ( x ,z,) contains therefore a point x, with x, z, = t, to- 7 or x, E K(z,, q). We have a(ro,x ) = a(ro,q)because x E K,(r, q) and conclude from (22.4) that xz, - T, -to--* - T or x x , = xz,- xg, = X % , - t ,

4- t - 7 0

--*

0

which shows x E lim inf K(z,, q). If a,, = a(r, p ) < 0 then K,(r, p ) intersects r a t the point z(-a,,), compare (22.4) and (22.9). If a,> 0, then K,(r, p ) will not intersect r. But if r lies on a straight line, we can find a subray ro of this line which contains r and such that the distance of the origins of r and ro equals a,,. If z 0 ( t ) represents ro, then z ( t ) = zo(t a,,). Because of (22.6) the point zo(0) lies on K(ro, p ) and hence by (22.10) on K ( r , p ) . This fact implies, of course, that in a straight space every limit sphere is a limit of spheres. In that case we show for completeness sake:

+

22.16


134

(22.16) Corollary: If z(t) represents a ray r in a straight space and 7,

p,

+

p,

then

+

00,

.

lim K(z(T,),p.) = K d r , PI. *+OD

For we may assume Km(r,p ) intersects r in a point q. If z, = z(t,) then 0 = a(r,p ) - a(r,q) = lim (zup- z,q) = lim (z,p, - z,q). Because the space is straight there is for every point of one of the two spheres K(z,, p,) and K(z,, q) a point (on the line through z ) on the other sphere whose distance from the first point is Iz,pv - z, qI. This implies lim K(z,, p,) = lim K(z,, q) = Kw(r,q). CO-RAYS

-

Let z(t) represent the ray r. A ray s with origin q is called a co-ray from q to I, if a sequence of segments T(q,,z(r,))With q, ---L q and t, co exists which tends to s. More precisely, if w ( t ) represents s and w J t ) represents the geodesic for which w,(O) = q, and the subarc 0 t q, z ( t J is T(q, , z(t,)), then w&) --* w(r) for t 2 0. The proof of (22.1) shows that every point q is ongin of at least one co-ray to r. The co-rays to r play the role of parallels to r. Our aims are to characterize co-rays to r (1) in terms of the function a(r, x ) , this is accomplishedby (22.16) and its converse (22.20); (2) as orthogonal trajectories, as it were, of the limit spheres Km(r,x ) . This is contained in (22.18).

<
0 if and only if q lies i n the interior of K,(r, and is a foot of p on Km(r,q).

p)

22.18

135


If a(r, p ) -a(r, q) = pq > 0 then q lies in the interior of Km(r,p ) and it follows from (22.6) and (22.9) that for x E K,(t, q)

fix 2 a(r, p ) - a (r, x ) = a(r, p ) - a(r, q) = pq. Therefore q is a foot of p on Km(r,q). Conversely, let q be in the interior of Km(r,9) and a foot of p on Km(r,q ) . Then a(r, p ) > a(r, q). On a co-ray $(T) from p to r consider the points p ( t 0 ) with to= a(r, +) - a(r, q ) . By (22.16), a(r, p ) - a(r, # ( t o= ) )to hence a(r, # ( t o= ) )a(r, q) and p(r0)E Km(r,q ) . Moreover to= pp(to) 2 p q because q is a foot of p . The definition of T~ and (22.5) show that to= pq. We can now prove: (22.18) I f z, is a co-ray to r with origin q then a n y point qo # q of 5 is a foot of q on Km(r,qo) and is the only foot of any third point q1 # q of 5 on Km(r,qo). Also q is a foot of qo on K,(t, q ) .

That qo is a foot of q is contained in (22.16) and (22.17). These two facts together with (20.6) show that qo is the only foot of q1 if (q qlqo). If (q qoql) take q2 with ( q q1q2) then also (qoq1q2). There is a foot f of q1 on Km(r,qo) because this set is closed. We conclude from (22.16) and (22.5) that qoqa = a(r, 40) -a(L

Qf

41

+

4142

42)

= a(r,

f ) -4rt

Qf

42)

42

G Qo41 + 4142 = 4042'

Therefore ( f q1q2) and fql = qoq,, and uniqueness of prolongation shows qo = f . Letting qo tend to q we deduce from the continuity of ~ ( r x, ) in x that q is a foot of qo on K-(r, q). The co-ray from a given point to a given ray need not be unique. An example is obtained by considering the surface 5 2 + ~ 2 =

1

for

(20,

c2=

[2+q2+

1

for

( 0 the subray w ( t ) = w o ( t 7) of w o ( t ) is a co-ray to r and the only one with origin w ( 0 ) = w0(q). I t is most remarkable that this is not a special property of the example, but holds in general.

+

+

22.19


136

(22.19) Ttmwem: If s is a co-ray from q to r and qo is any point of 0 different from q, then the co-ray from qo to r is unique and coincides with the sub-ray of 0 with origin q,,.

For let w ( t ) represent s and qo = w(to), to>0. If q ( t ) represents any co-ray to r with origin qo, then (22.16) yields for t' > to

a ( r , . ~ ( a o ) ) - a ( r , I ( t ' )= ) t ' - t o= a(r,qo) - a ( r , q ( t ' - t o ) ) . Therefore q(t' -to)E K,( r, ~ ( a ' ) ) . By (22.18). the point q ( t ' -to)is a ) ) , q(t' - to)= foot and I@') is the only foot of w(to) = qo on K:(r, ~ ( t ' hence w ( t ' ) , which shows that q(a) is a sub-ray of w ( t ) . We can now establish the converse of (22.16): (22.20) If w(a) represents a geodesic and

a( I, w ( t " ) ) - a(r, ~ ( t '= ) )t' - t"

for

T', t" 2 O

then w ( t ) represents for t 2 0 a co-ray to r. For ~ ( t )t ,2 0, is a ray because by (7.3) and (22.5) for t ' - t"

2 w(t") w ( t ' ) 2 a( r, ~ ( t "- a)( )r, w(t'))

t'>

t"

= t' - t".

+

I t follows from (22.17) and (20.6) that fqr positive y and t the point w ( y t) is the only foot of ~ ( qon ) K,(r, w(q t)), and from (22.18) that wq (t)= w(y a), t 2 0, is a co-ray from w(y) to r. Letting y tend to zero we see from the definition of co-ray that w ( t ) , t > O , also represents a co-ray to r.

+

+

I t will be shown that even in a two-dimensional straight space the fact, that s is a co-ray to r, does not imply that r is a co-ray to 0. I t is therefore important to know conditions under which the relation of co-ray t p ray is symmetric. Since each ray is a co-ray to itself the following statement contains a sufficient condition : (22.21) I f a(r, x ) - a(s, x ) is constant (that is, independent of x ) then every co-ray to one of the rays r, 0 is also a co-ray to the other.

The hypothesis yields a(r, x ) - a(r, y ) = a($, x ) - a ( 0 , y ) for any two points x , y . If w ( t ) is a co-ray to r then by (22.16)

a'-"'

= a(r, w ( t " ) ) - a ( r , w ( t ' ) ) = a(5. ~ ( 7 " ) ) - a ( s~ , (t')).

Now the converse (22.20) of (22.16) shows that w ( t ) is B co-ray to s. This result yields together with (22.8) and (22.19) the following facts:

23.1

137


(22.22) l'heorern: I f doo(r,0 ) = 0, then every co-ray to one of the rays r, 5 i s also a co-ray to the other. (22.23) If the origin of the ray s i s an interior point of the ray r and doo(r,5) = 0, then 5 i s a sub-ray of r. I f ~ ( tand ) y ( t ) represent two distinct straight lines with a common point then lim inf ~ ( ty(a) ) > 0. l r l +OD,

l u l +a

I t should be noticed that the condition dw(r, 0 ) = 0 for the rays r : z(t) and 0 : w ( t ) does not imply that the distances z(t)i or w ( t ) r are bounded for t 2 0. An example is furnished by the surface of revolution 1

t 2 + q a = - +etsin2c,

c

C>O,

in the ordinary (t,11, ()-space with the length of the shortest join on the surface as distance. The meridians of this surface are geodesics and straight lines in the sense of G-spaces. Let g, be the meridian 11 = 0, 6 < 0, ga the meridian q = 0, E > 0 and r k the sub-ray ( 2 1 of gk. Then r k contains the pointsq,, = ( ( - l ) ' ( v z ) - ' , O,vn), k = 1, 2 w i t h q , , q , , ~ z ( ~ ~ ) - ~ - ~ - ~ h e n c e dm(rl, r2) = 0. But rk also contains the points

23. Asymptotes and parallels in straight spaces. We now apply the results of the preceding section to straight spaces. In that case (22.19) may be extended as follows. (23.1) If in a straight space 5 is a co-ray to r, then any ray that contnins 4 i s a co-ray to r. The co-ray from any point to any ray is unique. A representation w ( t ) , t 2 0, of s may be extended to a representation w ( t ) , - co < t < 00, of the straight line containing 5. Let z(t) represent r. Since 5 is a co-ray to r there is a sequence qy w(0) and a sequence t, -00 such that w , ( t ) -,w ( t ) for all t, where w, (7) represents the straight line g(q,, z(t,)) with w, (0) = q,, I,(qv Z ( T , )) = z(T,). This implies for t' > t" and large Y, that t' - T" = w, (t") z(t,,) - w, (t') z(t,), +

w ( t ' )w , ( t ' )

+ w ( t " ) W"(T")2 It""

- [W(T")

z(t,)

- w(r') z(t,)]1;

23.2


138

hence by (22.6) a(r, ~ ( 7 " -a( ) ) r, ~ ( t '= )t ) '- t".

+

Since t', t" are arbitrary it follows from (22.20) that the ray wo(t)=w ( q t) is for any real 1 a co-ray to r. The uniqueness of the co-ray to r from a given point is a consequence of this and (22.19).

ASYMFTOTHS The usual terminology used for parallels in euclidean or hyperbolic space refers to oriented straight lines rather than to rays. A geodesic g represented by x ( t ) in a G-space may be traversed in two different senses, corresponding to increasing or decreasing t. We orient g by distinguishing one of these senses as positive, the other as negative and use g+ or g-as symbol for an oriented geodesic. x ( t ) is a representation of g', if x(t) represents g and traversing g+ in the positive sense corresponds to increasing t. The representations x ( t j?) of g and only these represent 9'. It is convenient to denote the other orientation of g by B-. If x ( t ) represents g+(g-) then x(- t j?) represents g-(g+). If x ( t ) represents the oriented straight line g+ then a positive sub-ray of g+ (or negative sub-ray of g-) is a ray of the form y ( t ) = x ( t p), t 2 0, and a negative sub-ray of g+ (or positive sub-ray of g-) has the form y(t) = x(- t j?), t 2 0. The oriented straight line 0: tends to the oriented straight line g+ if x, (t) -. x ( t ) for suitable representations x , ( t ) of g t and x (t)of g+. If g t tends to g+ or g- then g tends to g- or g + respectively. We may then express our results (23.1) and (22.18) as follows:

+

+

+

+

(23.2) Theorem: I n a straight space, i f ~ ( trepresents ) the oriented line g + , and qy -. q, ty .+ a,then the line g+(q,, x ( t , ) ) so oriented that x ( t , ) follows qy tends to an oriented line a+, which is independelit of the choice of the sequences { q y } and {t,}. The line a+ is called the asymptote to gf through q. The asymptote to g+ through any point q' on a+ coincides with a+. The feet of q on the limit spheres K,(r, x ) are unique and a+ is their locus. so that a is perpendicular to all K,( r, x ) , where Y is a positive sub-ray of g + .

In any straight Desarguesian space defined in a convex subset C of A", the oriented line a+ is therefore an asymptote to the oriented line g+, if points traversing a+ and g+ in the positive sense tend to the same point on the boundary of C. Thus we see:

23.5


139

(23.3) I n a straight Desarguesialz space asymptotes have the following two properties : Symmetry. I f a+ i s an asymptote to g', then g+ i s an asymptote to a+. Transitivity. I f a+ i s an asymptote to b+ and b+ i s an asymptote to c+, then a+ i s an asymptote to c'. We observe: (23.4) I n any straight space the Trairsitivity Property implies the Symmetry

Property. In a two-dimensional straight space the two Properties are equivalent. Proof. In an arbitrary straight space let a+ be an asymptote to g+ and p any point of g. If c+ denotes the asymptote to a+ through p , then transitivity yields that c+ is an asymptote to Q+. Since g+ is asymptote to itself it follows from (23.2) that cf = g+. Whether the converse holds for higher dimensions is an open question. To prove the converse in the plane case, let a+ be an asymptote to b+ and b+ an asymptote to c+. Since the assertion is otherwise trivial, we may assume that a t , bf, c+ are distinct. The line a does not intersect b nor does b intersect c. Moreover, a cannot intersect c, bhcause the Symmetry Property would imply that a+ and c+ are two different isymptotes to b+ through their intersection. For an indirect proof we assume that the asymptote b+ to cf through some point p of a+ is different from at and distinguish three cases: 1) b+ lies between a+ and c f . Because a+ is asymptote to b+ the line b must intersect b, hence b+ is not an asymptote to c'. 2) c lies between a and b. Since b does not intersect c it cannot intessect b, hence a+ cannot be asymptote to b+. 3) a lies between b and t. By symmetry t+ is an asymptote to b + . Now,'b t', b t are in the same position as a+,, ' b c+ under case 2). Hence bf is an asymptote to b+ but then a+ and b+ would, because of symmetry, be two different asymptotes through p to. ' b In order to show that neither symmetry nor transitivity hold in general straight spaces, it suffices to give an example for the absence of symmetry. We show: (23.5) The jollowing m a y occur in a two-dimensional straight space:

(a) There are oriented straight lines a+ and g+ such that a+ is a n asymptote to Q+, but neither is g+ an asymptote to a+ nor g- to a-. (b) There are oriented straight lines a+ and g+ such that a+ i s a n asymptote to g+, but g+ i s not an asymptote to a+, although g- and a- are asymptotes to each other.

140

23.5


H+ Gf GI+

\

two distinct points of the plane lie on exactly one curve of 2. I t was shown in Theorem (11.2) that the plane can be metrized such that it becomes a straight space with the curves in Z as straight lines. This furnishes an example for (b).

sponds to the positive sense. The curves H and G, intersect at q = (-1, 1). The curve in Z through p = (0, 0) and (- 1, Y) is an ordinary line which tends for v co to C, therefore Gf is the asymptote to C : through fi. Since H+ lies, from q on, +

23.7

141

PERPENDICULARS A N D PARALLELS

Theorem (11.2)the plane can be metrized such that it becomes a two-dimensional straight space with the curves in ,El as straight lines. Let Gt,Gi-,H + ,p and q be defined as before and denote by Hi the curve in .Zl through q obtained from H , by the translation t' = 6 2, 7' = T,I and so oriented that decreasing q corresponds to traversing Hi in the positive : and G- is an asymptote to G;, sense. This time Gf is an asymptote to G but the asymptotes to ' G and G- through q are H f and H:, hence different from G: and GT. If the straight lines g and a in a straight space have the property that, with proper orientations, a+ is an asymptote to gf and also a- to g-, then we call a Parallel to a. The two examples which we just constructed also show:

+

(23.6) The following may o c c w in a two-dimensional straight space: a i s parallel to g, bid neither orientation of g i s an asymptote to either orientation of a. a i s parallel to g, an orientation g+ of g i s an asymptote to a n orientation a+ of a, but g- is not an asymptote to a-.

THE PARALLEL AXIOMS With these examples in mind we are led to the following conditions as adequate extensions to straight spaces of the properties of parallels as they are found in the foundations of euclidean and hyperbolic geometry.

Parallel Axiom: The asymptotes have the symmetry property1 and: If a+ is an asymptote to gf, then a- i s a n asymptote to g-.

Hyperbolic Axiom: The asymptotes have the symmetry property and: If a t is an asymptote to of and different from g+, then a- is not an asymptote to g-. Some remarks on these conditions are in order. (23.7) The parallel .4xiom holds i n a straight two-dimensional space, if and only if /or a given line g and a given point p not on g, exactly oneline through p exists which does not intersect g.

Thus in the case of straight planes, the present Parallel Axiom is identical with the usual. The usual parallel axiom in space is only formulated when planes exist or the space is Desarguesian, and therefore implies transitivity. To prove (23.7) observe that if in any straight plane the lines a, b carrying the asymptotes a+ and b+ through p to the two orientations fl+ and 6- are different, infinitely many lines through p exist which do not intersect g, and that a =: b implies that a is the only line through p that does not intersect a.

142

23.8


Therefore the present parallel axiom implies the usual, and the usual axiom implies a = b and clearly also symmetry. The situation is somewhat different for the Hyperbolic Axiom. I t is not hard to construct similar examples as above which show: symmetry does not follow from the assumption that, if a+ is an asymptote to Q+ and different from Q+, the asymptote to Q- is different from a-. Scrutinizing the treatment of hyperbolic geometry found in the literature2, we find that axioms of congruence or mobility are introduced before the hyperbolic parallel postulate, and that symmetry is derived with the help of these axioms. If a ( t ) represents an asymptote a+ to Q+ # a+, then in euclidean and hyperbolic geometries a ( t ) Q is bounded for t 2 0. In hyperbolic geometry we have in addition a ( t ) Q -P co if T -C - a0 and a ( t ) Q + 0 if t -+ 00. We are going to show with examples that no statements of this type are possible in general, even under strong additional assumptions: (23.8) Let a ( t ) represent an asymfitote a+ to Q+ # a+ i n a two-dimensional Desarguesian space which satisfies the Hyperbolic Axiom. It may happen that

(a) a ( t ) Q-C

00

for

t+

co and a(.)

Q

is bounded for

(b) The sets S(Q, p) are convex for convex Q and a real T.

t

< 0.

> a ( t )Q > ,d > 0 for all

(c) The sets S(Q, p) ure convex for convex Q and a(.) Q .+ 0 for t + 00, whereas ~ ( tQ )is bounded for t < 0. The example a t the end of Section 18 proves (a). Our interest in (b) and (c) derives from the fact that slightly stronger convexity conditions will, without Desargues' Theorem, be shown to exclude x these occurrences. To obtain examples for (b) and (c) we take a convex curve C Y in A' which contains exactly one proper segment T and is not differentiable at a point q not on T . and as metric the Fig. 21. Hilbert distance in the interior D of C. If x and y are interior points of T, then the open segments Q+ from x to q and a+ from y to q oriented towards q furnish an example for (b). If we replace q by a point r on C - T where C is differentiable then the open segments Q from x to r and r from y

23.12


143

to r furnish an example for (c). Elementary arguments about cross ratios confirm these statements*. (22.10) shows that for any two positive subrays rl, r, of the oriented line g+ the limit spheres Km(rl,x ) and Km(r2,x ) coincide. We may therefore denote these limit spheres, also, by K,(g+, x ) . In euclidean geometry K,(g+, x ) = K,(g-, x ) . The Parallel Axiom does not generally imply this relation for (17.21) shows : (23.9) I n a Minkowskian G-space in which the spheres are not differentiable, there are straight lines g , such that K,(g+, x ) # K,(g'-, x ) for every point x . Also there are lines a+ and g+ # a+ which are asymptotes to each other such that K,(g.+, x ) # K,(a+, x ) for every x.

It suffices to choose g such that K(p, y ) is not differentiable a t y for and to choose as a+ any asymptote to g+ different from g+.

p, y E g,

By means of this example we conclude from (22.21): (23.10) Tht! following is a sufficient but not necessary condition for the (Symmetry and) Transitivity Property of the asymptotes i n a straight space: (23.11)

a(r, x ) -a(5, x ) is independent of x whert

5

is a co-ray to r.

If a(r,x ) - a(5, x ) is independent of x for two rays r, 5 of a G-space, then K m ( r j p ) = K m ( 5 , p ) for every p . For the equation of K,(r, p ) is a(r, x ) -a(r, p ) = 0 and this implies a(5, x ) - a(s, p ) = 0. If the space is straight, then an asymptote a+ to b+ is by (23.2) the locus of the feet of any point q of a+ on the limit spheres h',(b+, x ) = K,(a+, x ) . If b+ is an asymptote to c', then K,(c+, x ) = K,(b+, x ) = K , ( a + , x ) , hence a+ is the locus of the feet of q on K , ( c + , x ) , hence an asymptote to c'. Thus the Transitivity Property holds. That the condition (23.11) is not necessary follows from the fact that it implies K , ( a + , x ) = K,( a+, x ) when a+ is an asymptote to g+, which is not necessary because of (23.9). In euclidean space, if a+ is an asymptote to b+ and 1: is a positive subray of a+, and 5 is a negative subray of b+, then it is easily verified that (23.12)

a(r, x )

+

a(5, x )

is constant.

This leads to the question whether (23.12) is in G-spaces, and more particularly in straight spaces, connected with the existence of parallels. We show:

144


23.13

(23.13) I/ (23.12) holds /or the two rays r and s of a G-space then any co-ray to r (or s) is positive subray of a straight line gf, and every negative subray of fi+ is a co-ray to I (or r). The relation (23.12) implies a(r, x ) - a(r, p ) = a(s, p ) - a(5, x ) for any p and x , hence K,(r, p ) = K , ( s , p ) for any p . Moreover, the interior of K,(r, p ) coincides with the exterior of K,(s, p ) . If p1 and pa are co-rays from p to r and 5, they have no common point but p , because the interiors of K,(r, p ) and K,(I, p ) are disjoint. Any point al of p1 has by (22.18) $ as foot on K,(r, p ) = K,(e, p ) . We deduce from (22.17) and (22.20) that p lies on a co-ray from al to 5 , and then from (22.19) that p, is a subray of this co-ray. Therefore p1 and pa lie on a straight line fi which, in the orientation g+ for which p1 is a positive subray has the property that every negative subray of g+ is a co-ray to s. For straight spaces we obtain from (22.13):

+

a(e, x ) is constant for two rays r, 5 on the same line g, (23.14) I/ a(r, x ) then every point of the spacelies on a parallel to g (01,a/ a' is an asymptote to g+, then so is a- to g-). For (23.13) implies that for any orientation 0' of Q one of the rays t, s is positive and the other negative. The rest follows from (23.13) and the definition of parallel. The converse of (23.14) is not true since we saw in the proof of (23.13) that if a(r, x ) a(s, x ) isindependent of x , then K,(fi+, p ) = K,(g-,$) and this need - by (23.9) - not be true even when the parallel axiom holds.

+

24. Characterizations of tho highar dinreiisional Minkowskian geometry. It was explained in the introduction to the chapter why it is reasonable to expect interesting results if a space has convex spheres and satisfies the parallel axiom.

THE DESARCUESIAN CASE. We begin our investigation by proving: (24.1) l ' h e ~ ~ ~Ai Desarguesian . : space i n which the parallel axiom holds and the spheres are convex is Minkowskian. The parallel axiom implies that we deal with a metric defined in an entire affine space A". I t is to be shown that the affine mid-point of a segment T(Pl,p,) coincides with the mid-point in the sense of the given metric. By considering a (two-dimensional) plane through fi(pl,p,) we reduce the problem to showing that the metric in a given plane P is Minkowskian.

21.2

145


A circle K(q, a) in P is a convex curve. It suffices to see that K(q,a) possesses parallel supporting lines a t the end-points P I , p , of a given diameter T(p,,p,) of K ( q , a). For then q is by (16.7) the affine center of K(q,a), hence the affine midpoint of T(p,,p,) as well as in the sense of the metric. Supporting lines of K ( q , a) at p , and Pa are transversals to a($,, pa). Therefore (24.1) is contained in the following fact which holds without assuming Desargues' Theorem.

Fig. 22.

(24.2) If a two-dimensional straight space satisfies the Parallel axiom and has convex circles, then a line g which i s perpendicular to one of two parallel lines

bl, b, i s also perpendicular to the other. There is a Put p, = g n bi and' assume that g is perpendicular to .,)I transversal 1, to g a t p , (see (20.12)). If 1, = b, then nothing is to be proved - (23.9) shows that there may actually be a transversal b # ,)I when the assumptions of (24.2) are satisfied. If b # ,)I let qo = l,,nbl. We show that for any q with (plqoq)the point p2 is the foot of p , on I, the line joining ps, q. I t follows then for p,q 00 from the parallel axiom that p , must also be the foot of 9, on ba. If 3;' lies on I on the same side of p , as q, then T(p,, x ) intersects I, in a point xo with p , x > p,xo > fig,, hence x is not the foot of p1 on I. Let y be any point satisfying (q p , y ) ; through a point u with (plqu) draw the parallel g' to g. Since g' is the only line through u which does not intersect g it must be the perpendicular to bl at u. The line g' intersects I at some point u and the parallel to a($,, y) through q in a point w with (wou), hence WV < wu < wq. Therefore g ( p , , y ) cannot be perpendicular to I. For if it were, then g(q, w ) would be perpendicular to 1 as the only line through w which does not intersect g(p,, y ) . Hence fi is perpendicular to I, and (24.2) is proved. +

10

146

24.3


THE GENERAL

CASE

Theorem (24.1) shows that the main problem is to find conditions under which a straight space with the parallel axiom and convex spheres is Desarguesian. It may be that no additional conditions are necessary but this seems hard to prove. In higher dimensions we will see that it suffices to add a condition, which in smooth spaces amounts to differentiability of the spheres. We need the following: Lemma. Let C, be distinct from, and parallel to, G , and let L intersect G, at pi. Then an arc A with the following properties exists: (1) A connects points q, q' on L with (qpaq') and contains no other point of L. (2) A intersects G, in a point r (3) for every other point y of A the line g(P,, y ) intersects G,, in a point x.

Fig. 23. #

Let paGl -- 2 7 (>0 ) . The point PI divides G, into two rays R' and R". For x E R' define y by ($,yx) and p a y = 7. For x E R" define y by (xp,y) and p a y = 7. If p , is considered as point of R' the point y is the point q with (pgqpp,)and p,q = 7, and if p , is a point of R" the point y is the point q' with (P&') with Paq' = 7. If p,x co either for x E R' or x E R" the point y tends, because of the parallel axiom, to the same point r on G,. +

We apply this lemma to prove: (24.3) If C,, C, are parallel and intersect L at

p,, pa, and C, i s perpendicular

to L , then G, is perpendicular to L . For if A is constructed as in the lemma, then for reasons of continuity at least one point of A has Pa as foot on L, but all lines o(p,, y ) with y # r on A intersect G,, hence Y must have p , as foot. A second application of the lemma is:

If G, intersects L at p , and is not perpendicular to L then a given point Pa of L is foot of at least one point x, of G,. (24.4)

24.6


147

Let G, be the parallel to G, through p , # p,. We construct A according to the lemma. At least one point y, of A must have p , as foot on L. This time it cannot be r, because then G, would be perpendicular to L, and this would by (24.3) imply that G, is perpendicular to L. Since y, # r the line g ( p 2 , y,) intersects G, in a point x,. This has the important corollary. (24.6) If a , b are distinct points of K m ( g + ,q) then the parallel to g through a

(or b) is perpendiciilar to g(a, b). For if the parallel G to g through a were not perpendicular to g(a, b) then C would by (24.4) contain a point x whose foot on g(a, 6) is b, then x b but a is by (23.2) the foot of x on K,(g+, q).

< x a,

We now define differentiability of spheres:

If the sphere K(q, a) in a straight space is convex, then it is called “diflerentiable at x E K(q, a)” if no proper subset of the set W, formed by the points on the supporting lines of K(q,a) at x , decomposes the space. This implies because of (20.4) that W must coincide with the set of all points on the supporting rays of K(q,a) at x , and therefore in those spaces, in which we can speak of differentiability in the usual sense, our definition of differentiability is equivalent to the ordinary one. If K(q, a) is differentiable everywhere, we call it simply differentiable. As mentioned above, the following theorem then holds : (24.6) l’horern: A straight space of dimension greater than two i n which the spheres are convex and differentiable and the parallel axiom holds, is a Minkowski space with differentiable spheres.

To prepare the proof, consider any limit sphere K = K,(g+, 4). If a is a fixed point of K then a(a, b) is for all b # a on K transversal to the parallel L to 0 through a, see (24.6). The transversals to L at a are just the supporting lines of K ( p , a ) a t a, where p is any point of L different from a, compare (20.11). Therefore I< lies in the set W of all points on the supporting lines of K(P, a) at a. The limit sphere K decomposes the space, but no subset of K does, because a parallel to 0 intersects K in exactly one point and contains points on both sides of K. Since K ( p , a ) is differentiable a t a, it follows that Q(a,b) c W = K . Thus K contains with any two distinct points a, 6 the entire straight line g(a, a). Since K is closed it is, in our previous terminology, flat.

148


SPACESWITH

24.7

FLAT LIMIT SPHERES

The theorem (24.8) is therefore a consequence of the following: (24.7) l'hi?ormrb: A straight space of dimension greater than two is a Minkowski space with differentiable spheres, if and only if the limit spheres are Ilat.

The necessity follows immediately from (17.21). The sufficiency is more difficult. We decompose the rather long proof into several steps, (a) If the line L intersects K = K,(gf, p ) at a, but does not lie in I qq', because there is a point y on P n K ( q , 4') whose foot on g(q,q') falls on the same side of q as q'. Since the distance of y to a variable point of s(q, q') is peakless, ys > yq = qq', hence ss' = sM >/ ys > qq'. This con6 ) 3 A 3 s. tradicts the maximum property of q because S, 3

s(~,

26. Characterization OX the Minkowski plane I t is an open question whether Theorems (24.6) and (24.7) remain true when the restriction that the dimension be greater than two, is dropped. The difficulty derives, of course, from the complicated nature of Desargues' Theorem as compared to the existence of planes. But we will establish a theorem analogous to (24.6) under a stronger condition than the convexity of the circles, without assuming that the circles are differentiable. In these considerations some facts on convex curves are needed. The proofs are so similar to the usual proofs in euclidean geometry that we might leave them to the reader. However, these facts will occur again and again, and therefore it may be well to discuss them briefly once for all.

26


164

CONVEXCURVES

IN TWO-DIMENSIONAL

G-SPACES

As we saw previously in Section 13, every point of a two-dimensional G-space has a neighborhood D homeomorphic to Eawith the followingproperties : for any two points x , y the segment T ( x , y ) is unique, lies in D,and is contained in a unique segment whose endpoints lie on the boundary of D. The latter segment under omission of its endpoints we denote as the line G(x, y ) in D. If the space is straight we let D coincide with the space and G ( x , y ) = g ( x , y ) . A convex curve in D 1 is a component of the boundary of a proper convex subset of D. A convex curve is always the complete boundary of a convex subset of D. An arc is convex if it lies on a convex curve. A supporting line of a curve C is a line that contains at least one point of C, but does not separate any two points of C. The existence of a supporting line at a given point p of a convex curve C can be deduced from (20.3)but may also be seen as follows: Orient C, and let x approach p on C from the right. The line G ( p , x ) revolves (because of the convexity) monotonically about p , but does not pass through a position where it contains a point of C to the left of p. Hence G(p, x ) tends to a limit G, which is called the “right tangent” of C at p and is a supporting line of C. If G, is a supporting line of C at p,, and p, approachesp from the right, then limG, = G,. This follows from the fact that any line which can be h+m

approached by a subsequence of {G,} is a supporting line of C at p. If the right and left tangents of C at p coincide, then no other supporting line at p exists and we call C “differentiable” a t p and the unique supporting line a “tangent”. This definition applied to a convex circle K(q, p) of a twodimensional straight space agrees with the definition of the last section: any supporting line G of K(q, p) at one of its points p decomposes the space, and no subset of G does. Hence K(q,p) is differentiable a t p if, and only if, the supporting line at p is unique. The set of points where a convex curve is not differentiable is at most denumerable. The standard proof of this fact in E a uses parallels and is therefore applicable only with a modification. Let G, and GI be the right and left tangents of C at p. Let v be on the left, u on the right of p on C. If G, and G,, are supporting lines of C at u and v , then G,, ---c GI and G,, G, as u p and v p. Since G, and GI intersect at p , the lines G, and G, intersect when u and v are sufficiently close to p. About every point p we construct an arc f i for which C,,and G,,intersect at a point w. Since a countable number of such arcs cover C it suffices to +

-

-.

26.2

166


see that arcs of this type contain an at most denumerable number of points, where C is not differentiable. Let q1 and qa be two points on the arc where C is not differentiable, q1 to the right of q2. Traversing T(u,w ) from u towards w we meet in this order the intersections xl', xl", xz', xa" of T(u,w ) with the right tangent a t ql, the left tangent at ql, the right tangent at qa the left tangent a t qa. Therefore xl" # xl', xa" # xa', but x,' may coincide with xl". Thus T(xl', xl") and T ( x Z ' ,x2") have at most one endpoint in common and are non-degenerate. There can be only a countable number of such intervals on T ( u ,w ) . W

u Fig. 26.

We formulate our results as: (25.1) A convex curve in a convex domain of a two-dimensional G-space possesses at every point p right and left tangents. T h e y are supporting lines and coincide if and only if the supporting line at p i s unique. T h e set of points where they do not coincide, that is, where the curve is not differentiable, i s empty or countable.

Just as in euclidean geometry the existence of supporting lines is characteristic for convexity: (25.2) A closed Jordan curve C in D which has a supporting line at each point, i s convex.

The proof runs exactly as usual: If a segment T ( a , b) with a, b inside or on C existed, that contains a point c outside of C , let a' and b' be points on T ( c , a ) n C and T ( c ,b) n C respectively, and let p be a point inside C but

156

25.3


not on the line G(a, b). Then G(#, c) intersects C in a point q with (cqp) and in a point Y with (qfir). Obviously any line through q separates two of the three points a’, b’, I on C,contrary to our assumption. Finally we show : W (25.3) A convex arc A has finite

lertgth.

Since A is subset of a convex curve it can be covered by a finite number of arcs of the type constructed in the proof of (25.1). It suffices, therefore, to show that such an arc has finite length. If xo = v , xl,.. ., % ” _ I , x, = 24 follow each other from left to right on A , denote by yk the intersection of G ( X ~ -xk) ~ , with T(u,w ) . Then w ,yl, . . ., yy-l,u lie in this order on T(w,u) and

Fig. 27.

ow ’lY1

t-wy, 2 vy,

+ y1y8 2

+ YI-aYy--I2

~ , - 2 ~ ~ - 2

x,-1 y”-1+

= XOXl+ XlYl

XlY8

%’2

~ “ - 2x.-i

xv-2Yv-1

YI-124

+ ’2YQ

2 X,-1

+

xv-iY,-I

u = Xv-1 x,.

Ry addition of these relations we find vw

+ wu 2 xo

XI+

x1 x,

+ ... +

x,-1

x,.

+

Therefore the length of A cannot surpass vw wu. It is remarkable that an exact proof of (26.2) in EZ was already given by Archimedes, In fact, his proof is essentially the same as the present one. IMBEDDING OF STRAIGHT SPACES

We are going to establish our characterization of plane Minkowskian geometry by imbedding the plane in a three-dimensional space. We define the imbedding first, If R is a straight space, define a metric space R* with points x,, x E R, t real, by means of the distance

26.6

167

PERPENDICULARS A N D PARALLELS

Then R* is straight. The subsets t = const. of R* are isometric to R. This is the special case of (8.15)where R' is the real t-axis with the absolute value of the difference as distance. We add the observation; (25.5) If z(Lf) represents a straight line in R, z * ( t ) represents a straight line of R* which lies over z(Lf),and pz(Lf)"i s a convex function of Lf, then [p,z*(()]" is for a n y t,a convex function of t. For if z*(&) = zTi (ti), i = 1, 2, then

because the metric in the plane over z ( t ) is Minkowskian, compare Section 17. If 0 0 1 the convexity of pz(Lf)"and Ioi" yield

<
1 it suffices to consider the case a > 1. Denote the given space by R. Since Px(t)" is convex p x ( t ) is peakless hence the circles in R are convex, see (20.9).

168


26.6

Let a be a point on the circle K ( p , l), where this circle is differentiable, and let H be the tangent or unique supporting line at a. Let g+ be the orientation of g(a, p ) for which p follows a. Then H is the unique transversal to g at a and also supporting line of the limit circles K,(gf, a) and K,(g-, a). I t follows that H = K,(g+, a) = K,(g-, a). For, since K,(Q+, a ) decomposes the plane, it suffices to show that H 3 K,(g+, a). If b is any point of K , ( Q + ,a ) different from a, then g(a, b) is by (24.6) transversal to g, hence g(a, b) = H and b E H. \T

Fig. 28.

We now imbed R in the space R* constructed in (26.4,6) and identify R with the subset t = 0 of R* so that, for instance, a = a, and H C R*. Denote by *g+ the line in R* through a and PI, so oriented that p1 (that is, the point over p with distance 1 from p ) follows a and put K‘ = K,(*g*,u), K” = K,(*g-, a). We show first that

H c K’ n K”

= Kou(*g+,a) nK,(*g-, a).

To show H C K’ consider a sequence of points x:: on *g+ which tend in the positive direction to infinity. Then tv-.+ m. Since the points in R* over g form a Minkowski plane

P and #,a

= p p , = 1, we have

xvoa = xvo xi, = t,

26.6

159


and if b lies on the limit circle H = K m ( g + a) , in R 0

0.

In this, as in many other special cases, it is not convenient to put the motions in 5 in form of a sequence Zl, Z2,. .. If To= (0, 0), then {POW} is the set {xu, O}. Then becomes the set F o x < F x x , x # 0, where x = ( x , y ) , which is the strip - a12 < x < a/2. The motion Qi carries x = - a12

.

Fig. 30.

oriented in the sense of increasing y into x = a12 oriented in the sense of decreasing y. To obtain R we therefore have to "twist" the strip - u / 2 < x < a12 through 180 degrees and then paste the boundaries together, hence R is obviously a Moebius strip.

If instead of =

Po

we choose

qo= (0, b), b > 0

= (xu, (- 1)" b). The set ijoZ < qxZ,x

as starting point, then

# 0, then is the intersection

of the four open half planes containing ijo and bounded by the perpendicular bisectors of the segments T(qo,7-2),T ( q oq2), , - T(qo,q-J, T(qo,Q1),hence the boundary of H(%) consists of the rays r', r": x = - a, x = a, y 2 a2/4b

188

29.7


-

with origins U‘ = (- a, a214 b) and it’ = (a, ae/4b) and the segments 8‘,s” connecting T = (0, - a214 b) to u’ and The motion @%in 5 carries t’into ?’,these are therefore to be identified, To obtain and @ carries 8 I - l into TI’, and in particular u’ into 7 and 7 into a picture we first identify u’ with 7 and 7 with ’il” and then the two closed curves thus obtained with due regard to their orientation. The result must again be a Moebius strip but that is harder to visualize in the present case. This example also shows that the elements generating 5 according to part (i)in (29.4) depend on the choice of $1: For the first choice p1 = (0,O)=$,, the motions @ and 0 - 1 are the only elements obtained, for the second choice p 1 = (0, b) = Go we obtain @8 and @-a as well. In both cases we find that with any Ex also K1belongs to the EAdetermined by (i). This is, of course, a general feature of the procedure. ,‘I.

,”.

(29.7) The set of elements i n 5 determined by ( i ) i n (29.4) contains with a n y element its inverse. For if a point Zxwith FIZx = pxZx exists and EA= ql,then with Ti = X , q ’

&Za = & q

1

x,

=

pxq.,q

1

= $,Fa

hence SAis also one of the motions in (29.4 i).

30. Locally Minkowskian, hyperbolic or spherical spaces If every point p of a C-space H has a neighborhood S ( p , ufi),a,, > 0 which is isometric to a sphere S(p’, up) in a Minkowskian, hyperbolic or spherical space, then we call H locally Minkowskian, hyperbolic or spherical respectively. We did not postulate, because it follows, that S(p’,u,) may for all p be considered as lying in the same Minkowski space, the same hyperbolic space, or the same spherical space. For if p , q are any two points of K, then T(p,q) is contained in a finite number of spheres S(ri,u,), ri E T ( p ,q) such that two successive spheres overlap. The metric in the common part of two spheres determines the Minkowskian, spherical or hyperbolic metric in the entire union of the two spheres. SIMPLY CONNECTED SPACES

We know that the Minkowski spaces, the hyperbolic spaces and the spherical spaces (the latter only if the dimension is greater than one) are simply connected. The question arises whether they are the only simply connected spaces which are locally Minkowskian, hyperbolic or spherical. The following theorem, and (28.0) answer this question affirmatively.

I

i ~

30.2

189

COVERING SPACES

(30.1) Fheovem: The universal covering space of a locally Minkowskian, hyperbolic or spherical space (of dimension 2 2) is a MinkowskiaT8, hyperbolic or spherical space.

The proof is essentially the same in all three cases; the spherical case offers a slight additional difficulty. We treat here only the latter, since a proof for the other two cases, in addition to being simpler, is also contained in the construction of the universal covering space of a space with non-positive curvature given in Section 38. Let the sphere S ( p ,up),up> 0 of the G-space R be isometric to a sphere in a spherical space R with radius /?(or length of great circles 2 n 0). I t is clear that any sphere S(q,p) contained in S ( p , up) is then also isometric to a sphere By a standard argument, compare (7.5), we see that if a($) is the in least upper bound of those upfor which S ( p ,up)is isometric to a sphere inE, then a($) satisfies la(#)- u(q)l pq. If u(p) > np for some p , then we have nothing to prove. Choose an arbitrary point p in E and an arbitrary point p in R, denote by rR an isometry of S(3,a ( $ ) )on S ( p , a ( # ) ) . If a, = u ( p ) / 2 ,then Q induces an isometry 7j q of K ( 7 ,a), on K = K ( q ,uo).Denote by X,(t), x , ( t ) , t 0, the representations of the half geodesics in R and R for which ;,(O) = 3, x,(uo) = q, x,(uo) = q. Then X,(t)Q = x , ( t ) for 0 t a @ ) . We extend R to the association Tq(t) x,(t) for t 2 0.

x.

v,@. Put u = 1/3 min u(x) for fix p’. Then u > 0 because u ( x ) is continuous. I t follows from (8.11, 12) that there is an E > 0 such that qr < E , q, r E K , implies x,(t’)”(t’’) < u and X,(t’)”(t‘’) < u for 0 T’, t” p. Then

1. Now let L,,, be a flat of maximal dimension m through TI contained in F(&), (of course, m may be zero). Then the preceding argument applied to all rays T with origin p1 in L,,, shows that all j5,,must lie in the ( n - m)-flat ,L through TI perpendicular to L,. The totality of the rays with origin in L*-,,,nF(T,) is a convex cone C. The above argument shows that all FX,x 2 2 , must lie in the closed complementary cone C' to C. The intersection C ' n F(&) is bounded because C contains all rays with origin in F ( p l ) nL,, hence C' none. If C'n F(j5,) c S(&, p ) , then all F,, for which a point ZXwith ZXTX= X,,

F1

v1

v1

80.6

193

COVERING SPACES

exists must lie in S($l, 2 p ) , and there is only a finite number of such Fx.I t follows from (29.4i) that 8 can be generated by a finite number of its elements, which means, by definition, that 8 has finite connectivity. A general property of all spaces with non-positive curvature is that their fundamental groups contain no finite sub-groups, except the group formed by identity (see 39.3). Therefore the fundamental groups of locally euclidean or hyperbolic spaces have no finite sub-groups. It was already mentioned that there are too many locally euclidean or hyperbolic spaces to attempt an enumeration. We will however briefly discuss the two-dimensional locally euclidean spaces :

LOCALLY EUCLIDEAN

SURFACES

The only motions of E 2 without fixed points are the translations and the translations followed by a reflection in a line parallel to the direction of the translation *. In case the fundamental group 8 consists of the powers @ : x’ = x ka, = (0, 0) the y’ = y of the translation @ : x’ = x + a, y’ = y , then for fundamental set H ( 3 , ) is the strip - a/2 < x < a/2,and the surface with 5 as fundamental group is a cylinder.

+

+

If 8 consists of the powers@ : x’ = x -1ka, y’ = (- 1)’ y of @: x’ == x a, a > 0,y’ = - y we obtain the Moebius strip R already discussed in the last section. The group 8 contains as sub-group 6 the group of translations @% : x’ = x 2 ka, y‘ = y . The surface R’ belonging to 0 as fundamental group is a cylinder, and R’ is by (29.3)a two-sheeted8 covering space of R. We discuss briefly the geodesics on the Moebius strip R : If F(3,) is constructed for $, = (0,O) as in the preceding section, then its boundary is formed by the lines x = f a/2 which are to be identified such that the point (- a/2,y) goes into (a/2, - y). The segment - a12 x a/2,y = 0,yields on the surface a great circle x 4 2 , y = b and of length a . If b > 0 then the two segments - a/2 - a12 x a/2,y = - b form together on R a simple closed geodesic of length 2a. This is not a great circle, when 0 < b < a/2, because the arcs of the geodesic connecting (0, b) to (0, - b) have length a, and cannot be segments, because the segment in E 2 from (0, b) to (0, - b) yields in R ageodesic connection of length 2 b < a. For b 2 a/2 the geodesics are great circles. The lines x = c yield simple geodesics on R. The parts y 2 0 and y 0 yield rays, but the whole curve x = c is not a straight line, for, as we just

+

<
0,

b>O

in suitable affine coordinates x , y in E4, or the points Fk form a lattice. If the x-axis and y-axis are perpendicular, then F($,) becomes a rectangle and the surface H is clearly a torus. If the x-axis and y-axis are not perpendicular, then F(&) will be a hexagon and it requires a little discussion to see that R

30.6

196

COVERING SPACES

is a torus. However, we may use (30.4) and remetrize the plane with another euclidean metric in which the x-axis and y-axis are perpendicular, without changing the geodesics. Let = (0,O). Then F(&) is the rectangle - a / 2 x a / t , - b/2 y b/2. The lines x = const. and y = const.

<
- 0. We retrace the steps of the t )X ( t ) ! 2 and L the lineal first part of this proof: let g be the geodesic ~ ( = element represented by X ( T ) for It1 u u(x(0))/2. Since X(a)= X ( 0 )Ek the point %(a)coincides with x ( 0 ) . Denote by L" the lineal element given by ~ ( tfor ) I t - a1 u and by Z'' the segment over L" with center x ( a ) ,finally

<
i 1. The intersection bk of ro with T(pk,f k ) , h .: 1, goes under @ into the intersection bk@of rl with T(p,+1, fk+ 1)' and (bk@b k + l f k + 1) because rl lies on the left of ro. Moreover,

+

bkfk = b k @ f k @

= bk@fk+i>

bkilfk+i,

so that the sequence {bkfk} has a limit, and bk@fk+, has the same limit. I t follows that bk@bk+ +o, which Contradicts (22.23). X first corollary is: (32.13) I f a translation @ has two different axes fil and f i r , then any point between g1 and g2 lies on an axis. All axes of @ are parallel and any parallel to an axis is an axis. m

For if

p is any point between

g1 and g2 then u T(p@',PGi+l) bounds i=

-m

by (32.12) a convex set together with g1 as well as g2, hence must be a straight line, which is by (32.3) an axis. That any two axes are parallel follows from (32.4). The limit line of axes is also an axis. Unless the axes of @ cover the plane, there is a point q not on an axis, and an axis b closest to q. That q does not lie on a parallel to fi follows then from: (32.14) The asymptotes through a point q not on an axis of @ to the two orientations of the axis b of @ closest to q are distinct and have distance 0 from 6.

Let a be the asymptote through q to the oriented axis bf of @. The line a@-' is also an asymptote to $+ and lies for positive v between a and b (since q does not lie on an axis). Therefore a@-" converges for v a, to a line f. This line is invariant under @ and is therefore an axis of @. By construction f either separates b from q or coincides with 0. The first case is impossible since b was the axis of @ closest to g. Thus f = b. But a @ - " 4 t = b and a@-" b =: aI) implies ab = 0. -+

It follows more precisely, that there are representations a ( t ) of a+ and of b+ such that

y(t)

(32.15)

lim u(t)Y ( T ) = 0. 1-+m

33.16


214

Since I)-l is an axis of @-l we obtain the same result for the asymptote b through q to I)-. The lines a and b cannot coincide because aG-1 lies between a and q, whereas b lies between b@-' and $. The following theorem constitutes a very interesting addition to the theory of parallels developed in Section 23, because it shows that the existence of a translation along g excludes for the asymptotes to g many of the strange phenomena occurring in perfectly general straight planes. (32.16) Thewrem. The oriented asymfitotes to an oriented axis of a translation @ are all asymptotes to each other. dm(rl, ra) is finite for positive subrays rl, r, of any two of these asymptotes and doo(rl, ra) = 0 if # has only one axis.

There are three cases to consider. (1) The axes of @ cover the plane. (2) The axes of @ form a closed half plane. Let this half plane be bounded by the oriented axis I)+ of @ and denote the remaining open half plane by P. (3) The axes of @ form a closed strip S bounded by two parallel lines. Then let P and Q be the open half planes remaining after removing S. (3) includes as special case the possibility (Y), that S consists of only one line, which is then the unique axis of @. Case (1) is settled by (32.13). That dm(rl, ra) is finite for two positive subrays r,, r2 of two oriented axes g t and g i of @ is obvious: If xi(t) represents (32.17)

a+ then x , ( t ) x , ( r ) is periodic with period

x,(t)x,(t)

< p = max x , ( t ) x , ( t ) 0

for all

y(@), hence

t.

< < Y (@)

In case (2) let a and b be two distinct oriented asymptotes to I)', and let a+ be between 'b and.')I If ra, r,, rh are positive subrays of a', b+, $+ respectively, then d,(r,, rh) = 0 implies dm(ra,rb) = 0 hence the lines a', b+, I)+ are asymptotes to each other. It also follows from (22.22) that each oriented parallel gf to ')I is an asymptote to b+ and conversely. The finiteness of dm(r,,, rg) for positive subrays of b+ and g' follows from (32.17), applied to g+ and .I)', and dm(rb,rk) -. 0. In case (3) application of (22.22) shows immediately that all asymptotes to an oriented axis of @ are asymptotes to each other, and (32.16) and (32.17) yield the rest, also the additional statement that in case (3') any two asymptotes to the single axis have distance 0.

33.1

215

COVERING SPACES

33. Metries without conjiiga6e points on tho torus We consider a two-dimensional torus K whose universal covering space is a straight plane P. Why this case is particularly interesting is explained in the introduction, Section 26, to this chapter. We add here the following remarks: Morse and Hedlund [2] had proved the result of E. Hopf on the torus under the stronger hypothesis that there are no focal points, which means in our language (see Section 25) that the circles in P are convex. This result is contained in our theorems (35.15) and (33.1). The question whether Morse and Hedlund’s theorem can be extended to Finsler spaces by establishing that the metric is Minkowskian leads to the problem which we left unsolved in Section 25. Since we operate exclusively in P we denote its points by p , q,. . . (instead of F , q , . . .). The motions of the fundamental group of K may be assumed to be given by the translations

Y(m,n) : x’

=x

3 m,

y’ = y

+

71,

in properly chosen Cartesian coordinates in P. contained in the proof of Theorem (32.5):

m , n integers

The following facts are

Each Cartesian translation !P = Y(m, n ) # B is also a translation of P as straight plane. Every point of P lies on an axis of Y and PpY is constant. All axes of Y a r e parallel to each other. If a line L in P contains q and q Y ( m , n) it contains all points q!P(m,vn), v = f 1, f 2, . . . . A very important and not at all trivial property of the geodesics in P is: (33.1) Theorem. If the universal covering plaiie P of a torus is straight, then P satisfies the parallel axiom.

We assume again that the translations Y ( m ,n) are the elements of the fundamental group. I t will prove suggestive to call the line L rational if it contains two points of the form q and qY(m, n),( m , n) # (0,O). The rational lines are the axes of the various motions (# E ) of the fundamental group. For any point a not on L the axis of !P(m,n) through a is by (32.5) the only geodesic in P through a that does not intersect L. Thus the parallel axiom holds for all rational lines.

<
xo and i 2 1; hence the slope of L is at least p, and at most po. Since po was arbitrary, L has slope p. Statements (3) and (4) show that the parallel axiom follows from: (6) There is at most one line with a given slope ,u through a given point p l. For rational ,u this follows from the fact that the parallel axiom holds for rational lines. For if p = nlm and if the line L containing the points pY(vm,vn) has the equation y = f ( x ) , then any other line L' through p = (xo, yo) has an equation y = f ' ( x ) , with f ' ( x ) > f ( x ) for x > xo say. Because L is parallel to L'Y(0, l), which has the equation y = f ( x ) 1, the line L' must intersect L'Y(0, 1) for some x ' > x,. Then for a suitable v > 0 the point pY(vm, vn) Y(0,1) lies on L'Y(0, 1) and between L' and L. The line L" through p and pY(vm,vn 1) has slop:

+

vn

+ 1 >p

vm

+

and the slope of L' cannot be smaller than the slope of L".

Now, let p be irrational, and assume for an indirect proof that there are two different lines L , K through p with slope p. We may assume that p is the origin and that the two lines have equations of the form

=I(%), K : y = g(x) with g ( x ) >I(%) for x > 0. integral n > 0: L: y

Then for

0 < g(n)-fW

(6)

0 exists such that

g(n)-f(n) > 6

for

n 2 1.

For a given integral x 2 3 determine the integer m,, by (7)

Then

m,B 2 x

+ 1 > (m,,- 1) 6.

2 Si contains x -t1 points pi which represent the same point on the

i= 1

torus, or whose ordinates differ by integers. We distinguish two cases:

218

33.1


(a) There are for some x four points pi no three of which lie on the same geodesic. A familiar argument from elliptic functions shows that the convex closure of these 4 points in terms of P would then contain a "period parallelogram" Q whose sides are formed by segments of P as straight plane. Since the domain bounded by y = f ( x ) and y = g ( x ) for x 2 0 is convex, Q would lie in this domain; on the other hand Q would contain a point equivalent n) which was already seen to be impossible. to p . that is, of the form (m, (b) For every x at least x of the x 1 points pi lie on a geodesic H".Then H" is rational and has a rational slope px. Since no two of the x points lie on the same Sjthe abscissas n: of the x points are different. Let ny < n: + l. Then n: - n; x - 1, hence because of (7), n;/nE 1 - ( x - l)/m, 1 - 8/4.

+

(n, 0 independent of x , such that Ip,, -,uI 2 E. But if y = h(x) represents H",since the x points lie between L and K,

33.3

219

COVERING SPACES

which in conjunction with (8) contradicts Ipx -pl proof of (33.1).

2 E. This completes the

I t implies for the torus:

I n a metrization of the torus without conjugate points the closed geodesics are dense among ull geodesics. More precisely: If p(r) represents an arbitrary geodesic and N > 0 and E > 0 are given, then a representation q(r)of a closed geodesic exists such that p ( r ) q ( r )< E for It1 N . (33.2)

0 and denote by L, the curve in S containing the points pY(wn,vn). For any p , q the lines L, and L, are either parallel or identical. Let y = f p ( x ) represent L, (since n = 0 is admitted, f,(x) may be constant). That Y ( m n) ,

33.3


220

+

carries L, into itself implies f p ( x m) = f,(x) is periodic with period m and the area x*

d w . n ( p , 4)

+ n for all x , therefore/,(%) -I,(%)

f."

=J lip

-1,

(%)I

4

+

of the "parallelogram" Q bounded by L,, L, and x = xo, x = xo m is independent of 5. An arbitrary translation Y = !P(m', rr') carries Q into a parallelogram which has the same relation to P Y and q Y as Q has to and q. But Y leaves area invariant, hence #J

dm, n ( p , 4) = dm, n (pull, q p ) .

(9) Clearly

am, ( P , q) = (tm,,n (q, P ) and2 dm,n(p,9) = 0 i f and on.lv if L, = L,

(10)

The arbitrariness of xo yields

+

(11) dm,"(p,q) d,,,(q, I ) = d,,,(p, r) if and only if the line L, lies in the closed strip bounded by L, and L,; (12) d,,,(P, q)

+ d,,.(q,

r ) > d,,,,(p, r) if L,does not lie i n this strip.

Let 6 be the difference of the ordinates of p and q and determine the integer k by k - 1 161 < k. Then Y(0,f k) carries L, into a line L, for which L, (if different from Lp) lies between L, and L,. Then

0 for p # q follows from dm,,(p,q) = d,,,.(q, p ) and from (10) because for all pairs m,n. with the exception of at most one, the lines L, and L, (in the previous notation) will be different.

+

33.4

22 1

COVERING SPACES

Thus pq satisfies the axioms for a metric space. To see that the curves in S are the geodesics it mtist be shown: that for three different points p , q, r

+ qr -- pr if q lies on the segment u of the curve of p q + qr > p r if q does not lie on a.

(15) p q

(16)

S through p and

7.

If q lies on u, then for any m,n the line L, will either contain p and r or L , lies between Lp and L,. Hence it follows from (10) and (11) that (15) holds.

If finally q does not lie on a, let L be the curve in S through two arbitrary interior points q‘ and q” of the segments (in the sense of S) from q to p and r respectively. Then L separates u from q. If L contains for suitable m, n with m > 0 the points q’Y(vm,vn), then (16) follows from (12). If L does not have this property (i. e., is either a line x = const. or not rational) then the parallel axiom implies the existence of m > 0 and n such that the line L‘ containing the points q‘Y(vm,vn) is so close to L that it also separates u, and therefore p and r, from q. Then (16) follows again from (12). That the distance p q is equivalent to the euclidean distance is easily derived from either the analytic definition of p q or the geometric properties of S. The finite compactness of pq follows from its invariance under Y(m’, n’). EXAMPLES The construction of the distance pq in the preceding section happens to yield a Minkowski metric if the curves in S are the euclidean lines ax + by + c = 0. This is, however, accidental because other functions d,,,” ( p , q ) than the area could have been used. For instance, if p , = (xi, y i ) then (33.4) p1p2 = [ ( x , - x212

+ ( y l - y2)2~1’2+ )7y1+ sin 2 n y l - 7 y 2 - sin 2ny21

yields a metric for which the euclidean lines are the geodesics, because 7y sin 2 n y increases monotonically. Moreover, this metric is invariant under the Y ( m ,n). Instead of 7 y sin 2 x y many other functions could have been used, a similarly formed term in the .r, could have been added, the euclidean distance occurring in the definition of plp2 could have been replaced by an arbitrary Minkowski distance. Hence there is so much choice that the problem of determining all metrics which belong to a given system of curves becomes uninteresting. One might ask whether conditions I, I1 and I11 do not imply either IV or V. The examples under (23.6)show that this is not the cases.

-+

+

222

33.4


Finally we give an example which confirms the assertion of the introduction to this chapter that the curves of a system S satisfying conditions I to V need not be Desarguesian. To construct such a system S, we first define certain functions f , ( x ) , t >, 1 in the interval 0 x 1. Put

<
0 for

Then f,(1) = n and f:tl(x) - / : ( x ) increases. Moreover, put h ( x ) = (n

x

+ 1- t ) f n ( x )+ ( t - n)fntl(x)

# &,so that f,,+l(x)-f,,(x) if

n max (uTo,w To) or uTo = vTo> wTo. In the first case let x be the last point on ~ ( uw, ) at which %'To reaches its maximum for X I E 5(u, w ) . Then xTo 2 vTo, hence x # u, up. Choose xl, x2 on 0(u, w ) such that (xlxxa) and x,x,< E . Then xlTo < zTo, XaTo< xTo which contradicts (b). This implies for the second case that %To= uTo is the maximum oi x'To for E 5(u, v ) . If x is chosen as before we obtain the same contradiction. Applying this to the case where To degenerates into a point p we find that p x is a peakless function of u x for x E Q. But (36.18) states that p x is strictly peakless. To see this observe that the assumptions of (36.17) are satisfied in S ( p , q ( p ) / 2 ) . It follows from (20.5) that K ( p , u ) is strictly convex for sufficiently small positive u. If p x were not strictly peakless it would reach its minimum on a nondegenerate subinterval of Q. If T(x,, xa) lies on this interval and 0 < xlxa < E , then fixl = px2. Increasing the number n of the last proof, if necessary, we reach that pzy-' = u with strictly convex K(fi,0). If x"-' is the foot p on T( 0 such that (u, t)-+ q is topological for uuo< E and It - tol< E. For this purpose let a > p ( # ) / 2 be given. Put

P,

(a)

=

1 inf p ( x ) , where

x E S ( p ,2 a). Then

flu > 0.

Because of (8.12) x(u, t) is continuous in both variables u, t for and 0 t 2 a. Therefore an E, > 0 exists such that

2c

<
,50,) 2 k \< 8, < 5 < 2 k Vk(6)--*

00

for 6 + 6,,(p(2k

+ pk)

=:

V'(2 k

+ pkt

~ ~ ( convex, 6 )

+ ph) = .. . =V(m)(2k f pk) = 0.

On this surface (t,-1) represents an oriented geodesic 9'. The ordinary straight line through p and (th, - I), k = 1, 2.. ., is an oriented straight line I$ on S as G-space and 2k-1 t t, I - 4k2-4 k 2 1/4ka-4k 2

(v

+

+

1

is a representation of Qt. For k + 00 this representation tends to the representation (r,1) of the oriented geodesic I)+. I t follows that the ray t 0 of $+ is a co-ray to the ray t 0 of ~f and also conversely because the surface S has 7 = 0 as plane of symmetry. Nevertheless, neither of the two geodesics and g' is asymptotic to the other of any type.

>

>

COMPACTSURFACES

WITH NEGATIVE CURVATURE

For geodesics on compact orientable surfaces with negative curvature we have the following theorem implied by (34.11), (34.14), (34.15), (37.1), (37.2) and (39.14).

41.2

THE INFLUENCE OF THE SIGN OF THE CURVATURE ON THE GEODESICS

267

(40.6) l'heore9bz. O n a compact orientable surface with negative curvature the closed geodesics are dense among all geodesics in the following sense: Given a finite number of geodesic curves z , ( t ) , 0 t A, i = i, .. ., m, then a closed geodesic y ( t ) and numbers a, exist such that y ( z )z,(t - ai) < E for aj t ai + Aiand a l l i = 1,2, . . ., m. There are ale/ open geodesics y ( t ) with the property: Given a n y geodesic curve z(t), 0 t 1, a n y E > 0 and a n y N > 0, there exists a n a = = a (z, E , N ) > N szcch that y ( t ) z ( t - a) < E for a t a 1.

<
0 and the segments T(a,b) and T(b,c) are unique. Denote by b, and b, the directions with origin b which have proper common subsegments with T+(b,a) and T+(b,c) respectively. We define the angle abc by L abc = (ba, bc)''ex if b, and b, are not opposite, and put I abcl = labcl. If be and b, are opposite we put label = n. Three distinct points a, b, c in a neighborhood S ( p ,p ( p ) / 8 ) determine a triangle or simplex abc with vertices a, b, c, sides T(a,b), T(b,c ) , T ( c , a ) , whose interior is the simply connected set bounded in S ( p , p(p)/2) (which is homeomorphic to the plane) by T(a,b) u T(b,c) u T (c, a ) , provided the vertices do not lie on one segment. If the latter is the case we call the triangle degenerate. If abc is not degenerate a direction T f ( b ,x ) in L abc from b intersects T(a,c ) , similarly for the other vertices a , c. We define the "excess" of any triangle abc by

& ( a h )= label

+ lbcal + lcabl -n.

Degenerate triangles have excess 0. (42.8) If the angular measure is continuous and a,, b,, c, are distinct and tend to the same point p , then E(U,~,C,) + 0.

For a proof we may assume that the directions from a, through b,, from 6, through cy, and from c, through a, converge. If two of these directions tend to opposite directions, (42.8) follows from the remark on degenerate triangles. Otherwise the angle L c,a,b, and the vertical angles to L a,b,c, and L b,c,a, tend to three angles whose union is straight, Most important, although trivial, is the additivity of excess: (42.9) I f the triangle abc is simplicially decomposed into 4 (finite) number of triangles a+b,c,, v = 1, .. ., n, then m

&(@bC)=

2&(U,b"C,). v=l

42.11


279

We say that an angular measure on a G-surface R has positive excess if every non-degenerate triangle in R has positive excess. In the same way we define angular measures with negative, non-positive, non-negative and zero excess. SIGN

OF THE EXCESS AND PARALLELS

In non-euclidean geometry the sign of the excess is closely related to the theory of parallels. We are going to investigate briefly whether there are such relations in the case of general angular metrics. Let P be a two-dimensional straight G-space, hence homeomorphic to a plane. We begin with a trivial remark: (42.10) If P satisfies the fiarallel axiom then it fiossesses continuous angular

measures with zero excess. For we only have to choose an angular measure a t one point p according to (42.5) such that f ( s ) is a continuous function of s. At any other point q we determine the metric by putting the measure of an angle A equal to that of the angle at p whose directions are parallel to those in A . The converse is not true without further assumptions on the angular metric: If in the Klein Model of hyperbolic geometry, i. e., a Hilbert geometry (Section 18) inside an ellipse, the euclidean angular measure is used instead of the hyperbolic measure, the excess will be zero although the hyperbolic parallel axiom holds. Nor is it true, that the parallel axiom and constant sign of the excess imply that the excess vanishes: A hemisphere H' with diametrically opposite points on the bounding great circle identified is a well known model of the projective plane with the arcs of the great circles on H' as straight lines. If we leave the bounding great circle out, we obtain the projective plane without the line a t infinity, hence the euclidean plane H . If we use the spherical angles in H instead of the euclidean angles, we obtain an angular metric with positive excess. However, negative excess and the parallel axiom are incompatible : (42.11) If the excess is non-positive and the angular metric is continuous, then the parallel axiom irnfilies zero excess.

'-

Let abc be a non-degenerate triangle. If (abx)and x traverses g = g f ( a ,b) then lbacl lacx( + (cxal < n ; g + ( c , x ) tends to the asymptote bf to 0' through c. If y follows c on b+, then IacxI -+ lacy1 hence

+

lhacl

+ lacy1 0 A(vj

+ h) + S(E) + h) 2 A(vj) + S(E) (jlj

vi,

hence S(E) h 2 A(vj) - A(vj

(44.3)

+ h)

h > 0.

for

If Rj(vj)is not a closed geodesic and its angles in Tj(vj)are not concave, then Rj(vj)has a single vertex p with a convex angle in Tj(vi).Choose points ai, a, on the legs of this angle such that they lie on Rj(v,)and a, p = a, P < w(P, E ) . Consider the curve R‘originating from Rj(vj)through replacing T ( a l ,P) u T ( p ,a2) by T(a,, a,). The distance of R’ from Pi is at most vj a, P. Putting h = a1 and denoting the length of R’ by I’ we see that A(vi h) A’, hence

+ +
1 - 41a,/(a1 P + P a,). Because the angular measure is uniform at IC we have lalpa,l > n - e . We saw in the beginning of this section that (44.2) implies the following addition to (43.16): (44.4) A sphere with three holes does not possess an angular measure which i s uniform at x and has zero excess.

SIMPLE GEODESIC MONOGONS The metrizations of the plane which have interested us hitherto were principally those without conjugate points. If a G-plane is not straight, then questions like the existence of geodesics without multiple points, of simple monogons, etc. arise. I t will be useful to have a simple example ready for the theorems proved below.

2zy>2Zp-2py>zt-t.

If R, does not pass through (44.12)

?(t.

+ a)

~(f), then

for

II

+ 37
0.

Let h > 0 be less than min o ( q i ( s ) ,E ) and also such that the point bi of P, with q,(s) b, = h lies on the side of P, with endpoint qi(s). If ni = q,(s h), then lai q,(s) bil = n - 7: and a, q,(s) = bi qi(s) = h. The polygon P,,, is at most as long as the polygon originating from P, through replacing T(bi,q i ( s ) ) by T(b,,a,). Therefore

+

A(s

+ h) Q A(s) +

albl-

h 4-a, b, - h.

This yields together with (44.20) that

n - E ,

which proves (44.18) and the theorem because D s 3 D ' 3 M.

304

44.21

GEOMETRY O F GEODESICS

SURFACES WITH

POSITIVE EXCESS

We saw in (43.11)that a non-compact G-surface with positive excess is homeomorphic to the plane. Since E ( R )exists, see (44.13), the preceding considerations contain many interesting results on these surfaces if their angular measure is, in addition, uniform at n. We formulate them in one theorem.

(44.21) T/u?m*m~ Let R be a C-plane with non-negative, but not identically vanishing, excess and such that the angular measure i s uniform at 7c. Then a) E ( R ) 0 exists such that every point q with qN > p is vertex of a simple monogon containing N in its interior. g) R possesses no straight line and no closed geodesic with multiple points. h) If x ( t ) represents a geodesic Q with multiple points, then Q contains exactly one simple monogon m. If m i s given by x ( t ) for a t p. %(a)= x ( B ) , then each of the half geodesics represented by x(t) for t a and t /3 lies except lor T = a or T = p outside of the compact domain D bounded by m and does not intersect itself.

< <
t” 2 /I, but does not exclude ~ ( t= ‘ )%(TI’), for t‘< a < p < T”. Clearly (44.15) implies a). The excess of a compact domain bounded by a simple monogon is by (44.6)greater thann, hence two such domains must have common interior points. c) follows from (44.16). For the compact domain D bounded by a simple closed geodesic (44.6) yields E ( D )= 2n, hence E ( R )= 2n and the excess vanishes outside of D. This proves d). The surface F,, constructed on page 294 shows that a simple closed geodesic may exist, if the excess vanishes outside of a bounded set. To see f) we enclose N in a simple closed polygon which bounds a compact domain D’ with E(D‘)= n 37, 7 > 0. Such D’ exist because there are

+

44.21


305

by (44.10) domains D, bounded by a simple closed polygon and containing S ( N . Y). By the definition of total excess lim E(DJ= E ( R )> R. Hence D, may serve as D' for large Y . If D is any domain bounded by a simple closed polygon and containing D' then E(D)2 E(D')because excess is non-negative. Applying (44.9) to D as M yields a f3 such that every point q with q D 2 f3 is vertex of a simple monogon containing D in its interior. If D c S ( N , 0,) then p = 0 8, will satisfy f). A straight line Q on R and each of the half planes bounded by Q would satisfy the hypothesis of (44.17), but at least one of the half planes would have positive total excess, although Q has no proper vertices, which contradicts (44.17). The second part of g) is contained in h). h) is interesting because it furnishes far reaching information on the shape of the geodesics. Its proof is simple: g contains at least one simple monogon m which is not a closed geodesic. With the notation as under 11) the angle of m in D at %(a)is convex by (44.5) and b). Therefore the point ~ ( tlies ) outside of D for small t - p > 0. If there were a t > p with ~ ( tE)D , then a first such t, say towould exist. I t is impossible that ~(t')= x ( t " ) for p t' < t" < tobecause if t" is the first t following t' with ~ ( t "=) x(t'), then ~ ( trepresents ) for t' t t" a monogon whose interior has no common point with D in contradiction to b). = ~ ( t , then ), ~ ( trepresents ) for If tl is determined by a < tl p and %(to) tl t toa simple monogon m,. The compact domain D, bounded by m, must by b) have a common point with D, and hence contain the points ~ ( tfor ) a t Q tl. But then the angle of m, at ~(t,)in D, is concave which contradicts (44.5)and b). The fact that ~ ( tdoes ) not lie in D for t > fl precludes x ( T ' ) = x(t") for p t' < t", since this would again lead to a simple monogon whose interior has no common point with D.

+

k , where k < m . Then Qk-1

fiG@;-1

=@k

@k+l

@,

R,@;

@; =@k = R,.

@;+I

RG@;

= Ok

@I =

@k+l

Therefore

R,

.~

= @ , C D ~..

,

I

. . . @;

, ~ G ~ , @ , ~ l

in

S(z,a,).

Now @, maps Ha isometrically on Hi and aZleaves H t pointwise fixed, @& maps H i isometrically on H:, and so forth. Altogether, @,.. @ . , maps Ha isometrically on H:, hence H isometrically on a subset H , of Hi, and H, c H k c Do because H , c S(a, 2 a*) c S(z, 6 a,)n S(a, 6 a,) by (1). Then R, maps H , isometrically on a subset HL of and @; . maps isometrically on an open subsegment of Hf. This proves (k). We call R, the reflection of S ( z , a,) in G.

Hi,

..

Hi

320


47

47. Spaces with flat bisectors I1

From the fact that & is an isometry we will now deduce the euclidean, hyperbolic, or spherical character of the metric in S(z, aJ4) by establishing the trigonometric formulae for the right triangle. We first discuss congruenc? of triangles, then obtain the distinction between the three cases, and finally produce the formulae. ANGULARMEASURE,

CONGRUENT TRIANGLES

With the previous notations let a, b be any two distinct points on the circle K ( z , 6) where 0 < 6 < a,, and A one of the two arcs of K ( z , 6) from a to b. If c is an interior point of A , and d,, d, on A are such that the arcs of A from a to d, and from d, to c as well as the arcs from c to d, and from d, to b have equal lengths, then the reflection of S(z, a,) in L(z, d , ) carries K(z,6) into itself and a into c. The reflection in L(z, d,) carries c into b. The product of the two reflections is an orientation preserving isometry of S(z, a,) on itself, hence a rotation of S(z, a,) about z according to our definition in Section 42. We know from (42.7) that there is exactly one angular measure for the angles with vertex z, which is invariant under the rotations, and that this measure is proportional to the length of the arc on K ( z , 6). (There is no restriction on the size of 6 here since we know K ( z , 6) to be convex.) We use this angzclar measure henceforth. If A is a semicircle, i. e., (azb), and G is the perpendicular to L(a, b) at z , then & interchanges the two arcs of A determined by its intersection with C. These therefore have equal lengths, and it follows that perpendicular lines form angles of measure 4 2 , which we call, of course, right angles. A reflection of S(E,a,) about a diameter which carries a into a‘ carries a circle K ( a , E ) with 0 < E < a, - az into the circle K(a’, E ) , and since the reflection preserves length of curves, an angle with vertex a will have the same measure as the angle with vertex a’ corresponding to it under the reflection. Thus reflections and products of reflections, preserve angular measure. Put E = S(z,4 4 ) and let abc, a’b‘c‘ be two triangles in E for which either ab = a’b’, bc = b’c’ and ca = c‘a‘ or ab = a’b’, bc = b’c‘ and label = la‘b’c‘l. If m = ma, then mz < a,/4 and from (46.3)

E c S(m,a1/4

+ mz)C S(m, a,/2) C S (m,a,,,2-1 (1- $l).

Therefore the reflection Y of S ( m , a,,,) in the diameter perpendicular to the diameter carrying a and a’ is defined in E and sends a into a‘. Put b” = b Y, c” = c Y. Since ab < a1/2, ac < a,/2, it follows that the points b’, c’, b“, c”

4i.1

HOMOGENEOUS SPACES

321

lie in S(a', a,/2) C S(a', a,! 2-' (1 - & ) - I ) C S(a', a,,). Because of a'b' = a'"' and a'c' = a'c", both a reflection of S(a', a,,) in adiameter and a rotation of S(a', a,,) about a' exist which carry b" into b', and one of these will carry c" into c'. The triangles abc and a'b'c' are therefore congruent in the sense that all corresponding parts are equal. For instance, if (adc)and (a'd'c') with ac = a'c' then bd = b'd' etc. It also follows exactly as in elementary geometry that the two triangles abc and a'b'c' are congruent if bc = b'c' and labc) = Ja'b'c'J, lacbl = la'c'b'l. Speaking of congruent triangles instead of motions has the advantage that we only use the triangles, hence avoid discussing the range of a motion with the concomitant, cumbersome use of the triangle inequality. We may continue using motions whenever convenient, without troubling about this question since we can always reformulate the procedure in terms of congruence.

LAMRERT QUADRILATERALS A convex quadrangle abcd with three right angles is traditionally called a Lambert quadrilateral, because such quadrangles played an important role in Lambert's investigations on, what is now called, non-euclidean geometry. We imply in the notation that the angles a t a, b, and c are right angles and that d is opposite to b. All our figures will lie in E . We show first

(47.1) If abcd is a Lambert qzcadrilateral, then

JadcJ bc (or cd > ab) ladcl =n/2 if and only if ad = bc (or cd = ab) ladcl> n/2 if and only if ad < bc (or cd < ab) Let the perpendicular to L(a, b) a t m = ma* meet T(c,d ) in n. The reflection in L(m, n) maps c on the foot c' of n on L(a, d ) . (The point c' might lie outside E , but this does not matter because it lies in S ( z , a,).) Obviously c' = d or ad = bc if and only if ladcl = 4 2 . If (ac'd) then bc < ad and ladcl < 4 2 , otherwise the perpendicular to L ( a , d ) a t d would intersect T(c',n). If (adc') then ad < bc and ladcl > 742, otherwise the perpendicular to L(a, d ) a t d would again intersect T(c' n ) ; see Figure 63. We call a Lambert quadrilateral abcd rectangular, acute or obtuse according to whether ladcl = 4 2 , < 4 2 , or > 4 2 . I n order to distinguish the three elementary geometries we now prove:

322


47.2

(47.2) Either all Lambert quadrilaterals are rectangular, or all are acute, or all are obtuse. The proof is rather long; the first and main step consists in showing: If one Lambert quadrilateral is rectangular then all are. If ubcd is rectangular then - with the same notation as above - c' = d implies that L(m,n ) is perpendicular to L(c, d ) , hence mbcn is rectangular and, by (47.1). mn = ad = bc. It is clear that repetition of this argument yields for any point x on b C T(a,b) with bx/ba = p 2-' ( p , Y positive Fig. 59. integers) that the perpendicular to T(a,b) at x intersects T(d,c) in a point x' for which

Ul

Now let pqrs be any Lambert quadrilateral. Choose u with (puq) and v with (rvq) such that qu < ab, qu < bc and qp/qu and qr/pare positive integers. Let the perpendiculars to T(p,q) at u and to T(q,r) at v intersect at w. Choose x E T(a,b) with bx = qu and y E T(b,c) with by = qv. If z is defined as above then the two Lambert quadrilaterals ztquw and xbyz are congruent, and since the latter is rectangular, uqvw is too. Reflection of uqvw in L(u, I ) sends q, v into points ul,wl with (quu,), (vwwl) such that uluwwl and ulquwl are rectangular Lambert quadrilaterals. Reflection in L(ul, wl) carries u, w into points us, ws with (u,u,q), (w,wlw)

47.2

323

HOMOGENEOUS SPACES

for which u2u1wlw2and u2qvw, are rectangular Lambert quadrilaterals, Since qp/qu is a positive integer, the point u, will for some n coincide with p . Then (pw,s) and pqvw, is rectangular. We now reflect pqvw, in L(v. w,) and see, proceeding as before, that pqrs is rectangular. I t is now easy to finish the proof of (47.2). Let abcd be an acute (obtuse) Lambert quadrilateral. If (axb) and (byc) denote by z the intersection of the perpendiculars to L(a, b) a t x and to L(b, c) a t y . If xbyz were not acute (obtuse),then an obviouscontinuity argument would produce points x’ E T ( x ,a ) and y’ E T ( y ,c) for which the corresponding Lambert quadrilateral x’by’z’ is rectangular; but then abcd would be rectangular by the first part of this proof. Thus xbyz is acute (obtuse). If now pqrs is any Lambert quadrilateral we choose u with (puq), and v with (qvr), x with (axb) and y with (byc) such that qu = bx, qv = by. Then the corresponding Lambert quadrilaterals uqvw and xbyz are congruent. Therefore uqvw is acute (obtuse) and it follows that pqrs is acute (obtuse).

THEFUNCTIONS p(t) A N D

qe)

Rectangular, acute, and obtuse Lambert quadrilaterals correspond of course, to the euclidean, hyperbolic, and spherical geometries. Our next purpose is to actually derive these geometries from the mobility hypotheses, and we take the hyperbolic case as example, i. e., we assume that all Lambert quadrilaterals are acute. The arguments in the spherical case are strictly analogous. In the euclidean case they are somewhat different but considerably simpler, we will briefly indicate its treatment later. Let G and H be two lines perpendicular at w and Q one of the quadrants determined by G and H . If p E Q and x , y are the feet of p on G and H we put

px=p,

py=A,

wx=t,

wy=q,

pw=p,

ipwxl - 0 .

The quantities p and t for example, determine the others and if A or p are considered as functions of p and 6 we write A(p,t ) or p ( p , t ) to indicate this and use similar notations like 6 ( [ , p ) for other dependences. Let t 2 -t = t - t,> 0. t1 0, and put xi = (ti,0) (i. e., q = 0) p , = (ti,q ) (so that (p,pp,)), pi = p i xi. Let the perpendicular to L ( p , x ) intersect L ( p i , xi) in a;. Finally let p = m [p,, p ’ ] . Then the triangles alpl+ and a&’# are congruent and since

>

lxlalpl == lxe@&I = lP%P’l < 4 2

< IP,a2PI,

we have

47.2


324

(pp'p,)

and

(a) Since hence

€1

X1

X

Fig. 55.

Thus we see: A(€, q) and ~ ( 5q ,) are, for fixed q (or t),increasing convex functions of E (or q ) . Hence A(.!, q) possesses at .!= 0 a right-hand partial derivative ~ ( qwith ) respect to t :

exists and, because of the inequality (a),

The function A(.!, q)/t is for fixed 5 > 0 an increasing convex function of q which decreases monotonically to ~ ( qas) 6 decreases. Therefore ~ ( qis)

326

HOMOGENEOUS SPACES

47.2

a convex non-decreasing function of 7, and as such continuous. By Dini's ) Theorem (see Courant [l, p. 1081) the convergence of A@, 7 ) / t to ~ ( qfor 6 + O+ is uniform in q. (This also follows without using monotoneity from the convexity of A([, y)/E as function of 7.) Therefore

We will also need the information that q(q)is not constant. This may be seen as follows: let Y , y l , y2 be the feet of y on L ( x , P ) , L ( x l , P), L(x2,P ) and let L ( y , I ) intersect L ( x , ,Pi) at si. Then rsl = rs8 hence 2 YY = Y S ,

+

YS2>

YYl+

YY,.

Therefore y r is a concave function of [ whose right-hand derivative a t 0 equals p(7) because of (c). Applying (b) we find

Now let, for fixed 8, the line L(w,P ) intersect L(xi,9,) in qi. Then IwPxl 0,

see Picard [l, pp. 9-12]. THE TRIGONOMETRY

OF THE RIGHT TRIANGLE

We are now ready for the principal step, the proof of the Pythagorean Theorem : If abc is a triangle with a right angle at c, then cosh kab = cosh kac cosh kcb, or ~ ( a b=) ~ ( a c~ )( c b ) . Let (aalc), (accl) and alcl = uc. Choose b, on the same side of L(a, c) as b such that the triangle a,b,cl becomes congruent to abc. Then T(a,,b,) and

47.2

327

HOMOGENEOUS SPACES

T ( b , c) intersect at a point b,. Let (b,a,a,) and b,a, = ab. Choose c2 on the same side of L(a,, b,) as c (or cl) such that a,b,c, is congruent to alb,cl and hence to abc. Then au, = cc,

ula2 = bib,.

and

b

Fig. 66.

Put T(b,, c,) n L(a, c) = q and L(b, c) L(a,, c,) = p . The point p exists for small aa,. In this order let d , d,, e, el, A , h, be the feet of b on L(a,, bi), b, on L ( b , c), r = m M ,on L(a, b ) , r on L(a,, b,), s = on L(a, c), s on L(a,, c,). Then as u1 + a

Because of [earl = Jelalrl, ar = U,Y, [aerl = lalelrl = 4 2 the triangles aer and alel? are congruent, hence (erel) and eel -, similarly . . _ -- y aa, ar

hh h,s 2= a1 a2

t

a2s

so that lim

3 = lim

aa,

Then we obtain (using aal = ccl)

%a,

= S(lbac1) > 0.

328

47.2


But

so that the assertion follows from (f). The rest is now fairly simple. In a triangle abc with a right angle at c let f be the foot on L(a, b) of a point x with (cxa). Then (ufb). Put

b

0

a = kbc, E

/? = kca, y

= kaf,

= kab,

p = kxf,

6 = kax,

8 = Ibacl. c

Then the Theorem of Pythagoras yields the two relations

8-8

x

d

a

Fig. 57.

cosh a cosh (/? - 6) = cosh kxb = cosh p cosh (y - E ) cosh a cosh jl cosh 8 = cosh y cosh p cosh E. Subtracting the first equation from the second yields cosh a sinh /? sinh 6 = cosh p sinh y sinh E and division of this relation by the preceding one gives tanh /? tanh /? = tanh y tanh E or (g)

tanh E tanh k ac =--1im tanh k a b tanh6 -of

tanh --

E

tanh 6

- lim-k at = C(0). k ax

0, an which for a n y given distinct points a, a', x , y , z the relations a x = a'x, a y = a'y and xz + z y = x y imply az = a'z, then the universal covering space of R i s euclidean, hyperbolic, or spherical.

CHARACTERIZATION OF

THE ELEMENTARY SPACES

If the hypothesis of the last theorem is assumed in the large we obtain: (47.4) l'h?oi*errc. If each bisector B ( a , a') (i. e., the locus xu = xu') of a G-space R contains with any two points x , y at least one segment T ( x , y ) , then the space i s euclidean, hyperbolic, or spherical of dimension greater than 1.

Notice the corollary (47.5) The euclidean, the hyperbolic, and the spherical spaces of dimension greater than one, are the only C-spaces in which the bisectors are flat.

With the same method as for (46.b) we show first that under the hypotheses of (47.4) the bisectors are flat. Let a # a'. The set B ( a , a') is closed, hence finitely compact, and since it contains with any two points a segment connecting them, it is convex in Menger's sense. Uniqueness of prolongation is obvious because it holds in the whole space, To prove local prolongability it suffices

332


47.3

to show that B(a, a') contains with any segment T ( x ,y), where 0 < x y ba' because ba 2 ay- yb = a'y- yb = ba' and ba = ba' would imply (yba) and hence a = a'. For u E T ( y ,b) let v, be defined by vb = b and, if u # b, by (v,ux) and v,u: zy = bu :by. The point v, exists because v,w zy < p(y)/2 and xu vbd and my = az < a'z = a'vy there is a point w on T ( y ,a) with v, E B(a, a'). The point w cannot lie on B(a, a ' ) by the same argument that proved ba > ba'. But this is impossible because x , v, E B(a, a') and T ( x ,v,) is unique, and hence lies on B(a, a ' ) .

p. By (48.1) the sphere S (yI,min (d,, 6, - $ 4 ) ) and hence S(y,, p) permits rotation. If x E S(q,p) then x E K(q,t)for some t < p. Now (48.1) applied to q, yz, x instead of p , q, q1 shows that S ( x , p - q x ) permits rotation.

1, provides an example: every sphere about p = (0, 0, a-1) and q = (0, 0, - a-l) permits rotation. This explains why the elegance of Schur's Theorem in its usual form must be sacrificed if the goal is a statement with complete hypotheses and a definite radius v for which S(p, v ) is elementary. Our version is:

+ +

(48.4) .Tht?orertb. If O < pq < min (d,, p(p), 6,) and S(p, 6,) and S(q, 6,) permit rotation, then S(p, v ) , v = min (p(p),dP, 2 pq) is isometric to a sphere in a euclidean, hyperbolic or spherical space. For a proof put B = min (6, - pq, 6, - p7)/3. By (48.3) the sphere S(q, 8) is elementary. Let p = m,,, and r not on the same segment with p and q so close to p that V = T(q,I) u T(r, q') c S(p, dP). For every point x on V there is (becausepq < p(P))apoint y,eitherwith(y,xp)oron T(p,x)suchthatpy,=pq. Rotation about p carries S(q, p) into S(y,, /I). The number qy, changes continuously with x from 0 to 2 pq as x traverses V. For pyx< pq the sphere S(y,, p) lies in S(q, 6,). If z is given with either z E T(p, q) or (pqz) and qz < pq, then n E V exists such that rotation about q carries S(y,, 8) into S(z, 8). Now a rotation about p will send a suitable one among these points z into a given point w of S(p,v ) because v min (dP, 2 $4). Hence S(w, 8) is isometric to the elementary sphere S(q, p). Since also v < p(@) it follows as in previous proofs that S(p, u) is elementary. Notice the corollary:

0 exist such that S(q,, 6,) permits rotation,

then S(p, 6) is elementary. It follows from (48.3) that every point x in S(p, 6) except p has an elementary spherical neighborhood S ( x , p,). If 6, > @q for some t then (48.6) followsimmediately. If this is not the case let 0 < t < p(p)/3,so that p(q,) > t. We map S(q,, p,) isometrically on an elementary sphere and extend this mapping to a topological mapping of S(q,, p(q,)) on an elementary sphere U' such that each radius of S (q,, p(q,)) is mapped isometrically on a radius of U'.

336


48.7

By the argument used in the proof of (31.1) this mapping is locally isometric in the neighborhood of every point in S(q7,p(q7))nS ( p , 6 ) with the possible exception of p . Any segment T ( x ,y ) which does not pass through p is mapped on a curve of the same length, hence the distance of x and y equals the distance of their images. For reasons of continuity the mapping is isometric throughout.

THE HELMHOLTZ-LIE PROBLEM An immediate corollary of (48.6) is the important fact : (48.7) Thawem. I f every point p of a G-space R has a neighborhood S ( p , dP), Sp > 0 , such that for any points a, a’, b, b’ in S ( p , dP) with ap = alp, bp = b‘p and ab = a’b’ an isometry of S ( p , Sp) on itself exists which leaves p fixed and takes a into a’ and b into b’ then the universal covering space of R is euclidean, hyperbolic or spherical.

This theorem is the local version of the famous Helmholtz-Lie Problem whose history is briefly this: Riemann’s lecture [ 13 of 1854 ‘Ober die Hypothesen welche der Geometrie zu Grunde liegen” was answered in 1868 by Helmholtz [ 11: “uber die Tatsachen die der Geometrie zu Grunde liegen”. Helmholtz says that, while spaces of the general type discussed by Riemann are abstractly conceivable, physical space is distinguished by the free mobility of objects in it. A rigorous form of this property is this: If @ is an isometric mapping of the set M on the set M‘ (in the same metric space), then @ can be extended to a motion of the space. Helmholtz stated correctly, and believed to have proved under certain differentiability assumptions, that the only spaces with this property are the euclidean, hyperbolic and spherical spaces. A serious gap in Helmholtz’s proof was discovered by Lie [l, p. 4371, who reformulated the problem and solved it by using his theory of transformation groups, which required strong analyticity assumptions. Weyl [l] treats the problem completely under reduced differentiability hypotheses. In all these investigations the number of points in the sets M to which the mobility property is applied depends on 1 for an n-dimensional space. Our the dimension of the space, it is n or n last theorem shows that it suffices to consider point triples. This is particularly important in our case because we do not know whether an arbitrary G-space has finite dimension. On the other hand, G. Birkhoff [l, 21 showed, that the mobility property applied to denumerable sets M can be used to deduce finite compactness from the much weaker assumption of completeness.

+

4R.9

337

HOMOGENEOUS SPACES

The development of the foundations of geometry made differentiability assumptions appear altogether foreign to the Helmholtz-Lie Problem, and Hilbert [ 1, Appendix IV] eliminated them entirely in the two-dimensional case. Kolmogoroff [l] indicates without proof how Hilbert's idea can be generalized, but also works with (n 1)-tuples instead of triples. Recently Tits [ l ] indicated (with proofs forthcoming in the near future) how Kolmogoroff's approach can be completed, reducing at the same time the number of points from n 1 to 3. The work of Hilbert, Kolmogoroff and Tits uses much weaker hypotheses than ours (the modern results on Lie groups permit us to eliminate many of them). The space is not assumed to be a G-space, the groups of motions are shown to be isomorphic to those of the elementary spaces. It then follows that the elementary metrics provide the only M-convex metrizations. On the other hand, these methods definitely operate in the large and have hitherto not yielded theorems like our (48. 4, 6, 7). The latter results lead to the following solution of the Helmholtz-Lie Problem in the large within the framework of G-spaces :

+

+

(48.8) Tlmweatc. If a G-space possesses for a n y four points a , a', b, c with ab = a'b and ac = a'c a motion which leaves b and c fixed and carries a into a', then the space is euclidean, hyperbolic, or spherical.

We first show: If p , a , b and p , a', b' are isometric triples then a motion of the given space R exists which leaves p fixed and carries a into a' and b into b'. For there is a motion @ of R taking p , p , a into p , p , a'. If b@ = b* then pb* = pb = pb' and a'b* = ab = a'b'. Therefore a motion !P exists which sends p , a', b* into 9, a', b' so that @!Pcarries p , a , b into p , a', b'. This means that every sphere permits rotation, hence the universal covering space of R is by (48.7) euclidean, hyperbolic or spherical. We next prove a lemma which will prove useful also on other occasions: (48.9) I n a G-space R' let, for a n y two pairs a , b and a', b' with ab = a' b' < 7, where 7 i s a fixed positive number, a motion of R' exist which carries a into a' and b into b'. If x ( t ) and y ( t ) represent geodesics in R' then for a n y real a, tlrt2

4%)4 t 2 ) = Y b l +

+

a) Y(T2 a).

If R' i s not compact then it i s straight. The number p ( p ) is constant because there is a motion carrying p into an arbitrary other point. Let 0 < p< min (7, p ( p ) ) . There is a motion @, sending x ( O ) , x ( P )into y(a),y(a+P). Because T ( x ( O ) , x ( P ) ) and T ( y ( a ) , y(a+PI)

338

48.9


+

< < <
0, in which a triangle area i s defined and has the same valzle for isometric triples, then R is locally euclidean, hyperbolic or spherical.

The formulation of this theorem does not exclude the possibility that A(abc) has different values for a triangle abc in the overlap of two different neighborhoods S ( p , d,,), and the proof will show that it is not necessary to assume that the value be the same. On the other hand, it would be most unnatural to consider A(abc) without this consistency assumption. Since the result implies moreover that the values of A (abc) in the overlap of two triangles can differ only by a constant factor, we will agree to consider here, as well as in Theorem (60.6), only A(abc) which are consistent, because then we obtain a reasonable area on the whole surface. I t suffices to show that S(p,E ) with E = min (dP, &)/a) is elementary. Consider two distinct points a, a' in S ( p , E ) and two points cl, ca B(a, a') in S ( p , E ) which lie on different sides of the symmetric extension of T(a,a') of length p ( p ) . The triangle inequality yields immediately that T(cl, ca) intersects this extension in a point q with (aqa'). Let m = m-,. We are going to show

49

HOMOGENEOUS SPACES

343

that q = m. Assume m E T(q,a’). Then the propcrties l), 2), 3) of area and the hypothesis imply

+ A ( u c , ~ =) A (ac1c2) A (u’c,~,) A (u’clq)+ A (a’c2q) A (ac,q) < A (uc,q) + A (c,qm) = A (aclm) = A (u’c,m) < A (u’c,q) A(ac2q)< A ( ~ c 2 q+ ) A(c,qm) = A ( u c , ~= ) A(u‘c,vz)< A ( u ‘ c , ~ ) . A (aclq)

=

=

The first of these three relations implies equality in the last two, hence A(c,qm) = 0 and q = m by property 2). Let now c be any point in S ( p ,E ) and on B(u,u’)different from q. Then either T(c,,c ) or T(c2,c) intersects T(a,a’) and is therefore collinear with c1 and q or with c2 and q so that the points of B(a,a’) in S ( $ ,E ) lie on one segment. The a’ assertion follows now fiom (46.1).

k

A triangle area in the large, i. e., defined for all point triples in a twodimensional G-space is not a natural C2 concept except when the space is straight Fig. 60. or spherelike. A straight space satisfying the hypothesis of (48.16) is clearly euclidean or hyperbolic, and a spherelike space is, under the assumption of (48.16), spherical so that it is not necessary to require the existence of an area defined for all triples, although it can be verified in the euclidean and hyperbolic case, and with a little caution also in the spherical case, that such an area does exist. In the elliptic case the lengths of the sides of a triangle do not necessarily determine the triangle uniquely, and no formula for the area A(&) in terms of the sides valid for all triangles exists.

49. Involutoric motions Motions with period two play an important role in many parts of geometry. What is known about such motions in G-spaces forms the subject of this section. We begin with some observations on straight spaces and spaces of the elliptic type which are far from deep but amusing, because they show that phenomena with which we are well familiar in euclidean and non-euclidean geometry extend to much more general situations.

344

49.1


REFLECTIONS OF

STRAIGHT SPACES AND SPACES OF THE ELLIPTIC TYPE

We need some general facts on fixed points of motions. (40.1) The set F of the fixed points (if any) of a motion @ of a straiglt? space R is flat. For if @ leaves the points a and b fixed, it leaves every point on the straight line through a and b fixed, which therefore lies in F. Since F is closed, see (4.2), it is with the metric of R itself a straight space, hence flat.

The set F of fixed points (if a n y ) of a motion @ of a space R of the elliptic type is the union of flat sets F,, Fa, . . ., Fh such that a point in Fj i s conjugate to every point of 1; - F,. A point which i s not fixed lies on at most one line containing tuo fixed points. (49.2)

If F contains two non-conjugate points a, b then it contains the whole great circle through u and b. This shows that F has the form u Fj, where each Fj contains with any two points the whole great circle, and any point of Fi is conjugate to any point of I; - Fi. The number of Fi is finite because R is compact. Since F is closed each Fi is closed, so that each Fj is flat, and in fact again a space of the elliptic type. A great circle which contains two fixed points goes under @ into itself, a point lying on two great circles containing two fixed points goes, as unique intersection of the two great circles, into itself, hence is fixed. A motion @ is involutoric if G2 = E but G # E , where E is the identity. (49.3) A n involutoric motion @ of a straight space R i s uniquely determined by the non-empty set F of its fixed points: If p i s not in F , then the foot f of p on I; is unique and f = m [p, +@I.

The motion Qi is called the “reflection” of R in F. For because @ # E , there is a point p # p@. The midpoint f of p and p@ goes into itself because = p , so that F is not empty. Let g be any foot of p on F. Then the inequalities,

P@f = P f 2 Pg = P@&

Pf + fP@ = PP@

< Pg + gP@

show that g = f . An involutoric motion of a space of the elliptic type need not have fixed points; the mapping which maps every point of a great circle on its antipode shows this already. Less trivially,

40.6

345

HOMOGENEOUS SPACES

+ +

defines a distance preserving mapping of the sphere x: xi xi + xi = 1 on itself and induces in the three-dimensional elliptic space which it covers a motion without fixed points. (49.4) I f an involutoric motion @ of a space R of the elliptic type has fixed points, then the set F of all fixed points is the union of two flat sets F , and F,. Each point p E R - F lies on exactly one great circle gp intersecting both F, and F,. The point f, = g,,nF , i s the unique foot of p on F,, and f , , f a are the midpoints of the two arcs of g,, from p to pa.

Thus each of the sets F,, F , determines @ uniquely, and @ is called the “reflection” of R in F , or in F,. By (49.2) the set F is the union of flat sets F,, F,, . such that every point of F, is conjugate to every point in R - F , . By hypothesis there is at least one set F,. Let p C R - F , and let f be a foot of p on F,. Then f is the unique foot of any point y with (pyf), and if y f < p ( f ) then ( y f y @ ) by the same argument as above. Hence @ maps the great circle sp through p and f on itself (so that p@ E gp) and leaves therefore the point fa conjugate to f, on g,, fixed. f, cannot lie in F , because this would imply p E F,, so that R - F , is not empty. It is also clear that f, and f, are the midpoints of the two arcs of g, from p to p . A second foot of p on F , would yield two different lines through p which contain (each) two distinct fixed points in contradiction to (49.2). Hence F , determines @ uniquely. F cannot have more than two components because we proved that p E R - F lies on a great circle connecting a point of a given component Fi of F to a point of R - Fi, so that three or more components would lead to at least two distinct lines through p each containing two fixed points.

..

SYMMETRIC SPACES We now turn to involutoric motions of general G-spaces. The motion @ of the G-space R is called the “symmetry” or “reflection” in the point 9 if @ is involutoric and p is its only fixed point in a certain sphere S ( p , p ) , p > 0. An equivalent definition is: @ is a motion and there exists a positive p such that p = m [ x , x@] for x E S ( p , p ) - p . This and (28.8) show that the symmetry in p is unique, which we anticipated by using the definite article. If the symmetry in p exists for every point p of the G-space R then R is called symmetric. We notice (49.5) A symmetric space of the elliptic type is elliptic.

346


49.6

For by (49.4) the component of the set of fixed points other than p consists of all conjugate points to p and is flat, so that (49.5) follows for spaces of dimension greater than two from Theorem (21.2). In the two-dimensional case we observe that the reflection in p is also the reflection in the line formed by the conjugate points to p . Then the reflection in every line exists and the assertion follows from Section 47. There is no similar theorem for straight spaces. All Minkowski spaces are symmetric, the product [E' x HPla of a v-dimensional euclidean space and a ,u-dimensional hyperbolic space defined in Section 8 is an example of a symmetric Riemann space which does not have constant curvature, the hermitian hyperbolic spaces discussed in Section 53 provide other examples. The symmetric G-surfaces are determined in Theorem (62.8). Observe the following fact : (49.0) A symmetric G-space possesses a transitive group of motions.

For if x , y are two arbitrary distinct points of the given space K and m is any midpoint of x and y , then the segments T ( x ,m ) and T ( y ,m) are unique and form a segment T(x,y ) , hence the symmetry in m carries x into y. The symmetric Riemann spaces have been investigated by Cartan in several papers which are known for their depth, elegance, and importance for Lie groups; for a brief resum6 and the literature see Cartan [3, Chapter IV]. It ought to be pointed out that Cartan uses the term symmetric space in two non-equivalent meanings. One coincides with ours, the other leads to spaces which might be called locally symmetric. The G-space R is locally symmetric if every point p possesses a neighborhood S ( p , p), O < p < p ( p ) such that fi = m [ x , x@] and p@ = p define an isometric mapping @ of S ( p , p) on itself. Obviously a symmetric space is locally symmetric, but not conversely. Every space of constant curvature is locally symmetric, but only very few of them possess transitive groups of motions: the Moebius strip with a euclidean metric does not possess a transitive group of motions because of the existence of a distinguished great circle, see Section 30. None of the surfaces with constant negative curvature except the hyperbolic plane is symmetric (see Theorem (62.8)). Possibly the universal covering space of a locally symmetric space is always symmetric'. Doubtless many of Cartan's results both on symmetric and on locally symmetric spaces can be carried over toG-spaces. This optimism is based on the following observations :

HOMOGENEOUS SPACES

49.7

347

Assume that the C-space R is symmetric and compact. I t follows from of all motions of R is a Lie group and that R is a topological manifold. There is a Riemannian rnetrization R* of R invariant under see Cartan [3, p. 431. Since the symmetry C#J of R in a point p is involutoric and has p as only fixed point in a certain sphere S ( p ,p ) , the same holds for @ as motion of R*, hence it is also the symmetry of R* in p . Thus, in the compact case all results of Cartan on the structure of spaces which are symmetric (in the large) carry over to G-spaces. We observe that, locally, the midpoints m(x, y ) of x and y for R and R* coincide, hence the geodesics of R and R* coincide and along each geodesic the distances in R and R* differ only by a factor which in general depends on the geodesic. This will be so, for instance, on a torus with a Minkowskian metric. However, if the factors are known for the geodesics passing through one point, then they are known for all geodesics because every point can be moved into any other. The degree of freedom in the choice of the factor depends on the structure of the socalled isotropic subgroup of I‘, i. e., the subgroup of motions in rwhich leaves a given point p fixed. (49.0) and (52.4) that the group

r

r,

REFLECTIONS IN LINEAL

ELEMENTS

An involutoric motion @ of a G-space R which leaves every point of a segment T with midpoint p and a positive length 2 p < 2 p ( p ) fixed, and has no other fixed points in S ( p , p ) than those on T , is called the “reflection” in T . There cannot be more than one reflection in T . For if x is a point of S ( p , p/4) not on T then a foot f of p on T lies in S ( p , p / 2 ) and we see as in the proof of (49.3) that f is unique and f = m [ x , xa]. This determines @ in S ( p , p/4) hence in all of R . The length of T is immaterial: A motion which leaves a non-degenerate segment pointwise fixed leaves the whole geodesic g containing the segment pointwise fixed. @ cannot have in S ( p , pl(p)) other fixed points than those on g. For if q were a fixed point not on gin S ( p , p l ( p ) ) then T ( p ,q) would be pointwise fixed but contain points in S ( p , q) not on T. To be definite we may use the lineal elements introduced in Section 9. Whereas there are many types of symmetric spaces, there are only few which possess reflections in all lineal elements: (49.7) l’?tcovaem. A G-space R which can be reflected in each lineal element is euclidean, hyperbolic, elliptic or spherical.

Proof. Let T be a segment with center p and length 2 p ( p ) (a straight line q a point of S (p, p ( p ) ) - p and f i t s foot on T , moreover f = mqql. if p(p) = a),

49.7


348

If r E T , fir < p(f)/4then the reflection in T takes q into q‘ and leaves r fixed, so that rq = rq’. Also, if (4x1) then f is the unique foot of x on T and f = m[x, x @ ] , hence r x = rx@. This shows that f must be the foot of r on T(q,4’). Hence perpendicularity of segments is symmntric. If f = m[r, rO] and y E B(r, yo) n S (f, p ( f ) / 4 )- f then the segment T ( x ,f ) will be perpendicular to T. Let x , x’ be two arbitrary distinct points and m a midpoint of x and If (xym), ym < p(m)/4, and m = myyl (so that (my’x’)), choose a point z E B ( y , y’) - m with z E S ( p , p ( m ) / 4 ) . Such a point z exists unless R has dimension 1. Then the reflection in T(z,z’), where m = m,,,, takes y into y’, hence x into x‘. Thus the space possesses a transitive group of motions and p ( p ) is constant. Put 6 = p(p)/16. XI.

(a) If O < x y = x’y’< S then a motion exists that takes x into x’ and y into y’. We know there is a motion @’ that takes x into x’. Let y@‘ = y” # y‘. If (y’x‘y’’) then the reflection @“ in a segment perpendicular to T ( y , y“) a t x’ w ill take y” into y‘. If the relation (y’x’y’’)does not hold then the reflection @“ in T(x’,m y , y t t )will take y‘ into y“, in either case WW‘ satisfies (a). The main step is the lemma: (b) Let q1 # q2 and 6 > q,p = q& = a > 0, where p is the foot of q1 and qa on the segment T with center p and length 8 6. If m = m[q,. q2] # p , then p is the foot of m on T . We know that T ( p ,m) is perpendicular to Tq = T(ql,q2). Let x ( t ) represent Tq with x ( 0 ) = m and x(b) = ql, where /3 = m ql, and let y(u) represent T with y(0) = p . Then

y ( 2 a ) ~ ( t2) 2 a - p x ( t ) 2 a > pm

Y ( 4 xkk 8)z Y(0)x ( & P) > p m

for for

/ T I < fi 101

< 2a.

Therefore y ( u ) x ( t )is as a function of CI and t on the boundary of the rectangle IuI 2 a , It1 P everywhere greater than at (0,O) where it has the value pm. Consequently y(u)x(r)reaches a minimum a t some interior point (u’,t’) of the rectangle. Therefore the segment T’ = T(y(a‘),x ( t ’ ) ) must be perpendicular to both Tqand T. We are using here and in the following argument obvious extensions of (20.9 f) to perpendicularity of segments. The lemma will be proved by showing u’ = t’= 0. Let t’ 2 0. We extend x ( t ) and y(a) to representations of geodesics. Then the reflection @

bf 2 bc. Since also b’c’ 2 max (a’b’, a’c’) it follows that the footf‘of a’ on T satisfies (b’f’c’). Let af 2 a‘f’. Since bf fc = b’f’ f‘c’ we have either bf 2 b’f‘ or cf 2 c‘f‘. Assume bf 2 b’f’. Choose a,, on T(a,f ) such that aof = a’f‘ and b, on T ( f ,b) such that b,f = b’f’. By (a) and (c) there is a motion Y that takes a,, f , b, into a’, f ‘ , b‘ respectively. If a Y = a”, bY = b“, c Y = c”, then a”b” 2 arb” arbr because T(a’, /’) is perpendicular to T(b’f’). But a”b” = ab = a’b’, hence a” = a’, b” = b’ and the relation b”c” = b’c‘ implies c“ = c‘. Thus Y takes a, b, c into a‘b‘c’. In analogy to locally symmetric spaces we may consider spaces for which reflections in segments exist locally. Without proof we state the following theorem which is quite easily established by using (48.3) and the idea of the preceding proof:

+

+

>

(49.8) Let for every lineal element T with center p and length a(#) a n involutoric isometry of S ( p , o ( p ) / 2 )on itself exist which leaves the points on T and no others fixed. Then the space i s locally euclidean, hyperbolic, or spherical.

60. New characterizations of Minkowski spaces The problem of determining all G-spaces with transitive groups of motions is, except for two dimensions, much too general to be solved with the present means of mathematics. Only the abelian case is simple.

TRANSITIVE ABELIAN

GROUPS OF MOTIONS

The translations of a Minkowskian Geometry form a transitive abelian group, and a result of Pontrjagin [l, p. 1701 on more general spaces implies that every G-space with a transitive abelian group of motions must be locally Minkowskian. Since this result is of major importance for the theory of G-spaces it is desirable to establish it with the present methods. This can very easily be achieved with the help of our results on spaces with non-positive curvature.

60.4

36 1

HOMOGENEOUS SPACES

(50.1) Titeorewt,. A G-space which possesses a transitive abelian group of motions i s locally Minkowskian, and is, topologically, the product of a finite number of straight lines and circles.

We first prove two trivial, but frequently useful, lemmas on motions in general metric spaces : (60.2) If I' is a transitive abelian group of motions of the metric space R, and then xx@ is independent of x . Hence no motion in except t h identity, has fixed points. For if x and y are given, then a Y in exists such that y = xY and

@E

r,

r,

r

yy@ = xYxY@= XYX@Y== xx@. (60.3) A transitive group r of motions of a metric space R is simply transitive if and o d y if no motion in I', except the identity, has fixed points.

For if a motion @ in

r different from the identity has a fixed point f and

p # f , then a motion Y in r exists with pY = f . Then Y@is different from Y and both motions take p into f so that P i s not simply transitive.

r

Conversely let be transitive, but not simply transitive. Then two different motions @, Y in I' and two distinct points a, b exist such that a@ = b and a!P = b. Then @Y-' leaves a fixed and is not the identity. Applying the lemmas to the hypothesis of (50.1) we see that the given transitive abelian group is simply transitive, so that the motion in which carries a into b may be denoted by (a b). We observe first: = (a -+ c). (50.4) If a # c and the midpoint b of a and cis unique then (a + If a, b, c E S ( p , p ( p ) / 4 )for a suitable p and b = mu, then (a d)a = (a c) implies d = b. For the proof of the first part put @ = (a -+ b) and Y = (b c). Because of (50.2)' we have

r

-+

-+

-+

-+

aaY = bbY = bc = ac/2,

a Y c = aYa@Y = aa@ = ab

= ac/2

hence a Y = m, = b or Y = (a b) and (a c) = @Y= Under the assumptions of the second part of (50.4) there is a point c' with d = mw,. Moreover d is the unique midpoint of ac'. The first part of (50.4) shows (a + c') = (a + d ) z = (a c), hence c' = c and d = b. If we can show that for any three points a, b, c in S ( p , p@)/16) the relation -+

-+

-+

2 m,m,

= bc

352

b0.B


holds, then it will follow from (39.12) that the space is locally Minkowskian. Put b' = m,, c' = ma,, (a b') = @, ( a c') = Y, (b' -* c') = Q. Then 'Y = @Q and @Q@= Y@= YQ?P hence -+

-+

Now by (50.4) (b + d )

= (b + a )

(a + d ) = @-2@Q@ = Q and

(d +c) = (d + a ) (a + c) = Y-'QY1Y2= Q. Thus ( b -* c) = Qa = ( B bd = dc = b'c'

+

d ) a and (60.2) yields

< ab' + ac' 4 (ab + ac)/2< p ( p ) / 8 ,

ad

< ab + bd < p(p)/4

I t follows from the second part of (60.4) that d = nib, hence 2 b'c' = bc. The Moebius strip with a euclidean metric, compare Section 29, is an example which shows that not all locally Minkowskian spaces possess transitive abelian groups of motions. However, if T is the group of motions in the (Minkowskian) universal covering space i? of R which lie over motions of f , then theorems (28.10) and (28.11) show that 7 is transitive and abelian. Hence xx& is constant for any motion 5 in so that 8 is a translation of R. In particular, the motions of the fundamental group are translations, which implies that R is the product of a finite number of circles and straight lines. This could also have been deduced from the fact that R is a group space (because f i s simply transitive) and the general property of group spaces of having abelian fundamental groups.

r,

TRANSITIVE GROUPS

OF MOTIONS WITH MAXIMAL DlSPLACBMENTS

We saw in the last proof that xx@ is constant for each @ in I',and know from Theorem (32.6) that this condition is satisfied for a motion Q, of a straight space if there is a point a for which the displacement under @ is maximal, i. e., ZZQ,= sup xx@. Thus we are led to conjecture: (50.5) l'bwwb. I f a straight space R possesses a transitive group F of motions such that for each motion in f points with maxi& displacement exist, then the space is Minkowskian.

r

Since xx@ is constant for each @ in f no motion in except the identity has fixed points. By (60.3) the group r is simply transitive on R, and the motion in I' that carries a into b may again be denoted by (a + b).

60.6

363

HOMOGENEOUS SPACES

If a # b then @ = (a + b) # E and xx@ = ab implies, trivially, ab = min xx@. Hence, see (32.3),the line g+(a, b) is an oriented axis of ( a -. b) zc

R

and for any point x , because of xx@ = ab, the line a+(%, x @ ) is also an oriented axis of @. All these axes are by (32.4) parallel to each other. Since there is at most one parallel to a given line through a given point, each parallel to an axis of @ is an axis of @, and the parallel axiom holds. We show first that the space has the divergence property: If y(t), t 2 0, represents a ray r and C is a line through y ( 0 ) not containing the ray, then y(t) G co for t -. 00. For an indirect proof assume the existence of a sequence t,,+ 00 with y(t,,) C < M . If f,, is a foot of y(t,,) on C, then I,, # y ( 0 ) for large p and q,, = ~ ( 7 , )(/,,+ z ) has z as foot on G. Choose a subsequence { v } of { p } for which (4,) converges to a point q l. The line g (x(t,), qv)is an axis of (1, -* z) hence parallel to C, and tends therefore to the parallel C' to G through q. On the o t h x hand, the definition of co-ray implies that T(q,, y(t,)) tends to the co-ray from q to r, hence G' is also parallel to the line H carrying r. Since the concept of parallelism is symmetric G and H would be parallels to G' which is impossible. The distance of two axes of the same motion, hence of parallels is boundtd. Exactly as in Section 37 we deduce from the divergznce property that two lines C, and C, are parallel if, and only if, x,G, and xnC, are bounded for xi E G,. We now show that is abelian and thus reduce the present theorem to the preceding one. If @ # E and Y # E in r a r e given, we select an arbitrary point z and put z@ = p , zY = q, so that @ = ( z p ) , Y = ( z -.q). The relation @Y= Y@is obvious if z, p , q lie on one line. Assume therefore that g(z, p ) = C f H = g(z, q) and put H' = H@. For y E H the relation yy@ = zp shows that the distance of H and H is bounded, hence H' is parallel to H and an axis of Y, so that pY is a point p' of H' with zq = pp'. On the other hand, zq = z@q@, hence q' = @ =: p', so that @ = ( q q'), y/ = ( p 4'). I t follows that

.

-+

r

4

-

@Y= (2

+

p ) ( p --4') = (2 -4')

= (2

-

-4) (q - 4 ' )

= Y@

which proves our theorem. CHARACTERIZATION OF MINKOWSKIANGEOMETRYBY

PROPERTIES OF AREA

When we discussed the geometries in which the area of a triangle can be expressed in terms of its sides, the reader probably expected to see the euclidean geometry characterized by the property that the area of a triangle is determined by base and altitude. It is a curious fact that this property

354

60.6


does not single out euclidean geometry. We will show here that it characterizes the Minkowskian geometries with symmetric perpendicularity. There a triangle area A (abc)exists which has the form A(abc) = k * bc * afi(b, c) where k is a constant depending on the unit of area. In Section 41 we introduced a volume in Finsler spaces, which yields for a Minkowski plane: The Minkowski area is proportional to the area in an arbitrary associated euclidean geometry. The factor of proportionality is so determined that the Minkowski area of the Minkowskian unit circle becomes IC. Since the Minkowskian and euclidean distances are proportional on a fixed straight line, the Minkowskian area A(&) of a triangle shares with the euclidean area the property that A(&) depends only on a, bc and the line o(b, c). We are going to prove that the existence of an area with this property characterizes Minkowskian geometry. It will then follow easily that perpendicularity is symmetric when A(&) depends only on ~ ( bc), and be. We remember that every point of a C-surface possesses arbitrarily small convex neighborhoods. A line L(a, b) in such a neighborhood was the open segment containing a and b whose endpoints lie on the boundary of the neighborhood. (60.6) lNe@i~irc..If every point of a Gsurface R possesses a convez neighborhood Upin which a triangle area A (abc)is defined and depends only on a, L(b, c) and bc then R is locally Minkowskian.

We assume that A(&) is defined consistently, see the remarks after Theorem (48.16),although we actually do not need this hypothesis. If a, b, c are three non-collinear points of Up,then the additivity propert 3) of triangle area, and the fact that area is non-negative lead through itan .. ard arguments regarding the functional equation f(u) f ( v )= f(u v ) (see l'icard [l, pp. 1, 21) to the relation

+

(a)

A(axb) : A(axc) =; xb : xc

for

+

6'

X E L(b, c).

Consider three non-collinear points a, b,, b, in Up. Let m = m[b,, b,], (mqa) and put c1 =; L(ba,q) n T(b,, a ) , ca = L(b,, q) n T(b2, a ) , n = T(a,m) n T(c,, ca). Then (anq). The relation (a) yields A(ab,m) = A(ab,m) and A(qb,m) == A(qb,m), hence (b) so that by (a)

A (qbi4 = A ( @ a 4

60.6

HOMOGENEOUS SPACES

355

and similarly 44(@2c2)

- b2c2 - A ( q V 2 )

A(qc,a)

c2a

A(qb14

Therefore b,c, :cla = b2c2 :c2a and by (a) and (b)

(4

A (qb,c,)

= A (qb2c2)

A (qcla) = A (qc2a).

and

a

m Pig. 62.

Now (a) yields

a_n -- A(ac,n), A (qc1n) - Q a A ( q c z 4

A(ac,n) --

hence (c) implies A(ac,n) = A(uc,n); therefore by (a)

(4

cln = c2n.

We conclude from (a) and (d) that

+

2 A (b,n c,) A-(ac,b2) 2 A ( a n c,) =2 A (an b,) - A (a n c,) 4-A (b,n c2)

hence

Successive dyadic subdivisions of T(b,, b,) and application of (d) and (e) yield

(f)

If (b,xb,) and T(a,x ) n T(cl,c,) a y : ax

= acl

: ab,

and

=y

thett

cly :c2y = b,x : b,x.

60.6


356

This leads to (g) If z1 and za are points of L(b,, b,) then

A(Z1Zay) is constant when y varies on L(C1, ca). For ax __ A (alrix) A (ab1b3 --. A (ab,b,) - a*, - __ A(c,biba) cibi X Y A W4 A ( ~ b l b d

The general statement (g) follows from (a). U

W

b

1.

L

z

Y

a- (a,O)

I

I

4.

U

Fig. 8'3.

We call the line H parallel to the line L if for one pair of distinct points b,, b, on L the area A(yb,b,) is constant for y E L. We just proved that A(yzlza) is constant for any pair of points z,, Za of L. The relations (c) and (d) show that A(c,c,b,) = A(c,c,b,) so that L is parallel to H if H is parallel to L. We also find

(h) I/ H i s #arallcl lo L a d K lies between H and L and is Parallel to L , then it also is parallel to H . If for i = 1, 2, aiE H , hi€ K , C ~ EL and (a,b,c,) then alcl : blcl = a,c, :bncp. The proportion follows from (a) and this in turn implies that K is parallel to H . If T ( a ,c) and T(b,d ) are two non-collinear segments with the same center, then (h) shows that the opposite sides L(a, b) and L(c, d ) as well as L(a, d ) and L(b, c) are parallel. Thus every point in Up, in particular itself, is interior point of a parallelogram. In the interior P of any parallelogram ZMWV, where w is opposite to z, coordinates 6, q may be introduced as follows: Let the parallels to L(z, u) and to L(z, u) through a given point q in P intersect T(z,u) a t qu and T(z, u) at q,,. Then = zq,, q = zq, are the coordinates of q. A line L which intersects P intersects the boundary of P. Let for instance L n T ( z ,u) = u = (a,0). Choose any two distinct points q = (t,q) and

60.7

HOMOGENEOUS SPACES

357

qo = (to,q,) on L' = L n P which are different from a. If to= a then 5 = a and L' satisfies the equation E = a. If 5, # a let the parallel to L ( z , v ) through a intersect the parallels to L ( z , u) through q and q, in b and b, respectively. Then ( f ) and (h) imply

so that L' satisfies a linear equation.

Besides the given distance q1q2 we introduce in P the euclidean distance

Then (i) shows that e'(q,, q2) and q1q2 are proportional on a fixed line L' in P . The same holds then also for the more general euclidean distance

We consider a system of neighborhoods in R which is formed by such parallelograms. In the overlap of two parallelograms we have two different euclidean metrics. If two parallel lines in one of the two parallelograms have common points with the second, then they lie also on parallel lines in the second parallelogram. To see this we do not need the consistency of the area A(abc), we merely observe that, with the above notations, L(b,, b8) is parallel to L(c,, c2) if (ac,b,) and ac, : ab, = ac, : ab,. Therefore the euclidean metric in the second parallelogram can be changed affinely, i. e., in the way in which e'(q,, q2) was replaced by e(q,, q2) such that the metrics coincide in the overlap. We thus remetrize R as a locally euclidean space with the same geodesics as R in such a way that the euclidean distance and the given distance differ on each geodesic locally only by a constant factor. The universal covering space R carries then a euclidean metric in addition to the metric induced by R. The geodesics are in both rnetrics the euclidean straight lines, and the distances along the same straight lines are proportional. Therefore K is by (17.1) a Minkowski space which proves (50.6). The assumption that A(abc) be consistent induces a triangle area in 2 defined for all triples abc, which also depends only on a, g(b, c) and bc. We omit the easy proof of this fact and rather turn to the uniqueness of area. (50.7) I n a Minkowski plane there is, up to a constant factor, only one triangle area which defiends only on a , g(b, c) and bc.

368

60.8


For let H,L be two straight lines intersecting a t c. If a E H and b E L, then (a) shows that A(abc) is proportional to ac and bc so that

-

A(abc) = q ( H , L ) ac * bc,

(60.8)

where q ( H , L) = q(L,H)> 0. We have to show that the area of one nondegenerate triangle abc determines the area of all other triangles. The value of A(abc) determines with the above notation the value of q ( H , L). If now K is any line not parallel to H or L, and not through c, then K intersects H and L in points x and y , and

A ( c x ~=) q ( H ,L ) * xc

- YC = q ( K ,L )

*

XY

*

CY;

hence q ( H , L) determines q ( K , L). We can now replace L by any line K' not through K n L and not parallel to K or L and find in the same way that q ( K , L) determines q ( K , K'). It is now clear that the value of q ( H , L) determines q(C,G') for any pair G, G' of non-parallel lines. Since we know that an area proportional to an associated euclidean metric satisfies the hypothesis of (60.7),we have proved that these are the only triangle areas A(abc) depending only on a, g(b, c ) , bc.

AREA IN

TERMS OF BASE AND ALTITUDE

We now return to the question which led to the present discussion and show: (60.9) A straight plane possesses a triangle area A(abc) which depends only on the distances bc and ag(b, c ) , if and only if it is a Minkowski plane with symmetric perpendicularity.

We know that the plane is Minkowskian and that

A(abc) = p(H,L ) * uc * bc

where

H

= g ( ~ c, ) ,

L = g(b, c ) .

Under the present assumption q ( H , L) must have the same value a > O , whenever one of the lines H or L is perpendicular to the other. If a pair of lines H,L existed such that H is perpendicular to L at c but L not to H,then a transversal to L at a point b # c would intersect H in a point a # c and

-

A (abc) = a ac bc = a ab * bc

but ab

> bc

because c is a foot of a on L.

51.1

HOMOGENEOUS SPACES

359

It follows that the area of any triangle abc equals (50.10)

A(abc) = a.bc * ag(b, c ) .

Conversely, it must be shown that in a Minkowski plane with symmetric perpendicularity such a formula always holds. Because of (50.8) it suffices to see this for “right” triangles where the legs of the right angle have length 1. This means, that in an associated euclidean geometry all these triangles have the same area, and follows directly from the construction of all Minkowskian geometries with symmetric perpendicularity given at the end of Section 17. A detailed discussion is found in the author’s paper [el. If anything in the relation (50.10) distinguishes the euclidean geometry from the Minkowskian geometries with symmetric perpendicularity, it can only be the value of a. Indeed, Petty [ l ] proves: if area is normalized such that the unit circle has measure x, then a = ‘I2for the euclidean geometry and a> for the other Minkowskian geometries with symmetric perpendicularity. 61. Translations along two lines

A translation of a straight plane R along a straight line g, or with axis g, was defined in Section 32. Because of (10.11) there is for two points a , b of g a t most one translation along g that takes a into b. If this translation exists for arbitrary a , b on g we say that “all translations along g exist”. The results of Section 32 imply: (51.1) Let all translations Qi along the geodesic g of the straight plane R exist. The pnrallels to g coincide with the axes of @ and are equidistant jrom each other. The other equidistant curves to g (or its parallels) are convex curves which are concave toward g. Every point of R has exactly one foot on g (or any parallel to 9).

The last statement is not contained in Section 32 but is obvious: if p has two distinct feet f l , f a on g and ( f l f f z ) and the translation that takes f l into f takes p into p‘, then T ( p ,f,) would intersect T ( p ’ ,f ) in a point q which has both f and fi as feet on g, which contradicts (20.0). I t may actually happen that some equidistant curves to g are geodesics and others not, even in a Riemann space with non-positive curvature, see example following Theorem (37.12). Also, the parallel axiom may hold and all translations along one line (and hence its parallels) may exist in a non-Desarguesian straight plane. This is shown by the example constructed at the end of Section 33.

360


61.2

Finally, all translations along g may exist and all curves equidistant to 8 may be straight lines in a Desarguesian plane which does not satisfy the Parallel Axiom. The strip IyI 1 then ka < p(z). It follows from (32.5) that (zz,z2), hence (Z,-~Z,Z,+~) for all integers i. Since p(z,) = p(z) the segment T(z,,, zi+,) is unique and contains 2,. Now the existence of x ( t ) is clear and ~ ( trepresents ) a segment for ia t (i K) a because ka < p(zi). In particular x(O)x(ka)= z.@ = ka. We agree to use as distance for two motions @, Y of a compact space R the number d(@, Y ) = sup x@x!P. Then d(E,@) equals the maximal dis-

< < +

< < +

XER

placement under @. Before applying (52.1) to the proper subject of this section we notice the following interesting corollary : (52.2) l'teorcm. A one parameter group possesses api orbit which i s a geodesic.

r of

motions of a compact space

r)

We remember from Section 29 that the orbit O(p, of the point p under We assume that is represented in and choose E > 0 such that the standard form with @#,Gt,= @,,+', d(E,4) < inf p(x)/2 for It1 < E.

r consists of all points p@ with Q, E r.

r

xeR

For 0 , 2 Zz@u/2 = zz@u/z

+ ZQUl2

ZGU

Z z q

2 Z'Z'@,,

Thus z is also a point of maximal displacement for @u,2 and generally for Moreover (ZZ@,,/~Z@,,) and generally (ZZ@,,-I z@,,~-"). If ~ ( tis)the geodesic with x(0) = z which represents the unique segment T ( z ,z@J for 0 t zz@, = a' then (52.1) shows x(ia' 2-") = for all i and non-

<
d(a, a’) then segments T(a,b,) and T(a‘,b ; ) contain points b, b‘ respectively for which ab = a‘b’ = d(a, a’). There is a motion @ in that takes a into a‘ and b into b‘. Since b, and b;

p(fi) is constant.

r


386

64.2

are the only points x, x' which satisfy the relations bx = b'x' = ab, - 6(a, u'), (abx), (a'b'x') the motion # takes b, into b;. If ab, = a'b; < B(a, a') then we determine b and b' by (ablb), (a'bib') and ab = a'b' = 6(a, a') which is possible because B(a, a') < p(a) = p(a'). There is a motion # in T that takes a into a' and b into b'. Because ab = a'b' < p(a) = = p(a') the points b, and bi are the only solutions x, x' of ux = U'Z' = ab,,

so that bl#

T

(axb), and (a'db')

bi.

It will turn out that in the compact case not all the assumptions for a C-space are needed, it suffices to know that the space is M-convex, i. e., the last two axioms for a G-space, see Section 8 may be omitted. This rests on two facts, the first of which is the following: (64.2) Let R and R* be two equivalent metrizutions of the same set S as finitely compact, M-convex space. Let R possess a paim'se transitive grozlp of motions, which is also a group of motions for R*. Then the mtrics of R and R* differ only by a coltstant factor (so that is also pairwise transitive on R*).

r

r

For a proof we denote the distance of two points a, b in S as points of R by ab and as points of R* by 6(a, b). If ab = a'b' then contains a motion @ of R which takes a into a' and b into b'. Since @ is also a motion of R* it follows that d(a, b) = d ( d , b'). Thus there is a function f ( t )such that

r

6(a, b) = f(ab),

with

f(0) = 0,

>0

f(t)

for

z > 0.

If R is compact then there is a pair c , d whose distance equals the diameter

fi of R, and since a segment T(c,d) exists (see Section 6) the range of f ( t ) is

<
4 g(t2) -g(t1) g(t2 - t l )

0 that f(r)has for 0 t B the form k t, k > 0. Since, in the non-compact case, B was arbitrary the assertion follows.

THE RIEMANNIAN CHARACTER

OF THE METRIC IN THE COMPACT CASE

We now prove: (64.4) A compact, M-convex metric space R which possesses a eairwise transitive group of motions is a Riemann space.

r

If R is a G-space the assertion follows at once: By (62.4) the group of all motions of R is a Lie group, and a well known result of Cartan [3, p. 431 states that there is a Riemannian metrization of R invariant under r. By the preceding result the metric of R differs from this Riemannian metric only by a constant factor and is therefore itself Riemannian. The extension of this result to M-convex spaces rests on the following result of Wang [3]. It may be omitted by those interested only in G-spaces.

64.6


388

(64.6) l'im?rewt,. If the compact M-convex metric space R possesses a pairwise transitive group I'of motions, then every non-discrete, closed, normal subgroup A of I' is transitive on R .

We show first: given a point p and a positive E' there is a positive E < E' such that K ( p , E ) C O(p,A). The orbit O ( p , A ) is either finite or dense in itself. Because A is normal O(p!P, A) = O ( p , A ) !P for any !P in Since is transitive we conclude first that O ( p , A) cannot consist of one point, since then O ( p Y ,A) would equal #!P and A would consist of the identity only. Also O ( p , A) cannot consist of a finite number of distinct points p = pl, Pa,. ..,Pk, k > 1. For since A is nondiscrete, there is an element @ in A such that 0 < B(E,0)< m = min pipi,

r.

r

i#i

where 8(E,@) = sup xx#.

Since @ transforms O ( p , A ) into itself it must

XeR

r

leave every pi fixed. Because O(+!P, d) = O ( p , A ) P ! for Y E and pj!PpiY= p,pj 2 m the motion @ also leaves every point of O(pY,A) fixed, and being transitive, @ would be the identity, which contradicts 8(E,@)> 0. Since O ( p , A) is dense in itself it contains a point x with 0 < p x < E'. Put p x = E and let y E K ( p , E ) . Because I' is pairwise transitive it contains a motion ?Pwith PY = fi and x!P = y. The relation x E O ( p , A ) means the existence of a @ in A such that p@ = x. Then

r

y = p!P-l@?P E O ( P , A )

because A is normal, hence K($, E ) C O ( p , d). For the proof of (64.6)we define for any set M in R

O ( M , A )= u M @ = u O ( x , A ) . @€A

X€M

Then

O(M U M ' , A ) = O(M,A ) u O(M', A ) . By the first part of this proof there is for an arbitrary point sequence of positive ei with ei + 0 and K ( P , E ~ c) O ( p , A ) . If si = S(P,E j ) . s; = sj u K ( p ,E j )

p

a decreasing

then O(T,,A) is closed since A is closed and O(S,, A ) is open because S, is open. But 0(3j,A ) = O(Sj, A ) u 0 ( q p , E;), A)c (Sj,A ) u O ( P , A ) =: O(Sj,A ) so that O(Ti, A ) = R because R is connected.

64.6

389

HOMOGENEOUS SPACES

si

Let x be any point of R. Since x E O(Si,A ) , the sets O ( x , A ) and intersect for each i, so that O ( x , A ) n 3; is a decreasing sequence of non-empty compact sets. The intersection is not empty and is therefore the point p . Thus p E O ( x , A ) or x E O ( p , A ) which shows that d is transitive on R. We now turn to the proof of (54.4): there is a decreasing sequence of closed normal subgroups A,, A,, . . . of F whose intersection is the identity and such that r/A,is a Lie group (see Pontrjagin [l, Theorem 531, where subgroups of topological groups are by definition closed). The diameter of Ai is therefore from a certain subscript on, smaller than the diameter B of R. I t follows from (54.5) that A , is discrete for i 2 j , hence finite. Put y~ = min d(E, @). The group di0lies for a sufficiently large io > i in the &A I.-E

neighborhood 6 ( E ,Y) < rj of E , and thus consists of E alone. Therefore r/dio= is a Lie group. The remainder of the proof runs in the same way as for G-spaces. Under the assumptions of (52.4) spheres K ( p , p ) with a positive radius p < p(p) (notice that p ( p ) is constant) are homeomorphic to the (n - 1)-dimensional sphere if the space has dimension n.

r

DETERMINATION OF ALL

COMPACT SPACES WITH PAIRWISE TRANSITIVE GROUPS OF MOTIONS

The problem of finding all G-spaces with pairwise transitive groups of motions was completely solved in the compact case by Wang [3] who showed that they are the symmetric Riemann spaces of rank 1 whose groups of motions have simple Lie groups as identity components. However, Wang does not establish these properties directly, he obtains his result, so to say, by eliminating all other possibilities. Here we will aim directly at these properties. Let R be a compact Riemann space of dimension n > 2 with a pairwise transitive group of motions. Denote by r' the group of all motions of R. We decompose the proof into several steps

r,

(a) If i s a non-discrete normal subgroup of r' then every geodesic is orbit of a suitable one-parameter subgroup of TI. All geodesics of R are closed and without multiple points.

r,

Let Q7 be a one-parameter subgroup of (which may coincide with r') which is isomorphic to I), 1. By (52.2) there is a t least one geodesic x ( t ) which is an orbit of QT: for a suitable a > 0 we have x ( 0 ) = %(at). Since r ' is pairwise transitive it contains for a given geodesic y(t) a motion Y such that

390


64.6

~ (Y t )= y(t). Because rl is a normal subgroup of r ' the motion Y-107Y' lies in rl and y(0)Y-W7Y= x(0)O7Y= %(at)Y = y(at). hence y ( t ) is also an orbit of a one-parameter subgroup of Because @r is isomorphic to ID, the geodesic ~ ( tis )closed and as orbit it cannot have multiple points, see (9.4). The same holds then for y ( r ) . We notice in passing that this implies the transitivity of on R, a result proved above for M-convex spaces.

r,.

r,

(b) The identity component

r of

r' is pairwise transitive on R.

r

Since is a normal subgroup of r' it is, as we just saw, transitive on R. Let A' be the isotropic subgroup of r' belonging to an arbitrary but fixed point p . Because A' acts transitively on K = K(p,p), 0 0, let ~ ( t )~,' ( represent t ) segments of length 8, where is the diameter of R (compare (48.9), p. 337) with x ( 0 ) = a, x(ab) = b, x'(0) = a', x'(ab) = b'. There are motions Of Y in I' with a@ = p , pY = a'. If x(p) @ = a'', x'(p) Y-l = b" then a", b" lie on K, hence a suitable Q in A takes a" into b". Then @QYlies in and takes a into af and b into b'.

r

r is simple. If r were not simple it would contain a proper normal non-discrete subgroup (c)

rl of lower dimension. The group I' has a covering group F which is the direct product of compact semi-simple groups and of a compact commutative group (see Pontrjagin [l, p. 2861). The totality of elements in T which lie over elements of rlis a normal subgroup Fl of and is a direct factor of F .

r rl

s7

Any element of a one-parameter subgroup of T isomorphic to 3,which does not lie in ?; commutes with every element in Fl. Let cP7 be the image of in r. Then Or has an orbit x(ar) = x ( 0 ) G7 which is a closed geodesic. ) Put z = x(0). Let y ( t ) represent any geodesic in R different from ~ ( twith y(0) = z. By (a) the geodesic y ( t ) is orbit of a one parameter subgroup Y7 of r,: y ( p t ) = zY7. The motions O7and Y, commute for any t,0. The subset W of R consisting of the points z07Y,,is then intrinsically isometric to a torus with a euclidean metric. The proof is the same as that of Theorem (60.1). "Intrinsically isometric" means, that, while the distances in W are only locally the same as in R, the set U will in general not be flat, but the geodesics on W are geodesics in R. Clearly W contains open geodesics which contradicts (a).

s7

64.6

HOMOGENEOUS SPACES

391

(d) The rank of R equals 1. This is contained in the proof just given: The rank of the space (to be clearly distinguished from the rank of )‘l is the maximal dimension of a torus which can be generated in the way W was generated above. We need not go into the exact definition: the rank is a t least 1, and as soon as it is greater than 1, there are manifolds in the space which have dimension greater than 1 and which are intrinsically tori with a euclidean metric, see Cartan [l]. (e) The local spheres. A Riemann space is locally euclidean and a Finsler space is locally Minkowskian. In many cases it is useful to think of the local euclidean or Minkowskian geometry as imbedded in the given space E (we did this essentially in the proof of (41.12)): Let p be any point of E. In U = S ( p ,p ( p ) ) we define a new distance as follows: if a, b are two distinct points in U,consider the segments ~ ( ty)( ,t ) , 0 f t p ( # ) , with x ( 0 ) = y(0) = p , which contain a and b respectively. We put

v2+4v+2,

for

p>4.

The argument is similar for D/,.There we have to satisfy ,u(2pu1) = v 2 + 4 v + 2 ,

with

,u>v

+

+

and observe that p ( 2 p - 1)> va 4 v + 2 if p 2 6 and that p ( 2 p - 1) 2 is no square for p = 2, 3, 4, 6. dim 23, = 29. Finally we turn to b, acting on S’. There dim I‘= n The dimensions of I, B,,, a”, D, are not prime numbers which excludes 93, alone. B, x Dlis also impossible because then dim = 30 and 30 does not have the form v(2 Y l ) , v(2 v - 1) or v(v + 2). Similarly 23, x El leads to dim = 32 which does not have any of these three forms. In the case of the exceptional groups we notice that none has the dimension 29, 30, or 32. Thus, anticipating the fact regarding 0,,and using the list of Cartan [ l ] quoted in the last section we have proved Wang’s result:

+

r

+

r

(54.7) Theorem. A compact M-convex space with a paimise transitive group of motions is spherical, elliptic, hermitian elliptic, quaternion elliptic or the Cayle y elliptic plane.

394


64.8

In none of these spaces is the geodesic through two points unique except in the elliptic spaces. Therefore we have the corollary: (64.8) The elliptic spaces are the only compact G-spaces with pairwise transitive groups of motions i n which the geodesic through two points' is unique.

66. Odd-dimensional spaces with pairwise transitive groups o! motions There are various reasons why the non-compact spaces are much more difficult to treat; one is that the groups of motions are not compact, and we know much less about non-compact Lie groups. Another reason, which follows from this, is that we cannot establish the Riemannian character of the metric a priori, for this conclusion was essentially based on the result of Cartan [3, p. 431 which holds only for compact spaces. But even if we know that the metric is Riemannian there is no simple analogue to the conclusion that the rank of the space must equal 1, because the tori become now euclidean planes, and all geodesics are congruent. If a Finder space has a pairwise transitive group of motions, then we conclude from (64.6) that it is Riemannian and from Myers-Steenrod [ l ] that its group of motions is a Lie group. If the space has an odd dimension then the local euclidean spheres are even dimensional and we conclude exactly as in the compact case that the space has constant curvature (using the result on 6, which is derived at the end of this section). Therefore we have here a case where differentiability hypotheses mean a very considerable simplification. The reader who is satisfied with this case may omit everything preceding the discussion of Cayley numbers.

THESPACE

IS A MANIFOLD

We know that a non-compact C-space with a pairwise transitive group of motions is straight. We are going to show that any such space is a manifold, provided its dimension is finite'. We observe first (66.1) A sphere K ( p , p) with 0 < p < p ( p ) of a C-space of dimension greater than 1 is arcwise connected and locally arcwise connected. If x, y are any two distinct not diametrically opposite points of K(#, p) define x', y' by (px'x)and (py'y) and px' = py' = p/3. Then T(x', y') C S ( p , p ) . Because x and y are not diametrically opposite no point u' E T(x', y') coincides with p , so that there is a point u with (pu'lc)and pu = p. The points u form obviously an arc from x to y on K ( p ,p). If x and y are diametrically opposite

66.3

HOMOGENEOUS SPACES

396

we merely have to choose any third point z of K(P,p ) , which exists because the dimension is greater than 1, and connect x with z and z with y. The continuity properties of geodesics imply that for a given E > 0 a number S > 0 exists such that for any two points x , y with x y < 8 the above constructed arc lies in S( c, E ) . Keeping (52.4) in mind we now prove:

'

(55.2) A finite dimensional non-compact G-space with a pairwise transitive grozlp of motions is a topological manifold.

Consider a sphere K = K ( p , p ) (here p > 0 is arbitrary because the space is straight). The isotropic subgroup I'+ of the group of all motions of the space acts transitively and, as we saw in the last section, effectively on K which is finite dimensional, compact, connected and locally connected. I t follows from Montgomery-Zippin [l, Theorem 11 and Corollary 6'1 that I', is a Lie group, hence K is a manifold. The set U = S ( p , p ) - p is the topological product of K and a straight line, hence a manifold. It follows that R is a manifold, and if dim R = n, then dim K = n - 1.

ODD-DIMENSIONAL SPACES For odd dimensions we now establish the following analogue to Theorem (54.7) which is found in Wang [2]: (55.3) Theorem. A G-space whose dimension is finite' and odd (or two) and which possesses a pairwise transitive group of motions is euclidean, hyperbolic, spherical or elliptic.

Because of (48.9) and Theorem (54.7) we may assume that the space is straight. In addition we assume, of course, dim R = n > 2. We begin by showing that the spheres K = K ( p , p ) are homeomorphic to S"-'. We know already from the preceding proof that K and U = S ( p , p) - p are manifolds. Since S(P, p) can be contracted to p along its radii, U has for i < n - 1 the same homotopy groups n,as S ( p , p) so that n i ( U )= 0 for i < n - 1. Because U is the topological product of K and a straight line we conclude, that

n , ( K ) = 0,

i = 1,2, . . .,n - 2 .

Thus K is a simply connected homology sphere of the even dimension n - 1. I t follows now from another result of Wang [ l ] that K is homeomorphic to S"-'.

66.3


396

r,,

With the notation of the proof of (55.2) the identity component of is a connected compact Lie group acting effectively and transitively on K . Since dim K = n - 1 is even, is by Wang [l] isomorphic either to the group D,,l),2 of the rotations of the (n- 1)-sphere or to the exceptional group 6, which acts on S6.

c

But I': is not only isomorphic, but equivalent to D(,,-1),2or 6,, for see Montgomery-Samelson [l, Lemma 13, for 6, see Wang [l, Lemma 61. Equivalence is defined as follows: The topological transformation groups I' and d acting on the same space M are equivalent if a topological mapping Y of M on itself exists such that Y-lPP = d , i. e., d is identical with the set of mappings Y-'diY with di E I'. Isomorphic groups acting on the same space are not always equivalent. We know that D,,,,,, is with the ordinary spherical distance on S"-l pairwise transitive on S"-', and we are going to prove the same for 6,and S6. Taking the latter fact at the moment for granted and denoting ,,,D ,,, or 6, by we now finish the proof of (54.7) as follows: Since is equivalent to there is a topological mapping Y of K on itself such that Y-l$P = I'. If a(%,y) is a spherical distance on K invariant under r a n d we define E ( X , y) = a(#, y Y ) then E ( X , y) is invariant under because for di E

r

r

E(X0,

ydi) = U ( X Y Y - l d i Y , yYY-'diY) = o(xY, y Y ) -- & ( X , y).

c

The metric E ( X , y) is, of course, also spherical and has as pairwise transitive group of motions. Now is also a group of motions with respect to the metric x y on K given by the original space R. We cannot apply (54.2) because x y is as metric on K not convex. But we conclude again that a function f ( t )exists such that xy = f ( e ( x , y ) ) , f ( t= ) 0, f ( t> ) 0 for t > 0.

If x y = x'y' only for E ( X , y) = E(x', y') then I': will also be pairwise transitive on K for the metric x y . We have to show that f ( t )takes every value only once. If = f ( t )= /(a) with t > u > 0, then for a given point q E K =K@, p) the spheres H,:E(Q,x ) = t and H, : e(q, x ) = u on K would both lie on qx = v. The sphere H, separates p from H, on K . The cone V formed by the rays with origin p and containing a point x of H, separates H, from q. Therefore a segment T(q,y) with y E H, must cross V a t some point u. Let x be the point of H, on the generator of V through u. Then the two triangles p q x and pqy would satisfy the relations

66.3

397

HOMOGENEOUS SPACES

p x = py = p , qx = qy = 97 and T(q,y) would intersect T(p,x ) or its prolongation beyond x at u, which is impossible according to the proof of (10.11). Thus we see that is pairwise transitive on K . I t follows readily that for any two pairs a, b and a', b' with ab = a'b' a motion in I ': (now a motion of R ) exists that takes a into a' and b into b'. For the segments from p to --- a , b, a', b' or their prolongations intersect K in points a , b, a', b' respectively. If 2b > a'%'then the segments T ( f , x ) with x E T ( a , b) or their prolongations beyond x would intersect K in an arc from a to b which contains a point a* - -with a* b== a'b'. There is a motion in :'I that takes a 'into a* and b' into b. I t takes b' into b and a' into a point a", for which T(a,b) intersect T(p,a") or its prolongation.beyond a". The two triangles pba and pba" are impossible that takes for the same reason as before. Thus a b = a'b' and the motion in a into a ' and b into b' also takes a into a' and b into b'. Thus K satisfies the hypothesis of Theorem (48.10) and Theorem (56.3) is proved except for the assertion regarding the pairwise transitivity of 6, on SO. THECAYLEY N U M B E R S The group 6, is closely connected to the Cayley numbers. We list their properties, as far as they are needed here and refer for proofs to Freudenfhal [l], quoted as F , which is not very accessible but concentrates just on the questions which interest us here. While Cayley arrived at his numbers more or less by experimenting, we can now obtain them axiomatically together with the real numbers, complex numbers and quaternions, which elucidates the algebraic side of the results obtained in the last section. We consider a linear algebra W over the real numbers, i. e., a finite dimensional linear vector space over the real numbers, in which a product x y is defined for any two elements in H, lies in H, and satisfies the following distributive law (rx

+ s y ) (r'x' + s'y') = rr' - xx' + rs'

xy'

+ sr'

*

yx'

+ ss'

*

yy',

where r, s, r', s' are real numbers and x , y , x', y' elements of H. We also assume that I€ has a unit element e, such that xe, = e,x x for any x in H,but we assume neither the commutative nor the associative law for multiplication. Finally we postulate the existence of a real, symmetric inner product ( x , y) which is defined for any two elements of H, is linear: (rx sy,z) = r ( x , z ) I-

+

+

,

398

66.3


+ s ( y , z) for real r, s and any x, y, z in 23,is positive definite: + 0, and satisfies lxyl = 1x1 - Iyl, where 1x1 = + ( x , %)'la.

(x, x)

> 0 for

x

These postulates imply, see F, p. 6., (a) (ax, ay) = (xu, ya) = (a,a ) ( x , y) for any x, y, a i n H .

If the dimension of H is n, and we form an orthonormal base (eo, el, (notice that the first element is the'unit) which means (b)

(ej, ej) = 0

i # j,

if

. . ., en-l)

(ei, ei) = 1,

then the ei satisfy the relations a

if

(c)

e; = -eo

(d)

e;(eix) = - x

(e)

ei(eje,) = - ej(e;e,)

if

i > 0, i >0

eie, = -ejei

i>

i,

i,j

>o

i # j , i # k i, j , k > 0.

if

= ek(eiej)

if

The base may in addition be chosen such that each product e,ej is, except possibly for sign, again an e,. The advantage is a multiplication table which allows us to read off the product of two arbitrary numbers n-1

n-1

x =

2 riei

y=

and

i=O

2 siei. i=O

For any orthonormal base U-1

i- 0

is an inner product (but there may be others). All properties are obvious, except perhaps lxyl = 1x1 * IyI which follows from (a) and (b):

This relation is an identity of the form n-1

i=O

n-1

i=O

?+l

i-0

where the zi are bilinear forms in the xi and yi. It was shown by Hurwitz [l] with matrix theory that such an identity can exist only for n = 1,2,4, or 8.

55.3

390

HOMOGENEOUS SPACES

Thus the only systems Wsatisfying all our conditions are the real numbers, the complex numbers, the quaternions, and the Cayley numbers. A very elegant proof of Hurwitz’ Theorem is found in F, pp. 7, 8. The quaternions may be written in the standard form x = ro

+ r , i + r 2 j + r3k

with mu1tiplication table j2

ii = k,

ii = - k ,

jk

i 2 = k2

= i,

=- 1

k j = -i,

kj=j,

ik = -i

and this is, except for trivial variations, the only multiplication table. The Cayley numbers may be written in the form 7

x

+- 2 riei,

=-yo

i=1

but there are non-trivial variations between the different multiplication tables. We describe one firocess of forming a basis which will prove useful later: We choose el and e2 arbitrarily such that (eo, el) = (e,, e2) = (el , ea) = 0 and [ell= leal = 1. We put e3 = elea

(f)

-

then le3(= (el( (e2(= 1 and by (a) (g)

(e3,el) = (e1e2,eleo) = (e2, eo) = 0.

Similarly (e3,e2)=0. Moreover (e3,~o)= (ele2,eo) = (el(elea), eleo) = - (e2,el) = 0. Here we use (d) for e,, although we have no complete base, which is permissible because e,, el can always be completed to some orthonormal base. We choose e4 arbitrarily with (e4, ei) = 0 for i < 4 and leal = 1. The remaining elements of the base are now defined by (h)

e, = e1e4,

e, = -eae4,

e, = e3e4.

Clearly le,l = le,l = le71 = 1 and, as in (g), e, for example is orthogonal t o any e, that contains el or el as factor. The remaining orthogonality relations are proved as above:

(es, eo) =

eo) = (el(elel), eleo) = - (e4, el) = 0

400


66.4

and exactly the same for (e6, e2), (e6, e3). e6 and e7 are treated in the same way or as (ee, e3) = (elen, eleJ = (e4, el) = 0. The other products eiei can now be calculated, for instance using (e): ele6 = - e1(e8e4) = - e4(e1ea) = - e4e3 = e7 eses = e2(ele4)= - e4(ele2)= e7 t3e6

= e,(ele4) = - el(e3e4) = - ~ 1 e 7= - el(ele6) = e,.

We chose these three products because we now have the complete list of basic relations used by Wang [2] and Dickson [I, p. 141, (the list in F is different) although the last three relations are really superfluous because they follow from (f) and (h). THE GROUP 0, An automorphism 0 of the Cayley algebra leaves each real number fixed, preserves the value of the norm 1x1 (see F, p. 17) and therefore induces a topological mapping of the set : 7

7

(65.4)

u=

2 riei

r:

with

=1

i=l

i-1

on itself which in turn determines @ uniquely. Thus the group of automorphisms of the Cayley algebra may be regarded as a topological transformation 7

group of the six-dimensional sphere Z17 = 1 on itself. The identity component i-1

of this group is 8,, see F, pp. 17, 18. An automorphism 0 preserves the euclidean distance I x - y I since ( x - y)@ = x 0 - y@ and @ preserves the norm, hence 0 also preserves ( x , y) and the spherical distance Arccos ( x , y) on S6;this follows from ( x - y , x - y ) = =2(1-(x,y)). To complete the proof of (54.7) we must show that 0, is pairwise transitive on S'. Let x, y and x', y' with ( x , y) = (x', y') be given. The fact that the points lie on S6 implies ( x , 1) = (x', 1) = (y , 1) = (y', 1) = 0.

Put x =: ul, x' = ui. If y = f x , then y' -- & x' and we complete 1, u1 and 1, ul' to bases (1, ul, . .,u,), (1,ul', . . .,u7')by the definite process described above. Then the multiplication table for the two bases are identical, hence

.

86.4

401

HOMOGENEOUS SPACDS 7

7

i=l

i=l

defines automorphism of the Cayley algebra’that sends x into x’ and y into y’. If y is not dependent on x we put

Then (u,,ul) = (u,,1) = (u,’,ul‘) = (u2’,1) = 0, lu21 = I u , ’ ~ = 1 and we can again extend 1, ul,u, and 1, ul’,u,’ to complete bases by the above process because the first two elements el, e2 were arbitrary (except for orthonormality). If @ is defined as before it will again be an automorphism. To make sure that @ is in 0, we ought to ascertain that the vectors ul, . . ., u7 and ul’, . ,u7’ determine the same orientation of S6, but we can avoid this problem by observing that -ul’,. . ., -u7‘ determines the opposite orientation from 2t1’, . . ,, u7‘ and has exactly the same multiplication table. Because ( x , y) = ( x ’ , y’) and

..

1, = (y - ( x , y ) 2, y - ( % y ) x ) = 1 - ( x , r), it follows that y = ( x , y ) u1+ AU,,

y’ = ( x , y ) 211‘

+

hence x@ = x’ and y@ = y‘. This finishes the proof of (56.3) and also the proof of (54.7) in the case where 0, acts on S6 in the local euclidean geometry of a seven-dimensional Riemannian space. To complete the proof of (54.7) we have to discuss the exceptional case where 0, is the group of motions of a &dimensional Riemann space and % is its isotropic subgroup. We obtained the combination 8,. 2& simply by counting parameters. That Ir, is really the isotropic subgroup of 0, as transformation group of S6, may be seen as follows: We consider the automorphisms of the Cayley algebra H which leave e7 fixed. Let H’be the

.

subspace of H spanned by el, . . , e,.

6

For any element x = Z . rieiof

we

in1

introduce complex coordinates zl,z,, a, by 21 = -11

+ ir,,

z2 - - r,

+ ir,,

a, = - Y,

+ ir,,

where we use the same multiplication table as above. If x = (al, za, a,) and y = (w,,w,, w,) are elements in H’then

402

66.4


where a and is are, respectively, the real and imaginary parts of

z1G1*+ z,Eg

+ zaEa.

Thus, a unitary unimodular transformation of H' with the coordinates zl, z,, z3 on itself induces a transformation of H which preserves the product hence an automorphism of W ,and we see that the isotropic subgroup of 0, as transformation group of S8 contains % as subgroup. Conversely, an automorphism Qi of E which leaves e7 fixed induces a real linear transformation of H' on itself. This follows from the relations (e7x)Qi = e7Qix@ ==e7(xQi) for x E H'and ~~e~ e,,

e7e, = e,,

=L

e7ea = c,,

e7e, = - ear

e7e, = - e,,

elea = - el

which are contained in the above multiplication table. Also, Qi preserves the norm zlZ, a,X, + 2878, therefore the transformation induced by Qi lies in 9l,, so that 4 is the isotropic subgroup of 0,.

+

There is no group different from (5, and locally isomorphic to 0,, whereas there is a group locally isomorphic to, and different from, q.Corresponding to these two cases the coset space 0,& is, topologically, the six-dimensional sphere or projective space. To see that the metric of the space is spherical it suffices to prove: If the sphere .S8is metrized as a G-space H such that a group F isomorphic to 0, is a group of motions of R, then R is spherical. But this is clear; for we just saw that 0, may be interpreted as a subgroup of the rotations of 9.Since 01 and are equivalent we produce, by the process described above, a spherical metric invariant under F, and 'I is pairwise transitive for this spherical metric. Now (64.2) shows that the metric of R is also spherical.

r

According to Tits [l], in even-dimensions the hermitian and quaternian hyperbolic spaces and a hyperbolic analogue to the Cayley elliptic plane are, besides the euclidean and hyperbolic spaces, the only non-compact spaces with pair-wise transitive groups of motions. Finite dimensionality is no longer assumed, but is deduced from the recent results of Yamabe on Lie groups. Moreover, Tits also omits the last two axioms for a G-space and indicates that M-convexity may be replaced by a weaker hypothesis. As mentioned before, proofs for these statements have not yet appeared.

Appendix: Problems and Theorome We conclude the book by listing some problems, many of which were already mentioned in the text, and a few theorems which either indicate possible further developments of the theory of Finder spaces or elucidate points not fully discussed in the text. P, 15 means a problem and T, 16 a theorem connected with material treated in Section 15. Finite compactness is not a topological concept, but it is easy to characterize the spaces which possess finitely compact metrizations: A Hausdorff space can be metrized as a finitely compact space if and only if it is locally compact and has a countable base. See Vaughan [l],also Rusemann 12, p. 2051.

Are all geodesics in a finitely compact, M-convex metric space R straight lines, if R satisfies the Axioms of local prolongability with p ( p ) E Q) ? Does p,(p) Q) imply p(p) Q) ? Or in words: In a simply connected space does uniqueness of the shortest connection imply the absence of conjugate points ?

Is a circle K ( p , 6 ) . 0 < d

< p(p). in a two-dimensional G-space rectifiable?

The problem whether every G-space is a topological manifold is at present, inaccessible. The following problems which go in this direction may be accessible.

Is a three-dimensional G-space a topological manifold ? Is a G-space locally homogeneous in the sense of Montgomery [l] ? This paper shows that local homogeneity has strong topological implications for finite dimensional spaces. Has every G-space a finite dimension ? Does the property of domain invariance (see Section 58) hold in every G-space I The answer is affirmative if the dimension is finite (communication from Prof. D. Montgomery). (9)

(10)

P. 11 To determine the systems of curves which occur in metrizations of the projective plane as a space of the elliptic type. P, 14

Can a two-dimensional Desarguesian space of the elliptic type be imbedded in a three-dimensional Desarguesian space ? The discrepancy between the sufficient conditions for a G-space given in Section 16 and necessary conditions is very great. It seems doubtful whether a considerable reduction can be obtained by the classical methods. The methods of Busemann and Mayer [l] may prove helpful, although this paper does not aim at uniqueness, but at differentiability properties. 403

404

APPENDIX: PROBLEMS AND THEOREMS

(11) P, 16

Find weaker conditions on the integralid F ( x , 6) than in Section 15 which imply that the space is a C-space. To find necessary and sufficient conditions is probably not a reasonable problem. The same problem may be approached from a different angle, namely, by imposing conditions on a G-space which make it a Finsler space. Two results are known in this direction, see Busemann [I, Chapter I1 p. 63 and p. 701 of which the first is this:

(12) T, 16

For a, b < a(a) (Section 9) denote by r(a, b) the segment with origin n through b of length u( a) . Let the following hold: if a,, b,, c, tend to p such that r(a,, b,) and r(a,, c,) converge, but to different segments, and if lim a,b,/a,c, exists (a, admitted), then r(b,, c,) converges. and lim (6,~. a,,c,)/b,c, = 1 if and only if lirn r(b,, a") = lim t(b,, c,) or lim r(a,, c,) = = lim r(b,, c,). Then a Minkowskian metric m(x, y) (equivalent to x y ) can be introduced in S ( p , p ( p ) ) such that m(p, x ) = p x and x,yy/m(xv. y J +1 if x,+ p. yr+ p and ( x l p pyv)/x,y, is bounded.

+

+

+

(13) T, 18

A Hilbert geometry with symmetric perpendicularity is hyperbolic, Kelly and Paige [l].

T, 18

In a Hilbert geometry let for any two opposite rays r, 5 (i. e., r and $ have only their origin in common and lie on the same line) the locus a(r, x ) = a(5, x ) (see Section 22) be flat. Then the metric is hyperbolic.

(14)

In all spaces considered in this book the distance is symmetric ( x y = y x ) . Many problems remain significant, or can be reworded so as to become significant, for spaces with non-symmetric distances. Only the simple results corresponding to our Sections 1 to 8,22, 27, 28 have been extended to non-symmetric spaces, called E-spaces, in Busemann [3]. (16)

P, 20-66 A systematic investigation of non-symmetric spaces by the present methods. Since (6.9) is a best estimate even in straight spaces there is in general for two distinct points a, b no line through b whose distance from a exceeds ab/2.

(16) T, 20

(17)

Given two points a, b in a straight plane, there is a line through b with distance at least ab/2from a.

P, 20 Given a line in a straight space of dimension greater than 1, are there always points with arbitrarily large distance from the line ?

(18) T, 21

In a two-dimensional space of the elliptic type there is for every point p exactly one geodesic flp with maximal distance from p. The mapping p is topological, Busemann [2].

(19)

+flp

P, 22 Do two straight lines in a straight space coincide, if they have representations

~ ( t Y)( . T)

which satisfy

lim T+m,

x ( 7 ) y(a) =

a+m

7-b-

lim

X(T)

m, a + -

m

y(o) = 0 ?

APPENDIX : PROBLEMS AND THEOREMS

406

For general G-spaces the answer would be negative, e. g., for the surface a

=

[l

+ x4 + y”-1.

(20) P, 23

Does symmetry of the asymptote relation imply transitivity in straight spaces of dimension greater than 2 ?

(21) F, 24

Are Theorems (24.6) and (24.10) true if the symmetry condition is omitted from the parallel axiom? ( I t is used only in (24.4).) More generally, does the second part of the parallel axiom imply the first ?

T. 24 and 30 Let Rn denote the G-space originating from the G-space R by multiplying all the distances by I > 0. If R1 is isometric to R for all I > 0 or only for all 1 in an interval of positive length, then R is Minkowskian. One proof is obtained by using Theorem (39.12). P, 24

To find a simple characterization of some Desarguesian space other than the Minkowskian, hyperbolic and elliptic spaces (for instance through a metric property of their planes as in (24.7)). The following problem goes in the same direction (compare (14)):

To determine the straight spaces in which the loci a(r. x ) = a(s, x ) are flat for any two opposite rays r,s.

P, 24

If the dimension exceeds 2 and the spheres are differentiable the Desarguesian character follows as in the proof of (24.6). (25) P, 25

Is a straight plane Minkowskian when its limit circles are straight lines?

T , 26

The parallel axiom holds in a straight plane if the loci of the points equidistant to a given line consist of two straight lines.

(26)

There is a remarkable example of Funk [2] which shows that this is not necessarily the case when the distance is not symmetric, even if the plane is Desarguesian. P, 27

To find conditions (other than (27.16)), in particular conditions applying to an ordinary cylinder, under which a non-compact space has the property that every locally isometric mapping of the space on itself is a motion.

P, 31

Is an odd-dimensional G-space in which all geodesics are straight lines or great circles, necessarily straight, sphere-like or of the elliptic type ? The hermitian elliptic spaces discussed in Section 53 show that the answer would be negative for even dimensions greater than 2. If it should be positive in odd dimensions, a proof is a t present hopelessly difficult.

’

Owing t o the topological character of M. Morse’s methods a great number of his results on the calculus of variations in the large, see Morse [2], extend to G-spaces. Similarly the methods of Lustermik and Schuirelmann should prove applicable to G-spaces.

APPENDIX: PROBLEMS AND THEOREMS

406

P, 32 To extend the methods of the calculus of variations in the large t o G-spaces. P, 33

On a (two-dimensional) torus without conjugate points, is each non-closed geodesic everywhere dense ?

P, 33 To find necessary and sufficient conditions for curve systems on a cylinder or compact surfaces of higher genus to serve as geodesics in metrizations of the surfaces without conjugate points.

It was mentioned in the text that the results of Nielsen (Section 34) have been extended in many directions to more general Riemannian surfaces, principally by Morse and Hedlund. Much of this theory holds for G-surfaces. (32) P, 34

To extend toG-surfaces the work of Morse and Hedlund concerning geodesics on Riemannian surfaces. The methods of Morse [ l ] apply without change to G-surfaces. In fact. some proofs can be considerably simplified by extending the concept of axial motion to general G-spaces and applying the present methods. The central result in Morse [1] is the following: Let a Poincar6 model of a compact orientable G-surface (not necessarily without conjugate points) be defined as in Section 34. For any two distinct points of the unit circle there is at least one straight line in P with these points as endpoints, and a constant B exists such that any straight line g in P and the hyperbolic straight line gh with the same endpoints satisfy o h c s(fin p) and 0 c S ( g h The corresponding result for the torus is found in Hedlund [l] and be extended to G-surfaces. can - with the same simplifications as above

-

(33) P, 34

Does the Divergence Property hold in a straight space which is covering space of a compact space ?

(34) P, 30

Is a Hilbert geometry with negative curvature hyperbolic ?

(35) P, 37

Is the asymptote relation transitive or symmetric in a straight space with convex capsules ?

P, 40

In a G-space with non-positive curvature let 4 7 ) . y ( 7 ) represent geodesics which are asymptotic of some type, and are rays for T 2 0. Are these rays co-rays to each other ? Does the converse hold in spaces of finite connectivity ?

(37) T, 40

A cylinder with non-positive curvature is simply covered by a suitable family of straight lines. A cylinder with negative curvature either carries exactly one great circle g and is simply covered by the straight lines perpendicular t o g, or it contains no closed geodesic and its straight lines are all asymptotes t o each other, Busemann [5, p. 3001.

APPENDIX : PROBLEMS AND THEOREMS

(38)

407

P, 38-41 To investigate G-spaces with non-negative or positive curvature, where these concepts are defined analogously to (36.1,3). In particular, some of the results of E'reissmann [I] on Riemann spaces with positive curvature should prove extendable.

(39) P, 41

In an n-dimensional space with non-positive curvature and suitable differentiability properties, are J S ( p a)/, , (or even IS@, a)/,, - dn)u " ) and IK(p, u)I,,-l convex functions of u for 0 u 6 > 0 and 0 < [ < E'. This leads readily to p(6, ( ) I t + C-' (0) and p(d, E ) / [ - T(0) for 6 + 0 and € + 0 I t is easily seen that this is so, when the universal covering space i s straight, in particular therefore, if the curvature is negative. - While the present hook was in print,Kostant proved this conjecture for all Riemann spaccs. q is different from z because of (22.23). This follows from Theorem (52.9). This may also be proved without using Brouwer's theorem by referring to an elementary theorem on surfaces of the elliptic type, see (18) in t h e Appendix. Replacing the 6 in Theorem (51.2) by r1 amounts t o replacing the arclength on gt by a parameter proportional to the arclength. A detailed discussion of thehermitian metrics in Ca is found in Cartan [a]. The existence of such subgroups is wellknown (in standard terminology the rank of is a t least 1) and may be verified by forming the closure of any one-parameter subgroup. This group is abelian and hence the direct product of groups isomorphic to ID,. According to Tits [ I ] the assumption of finite dimensionality issuperfluous.

r,

+.


A Accumulation point, of set, 5 of sequence, 5 a-function, 131 Angle, concave, 274 convex, 274 straight, 274 Angular measure, 236, sec. 42 continuous, 278 non-extendable, 280 uniform at II. 293 Antiprojectivity, 381 Arc, 25 Arc length, 22 as parameter, 21 Area of triangle, 342, 354, 358 Archimedes, 156 Ascoli’s Theorem, 8, 1) Associated euclidean metric, 94 Asymptote, 60, 138. sec. 23 distinguished, 370 symmetry and transitivity of 139, 240, 281 of type c. 263 Asymptotic geodcsics. 263 Axial motion, 207, sec. 32. 254, 253 Axioms, for G-space. 37 hyperbolic axiom, 141 local axiom of Pasch, (10.3) of local prolongability, (7.4) parallel axiom, 141 I’asch’s Axiom, 50 of uniqueness of prolongation, (8.1) Axis, 207, 212, 227

Bernald, L., 270 Betweenness, 28 Birkhoff, G., 336 Bisector, 309 flat, sec. 46, sec. 47 Bisector theorem, 310 Blaschke, W., 93, 103 Bliss, G. A., 162 Blumenthal. L. M.. 407, 413 Bolyai, J., 307 Bolzano-Weierstrass Theorem, 1, 6 Borel, A., 302 Bounded set, 6 Brouwer, I.. E. J., 254, 365

C Calculus of variations, inverse problem of sec. 1 1 relations to, 161--163 Capsule, 243 convex, 235, 243-248 strictly convex, 236, 244-248 Cartan. E., 254. 258, 307, 334, 346, 347, 381, 382, 384, 391, 394 Cayley, A., 397 Cayley numbers, 384, 397-402 Cayley elliptic plane, 384 Center of curve, surface, 89 Central ray, 132 Characteristic, sec. 43 Closed set, 5 Closed geodesic, 45, 166, sec. 32 Closed limit, 10 Closure, f i Cohn-Vossen, S..235, 237. 293, $05, 296, 301, 305, 407 Cohn-Vossen’s Theorem, 288 Compact, sec. 2 Congruence, 15 Congruent triangles. 320

B Beclcenbach, E. F., 407 Behrend, F., 90 Beltrami‘s Theorem. (15.3),66, 85, sec. 15, 115 417

418

INDEX

Conjugate points on great circle 36 Continuity, 7 Convergence of, geodesics, 40, 48 motions, 17 sequences, 5 sets, 1, sec. 3 Convex curve on G-surface, 184-156 differentiable, 154 Convex function, 99 strictly, 99 strongly, 99 Convex set, in A", sec. 16 in C-space, 54. 117 strictly, 88, 117 Convex surface, 89 Convexity of spheres, (6.9), 115, sec. 20. 124 Co-ray, 116. 134 symmetry of, 136, 137 uniqueness, 136 Cosine inequality, 268 Countable, 413 note 3 to sec. 2 Covering space, CH IV, 171 Curvature. CH V negative, 235, 237 non-positive, 235, 237 in standard sense, 260 in two-dimensional Finsler space, 270 zero, 237, 261 Curve, sec. 5 geometric, 23

Distance, a t infinity, 131 for motions, 16 for sets, 11 Ilistinguished asymptotes and limit circles, 370 Divergence property, 230. 240 Domain invariance, 234 Douglas, J., 63

B Effective action, 368 Elementary spaces, 307 characterization of, 331, 337 local characterization of, 310, 335 Elementary sphere, 334 Ellipse in straight plane, 161 Ellipsoid, characterizations of, 90-93 Elliptic space 115,124, 125, 345, 375, 395 Elliptic type, space of the, 48, 124,125,199 Endpoint of axis, 225 Equicontinuity, 8 Equidistant loci, 107 Equivalent, curves, 24 topologically, 12 topological transformation groups, 396 Euclid, 309 Euclidean metric, (17.24 associated, 94 Excess, 278, sec. 43 ncgative, non-negative. 279, 286 positive, non-positive, 279, sec. 44 total, 299. sec. 44

D Dekker, 1). H., 413 Denumerable, 413 note 3 t o sec. 2 I)esargocs' Theorem, 63, 67 significance of, 80 Desarguesian spaces, CH 11, 65, sec. 13, sec. 14 Desargues property. 67 Dickson. L. E., 400 Dimension, 46 Dini, U., 328 Direction, 274 Distance, 1, sec. f continuity of, 4 in I?, I)

F Finite connectivity, 186 Finitely compact, 1, sec. 2 Finsler. P., 186 Finsler space, sec. 18. 162, 235, 370 defined, 84 Fixed point, 15 Flat, 64 Foot, 9 Forder, G. H., 307 Foundations of geometry, sec. 13 Fourth harmonic point, 68 Free homotopy, sec. 32 Free mobility, 336

419

INDEX

Freely rotatable. 278 Freudenthal, H., 397, 399, 400 Functions, convex, 99 peakless, 109 positive homogeneous, 99 strictly convex, 99 strictly peakless, 109 strongly convex, 99 Fundamental group, 165, 179, sec. 39 Fundamental set. 166, sec. 29 Funk, P., 405 Q

C-space, 37 C-surface, 212 Cans, D., 368 Gauss-Bonnet Theorem, 236, 282, 408 Geodesic@), 2, sec. 7, sec. 8, sec. 9 asymptotic, 263 closed, 45, 168. sec. 32 convergence of, 40, 48 existence of, 34 oriented, 138 representation of, 32 transitive. 166, sec. 34 uniqueness of, 38 without multiple points, 45 Geodesic curve, 189 Geodesic monogon, 210, sec. 4 Geometric curve, 23 Geometric geodesic, 32 Gleason, A. M., 369 Great circle, 35, (9.8) Green, I.. W., 183, 187 Groups of motions, sec. 4 transitive, 178, sections 51-55

H Hadamard, J.. 235, 236, 262, 263, 264 Hamel, G., 66, 67 Hamel’s Theorem, 74 Hausdorff, F., 1, 10, 14 Hausdorff’s limit. 1, 10 Hedlund, G. A., 182, 163, 187, 215, 406 Heine-Bore1 Theorem, 8 Helmholtz, H. von, 336 Helmholtz-Lie Problem, 307, sec. 48

Hermitian elliptic space, sec. 53 Hermitian hyperbolic space, sec. 53 Hessenberg, G . , 81 Hilbert, D., 83. 88, 105, 337 Hilbert’s geometry, sec. 18 Hjelmslev, J., 87 Homeomorphic, 7 Homogeneity of C-spaces, 49 Homotopy, 172 free, 204 to c along ( p , q), 210 Hopf. E., 167, 215 Hurewicz, W., 53 Hiirwita, A,. 398, 399 Hyperbola in straight plane, 181 Hyperbolic axiom, 141 Hyperbolic geometry. 376, see also elementary spaces Klein’s model, 105 E’ohcar6’s model, 223, 380

I Imbedding of straight desarguesian spaces 81 Intrinsically isometric, 390 Inverse curve, 26 Involutoric motion, 344, sec. 49 Isometric, 14 Isometry, sec. 4 J Jessen, B., 152

K Kelly, P. J., 404 KerbkjArt6, B. von, 370 Killing, W., 188, 180, 192 Klein bottle, 196 Koebe, P., 167 Kolmogoroff, A,, 272, 337

L Lambert quadrilateral, 321 Leibniz, G. W.,309 Length of curves, 2, sec. 5 additivity of, 20 semicontinuity of, 20

420

INDEX

Levi-Civita, 269 LHuilier, S., 341 Lie, S., 336 Lie over, 167, 177 Limit, for sets, 10 for motions, 17 Limit point, 5 Limit sphere, 102, 116, 132 interior and exterior of, 132 Lineal element, 44 Linear mapping of segments, 238 Lobachevsky, N. I., 307, 342 Local axiom of Pasch, (10.3) Local prolongability, (7.4) Locally, isometric spaces, sec. 27 euclidean surfaces, 193-196 hyperbolic surfaces, 196-199 Minkowskian spaces, sec. 30 spherical spaces, sec. 30 symmetric spaces, 346 Loewner, C., 90 Lower limit of sets, 10

M M-convex, 2, 8 Mapping, in, on, 7 Mayer, W.. 83, 403 Menger, K., 27, 29, 46 Metric space, 3 Metrics without conjugate points, 162 in plane, sec. 11 on torus, sec. 33 Metrization of surfaces, sec. 11, sec. 33, (39.14). (43.6), (43.11)

Minkowskian space, geometry, metric, 66, 84. sec. 17 defined, 94 characterizations of, 117, sec. 24, sec. 26, sec. 60, 364 Minkowski's inequality, 42 Mobility, free, 336 Moebius net, 70 Moebius strip, 187, 193 Monogon, geodesic, sec. 44, 210 Montgomery, D., 264, 369, 392, 396, 396, 403

Morse, M.,1, 162, 163, 167. 216, 406, 406

Mostow, G. D., 369, 370 Motions, 1, sec. 4 axial, 207, sec. 32, 262, 263 involutoric, 344, sec. 49 with maximal displacements, (32.6), 362 of Minkowski space, 96, 101 Multiple point, 44 Multiplicity, sec. 9 Myers, S. B., 394

N Neighborhood, 4 Nielsen, J., 167, 226, 231 Non-desarguesian systems, 63, 139, 222 Non-conjugacy hypothesis, 162 Non-extendability, 236, 280 Non-focality hypothesis, 163 0

Open set, I5 Orbit, 181 Orientation, of curves, 20 segments, 27 geodesics, 138

P Paige, L. J., 404 Pairwise transitive groups of motions, 308 sections, 63-66 Pappus' Theorem, 67 Parallel axiom, 116, 141 Parallel(s). 141 theory of, sec. 23, 116, sec. 37, 279-281 Pasch's Axiom, 60, 67 Peakless function, 109 Pedersen, F. P., 106, 270, 416 Permits rotation, 333 Perpendicular(s), 103. sec. 20, 121 asymmetry of, 103 Petty, C. M., 89, 369 Poincar6, H., 262 Poincare model, sec. 34 Pontrjagin, L., 166, 360 Positive homogeneous, 99 Preissmann, A., 236, 269, 260, 407 Product of G-spaces, 42 Product of motions, 1B

INDEX

Projective space, 74, 375 complex, 375 quaternion, 375 Prolongation, possibility of, 2, (7.4) uniqueness of, 2, (8.1) Properties A , B , 244

Q

Quasi-hyperbolic geometry, 308, 363, 407 characterization of, 371 Quaternion projective space, 375 Quaternion elliptic space, 383

R Ray, 116, 130, 138 Rectifiable, 19 curve in geometric sense, 24 reflection, 344 in lineal element, 347 Representation, of geodesic, 32 of segment, 27 Riemann, B., 336 Riemann space(s), sec. 15, 163, 235, 268, 269, 282, 293

Rotation, 276

S Samelson, H., 392, 396 Schur, F.. 334 Segment, 2, sec. 6 existence of, 29 uniqueness of. (8.6) Selection theorems, 8, 9 Separable, 6 Shortest join, 25, (5.18) Simple point, 44 Simply covered, 59 Simply transitive, 178 Spaces, G-, 2, sec. 8 abelian fundamental groups, with, 211,

421

Spaces, flat bisectors, with, 124, sec. 46, sec. 47 flat limit spheres, 148, 162 homogeneity of, 49 homogeneous, Ch VI locally isometric, sec. 27 negative curvature, with, sections 35-41

non-positive curvature, sections 35-41 one-dimensional, 46, (9.7) products of, 42 spherelike, 129 straight, 2, 38, sec. 11, sections 23-25, 162, sec. 37 straight universal covering space, with, ( 3 2 . 6 1 1 ) . (34.10), (34.14). sec. 38 symmetric, 345 two-dimensional. sec. 10 with unique geodesic through two points (gag), (13.1), (14.1), 166, sec. 31

Sphere, 25, 102 convex, strictly convex, 119, (20.9) differentiable, 147 permits rotation, 333 Spherelike space, 129, sec. 31 Standard representation, 21 Steenrod, N. E., 394 Straight line, 35, (9.6) Straight plane, 212 Strictly convex set in G-space, 117 Strictly peakless function, 109 Strongly convex, 99 Supporting line. 93, 118 Supporting plane, 87 Supporting ray, 119 Symmetric, 345 locally, 346 Symmetry of asymptotes, 139, 140. 249, 381. 405

259

contain planes, which, sec. 14 convex capsules, with, 235, sections 36-41

covering, Ch IV Desarguesian, Ch 11, 65, sec. 13, sec. 14 elliptic type, of the, 48, 124, 125, 199

T Tangent cone, 89 Tits, J., 308, 337, 385 Topological mapping, 7 Topologically equivalent metrics. 12

422 Torus, 104, 196, sec. 33, 216, 261 with hole, 198, 199 Total excess, 116, sec. 44 Transitive geodesics, 167, sec. 34 Transitive set of transformations, 178 simply, 178 Transitivity of asymptotes, 139, 140, 281, 249, 405 Translation(s), along line, 212-214, sec. 51 along great circle, 365 Transversal, 123 Transversals of a triangle, 339 Triangle area, 342 Triangle inequality, 3 Tube, 199, 287, sec. 43, sec. 44 bulging, 291 contracting, 290 expanding, 291

INDEX

u

Uniform at n, 293 Uniform continuity, 7 Uniqueness of prolongation, (8.1) Universal covering space, 165, sec. 28 Upper limit of sets, 10' Uryaohn, P. 40

V Volume, 271-273 Vaughan, H. E., 403

\v Wang, H. C., 308, 339, 386, 387, 396,400 Weyl, H., 336, 381 Whitehead, J. H., 162

T Yamabe, H., 402

x Zippin, L., 369, 395

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Automata, Languages and Machines. Volume B. (Pure and Applied Mathematics)

Topology (Pure & Applied Mathematics)

Categories and Functors (Pure and Applied Mathematics)

Entire Functions (Pure and Applied Mathematics (Academic Press), Volume 5)

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Set theory, Volume 79 (Pure and Applied Mathematics)

Automata, Languages and Machines. Volume B. (Pure & Applied Mathematics)

Functional Analysis (Pure and Applied Mathematics)

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Classical Complex Analysis (Pure and Applied Mathematics)

Homotopy Theory (Pure and Applied Mathematics 8)

Sobolev Spaces. Pure and applied Mathematics

Noncommutative Distributions (Pure and Applied Mathematics)

Linear Algebra. Pure and Applied Mathematics

Dynamic Programming (Pure and Applied Mathematics)

Complex Analysis (Pure and Applied Mathematics)

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