Projective Differential Geometry of Submanifolds (North-Holland Mathematical Library)

PROJECTIVE DIFFERENTIAL GEOMETRY OF SUBMANIFOLDS

North-Holland Mathematical Library Board of Advisory Editors:

M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. Hormander, J.H.B . Kemperman, H.A. Lauwerier, W.A.J. Luxemburg, L. Nachbin, F.P. Peterson, I.M. Singer and A.C. Zaanen

VOLUME 49

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

Projective Differential Geometry of Submanifolds

M.A. AKIVIS Moscow Institute of Steel and Alloys Department of Mathematics Moscow, Russia

V.V. GOLDBERG New Jersey Institute of Technology Department of Mathematics Newark, NJ, USA

1993

NORTH-HOLLAND AMSTERDAM LONDON NEW YORK TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands

L i b r a r y of C o n g r e s s Cataloging-in-Pub

lcation D a t a

Akivis. M. A . ( M a k s A i z i k o v i c h ) Projective dicserential geometry o f subman folds M.A. h k l v i s . V.V. Go1 'dberg. p. cm. -- ( N o r t h - H o l l a n d m a t h e m a T i C a l librz-) ; 49) Includes bibl.2graphiCal references.

ISBN 0-444-89771-2 1 . Subnanifalzs. I.G o l ' d b e r g . V. V . .

2. P r o l e c t i v e d i f f e r e n t i a l g e c - e t r y , ( V l a d i s l a v V i k t o r o v i c h ) 11. - t l e .

III. S e r i e s . 1993 5'6.3'62--0C20 ZA649.A38

93-10725

CIP

ISBN: 0 444 89771 2

0 1993 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper Printed in The Netherlands

V

Preface 1. The projective differential geometry of submanifolds of three-dimensional projective space was intensively developed during the first half of this century. Its main results were presented in the monographs [Wi 061 by Wilczynski, [Tz 241 by TzitzBica, [Ce 261 by Cech, [FC 261, [FC 311 by Fubini and Cech, [La 321, [La 421 by Lane, [Fi 371, [Fi 501, [Fi 561 by Finikov, [Bo 501 . by Bol, [H 451 by Hlavatf, [Mi 581 by Mihailescu, [Shc 601 by Shcherbakov, [KO631 by Kovantsov, [Go 641 by Godeaux, [Shu 641 by Shulikovskii, [Sv 651 by Svec, [Su 731 by Su Buchin, [Kf 801 by Kruglyakov, and [Sto 92a] and [Sto 92b] by Stolyarov. Many papers of E. Cartan were also devoted to projective differential geometry (see [Ca 161, [Ca 181, [Ca 191, [Ca 2Oa], [Ca 20b], [Ca Zqc], [Ca 20d], [Ca 241, [Ca 271, [Ca 311, [Ca 441 and [Ca 45bl). Moreover, E. Cartan was among the first geometers to study multidimensional projective differential geometry systematically. Following 8. Cartan, in the 1930’~-1940’s many geometers (Bol, Bompiani, Bortolotti, Brauner, Chern, Finikov, Fubini, Hlavatf, Kanitani, Muracchini, Norden, Vangeldkre, Villa, Vaona and others) also considered some problems in multidimensional projective differential geometry. During these years, many papers on multidimensional projective differential geometry were published in the Soviet Union. However, these papers remained unnoticed by western geometers. Many of these papers were authored by the participants in the seminar on classical differential geometry at the Moscow State University. The seminar was under the supervision of S.P. Finikov (1883-1964), G.F. Laptev (1911-1972) and A.M.Vasilyev (1923-1987). In 1978 S.S. Chern in his scientific autobiography [C 781 emphasized the importance of projective differential geometry. He wrote: “I wish to say that I believe that projective differential geometry will be of increasing importance. In several complex variables and in the transcendental theory of algebraic varieties the importance of the Kahler metric cannot be over-emphasized. On the other hand, projective properties are in the holomorphic category. They will appear when the problems involve, directly or indirectly, the linear subspaces or their generalizations.” Note that projective differential geometry is a basis for Euclidean and nonEuclidean differential geometries since metric properties of submanifolds of Euclidean and non-Euclidean spaces should only be added to their projective properties. This fact was noted by E. Cartan in his paper [Ca 191.

vi

PREFACE

In recent years the interest in multidimensional projective differential geometry increased again. Many interesting works devoted to different problems of multidimensional projective differential geometry were published (see [A 82~1, [A 83b], [A 84a], [A 84b], [A 881, [A 92a], [A 92b], [Cha 901, [Gr 741, [GH 791, [JM 921, [Kr 801, [P 901, [Sto 92a], [Sto 92b], [Sas 881, [Sas 911, [Wo 841, p a 851, Tya 921, etc.). However, there is as yet no book in which the multidimensional projective differential geometry has been systematically presented. The present book will fill the indicated gap in the literature on differential geometry. In particular, this book reflects the content of many papers by Soviet geometers in multidimensional projective differential geometry. In this book we give the foundations of the local projective differential geometry of submanifolds and present many results obtained after World War 11. In particular, we investigate here a series of special types of submanifolds with a “special proj,ective structure”. The problem of studying such submanifolds was posed by E. Cartan in [Ca 191. The authors of this book were very much influenced by the paper [GH 791 of Griffiths and Harris where the relationship between local differential geometry and algebraic geometry was stressed. In our exposition we emphasize a projective base of some problems which were usually considered in affine, Euclidean and non-Euclidean geometries. Among these problems are: the theory of conic conjugate nets (Section 3.6), the theory of parabolic submanifolds without singularities (Section 4.7), the construction of the generalized Koenigs nets on a hypersurface (Section 5.5), some aspects of the theory of normal connections on submanifolds (Sections 6.3 and 6.4), the projective interpretation of the notion of affine normal (Section 7.3) and hypersurfaces with the parallel second fundamental form (Section 7.2) and the projective theory of space hyperbands (Section 7.6). A series of other problems of this nature, for example, the projective interpretation of the Egorov transformations (see [A 84bl) is out of the scope of this book. As a rule, we use the index notations in our presentation. In our opinion, this allowed us to obtain a deeper understanding of the essence of problems of the local differential geometry. As a rule, we also do not distinguish between the presentation of material in a real domain and in a complex domain. This is the reason that in the book we use the notation GL(n) for the general linear group instead of GL(n,R) and GL(n, C) as is often done. However, in some places, namely in those where we are forced to solve algebraic equations or find intersections of algebraic images, the assumption that the objects under consideration be complex becomes essential. Note also that if we impose a restriction on a submanifold, then, as a rule, we assume that this condition holds at all points of this submanifold. More precisely, we consider only a domain on a submanifold where this restriction holds. 2. We will make a few general remarks for readers of this book. First of all,

PREFACE

vii

note that the book is intended for graduate students whose field is differential geometry, and for mathematicians and teachers conducting research in this subject. This book can also be used for a few special courses for graduate students in Mathematics. In our presentation of material we use the tensorial methods in combi?ation with the methods of exterior differential forms and moving frames of Elie Cartan. The reader is assumed to be familiar with these methods as well as with the basics of modern differential geometry. Many notions of differential geometry are explained briefly in the text and some are given without any explanation. As references, the books [KN 631, [St 641, [Ca 371, [CaH 671 and [BCGGG 911 are recommended. For Russian readers the books [A 771 and [Vas 871 can be also recommended. All functions, vector and tensor fields and differential forms are considered to be differentiable sufficiently many times. The book consists of eight chapters whose subjects are clear from the Table of Contents. Sections, formulas and figures are numbered within each chapter. Each chapter is accompanied by a set of notes with remarks of historical and bibliographical nature. A sufficiently complete bibliography, a list of notations and index are given at the end of the book. A large portion of the book was written during the summers of 1991 and 1992 in the Mathematisches Forschungsinstitut Oberwolfach (MFO), Germany. We express our deep gratitude to Professor Dr. M. Barner, the Director of MFO, for giving us the opportunity to work in MFO using its excellent conditions and facilities. We also are grateful to the Mathematics Departments of the Moscow Institute of Steel and Alloys, Russia, and New Jersey Institute of Technology, USA, where we are working, and t o the Department of Mathematics and Computer Science of Ben-Gurion University of the Negev for the assistance provided in the process of our writing the book. We express our sincere gratitude to G.R. Jensen, V.V. Konnov, J.B. Little and J . Vilms for reading some of the book chapters and making many useful suggestions, and to L.V. Goldstein for her invaluable assistance in preparing the manuscript for publication.

Moscow, Russia Livingston, New Jersey, USA

Maks A. Akivis Vladislav V. Goldberg

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ix

Table of Contents Preface

V

CHAPTER 1 Preliminaries 1.1 1.2 1.3 1.4 Notes

Vector Spaces . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . .. Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Algebraic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . .

. . . . . .. . . . . . ., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5 17 26 31

CHAPTER 2 The Foundations of Projective Differential Geometry of Submanifolds 2.1 2.2 2.3 2.4 2.5 2.6

Submanifolds in a Projective Space and Their Tangent Subspaces . . , . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Second Fundamental Form of a Submanifold . . . . . . . . . Osculating Subspaces and Fundamental Forms of Higher Orders of a Submanifold Asymptotic and Conjugate Directions of Different Orders .........,... on a Submanifold . . . . . . . . . . . Some Particular Cases and E Classification of Points of Submanifolds by Means of the Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . .

Notes . .

....

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

33 38 42 47 53 61 70

CHAPTER 3 Submanifolds Carrying a Net of Conjugate Lines 3.1 3.2 3.3 3.4 3.5

Basic Equations and General Properties . . . . . . . . . . . . . . . . . . The Holonomicity of the Conjugate Net Cz . . . . . . . . . . . . . . . Classification of Conj Some Existence Theorems . . . . . . The Laplace Transfor Generalizations . . . . . Conic rn-Conjugate S

3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 76 81 85 90 103 111

TABLE OF

X

CONTENTS

CHAPTER 4

Tangentially Degenerate Submanifolds Basic Notions and Equations . . .. . . . . . .. . . . . . . . . . .. . . . . . . . Focal Images . . . . . . . . . . . . . . . . . . . . . . . Decomposition of Focal Images . . . . . The Holonomicity of the Focal Net . . . . . . . . . . . . . . . . . . . . . . . Some Other Classes of Tangentially Degenerate Submanifolds . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manifolds of Hypercones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 4.7 Parabolic Submanifolds without Singularities in Euclidean and Non-Euclidean Spaces. . . . . . . . . . . . . . . .. . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5

113

122 126 130 132 141

CHAPTER 5

Submanifolds with Asymptotic and Conjugate Distributions 5.1 5.2 5.3 5.4 5.5

Distributions on Submanifolds of a Projective Space . . . . . . Asymptotic Distributions on Submanifolds . . . . . . . . . . . . . . . . Submanifolds with a Complete System of Asymptotic Distributions ............................................. Three-Dimensional Submanifolds Carrying a Net of Asymptotic Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Submanifolds with a Complete System of Conjugate Distributions .............................................

Notes .........................................

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

143 145 148 151 165 171

CHAPTER 6

Normalized Submanifolds in a Projective Space The Problem of Normalization of a Submanifold in a Projective Space . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Affine Connection on a Normalized Submanifold . . . . . . 6.3 The Connection in the Normal Bundle . . . . . . . . . . . . . . . . . . . . 6.4 Submanifolds with a Flat Normal Connection . . . . . . . . . . . . . 6.5 Intrinsic Normalization of Submanifolds . . . . . . . . . . . . . . . . . . 6.6 Normalization of Submanifolds Carrying a Conjugate Net of Lines . ......, ......... ....... ........ .... . ... . ... .. .. . . Notes . . . . . .. ... .. . . .. . . ....... .. . . . .. . . . . . . . . .. . . . . . ... .. .. . . . . . 6.1

173 178 182 188 192 199 205

TABLE OF CONTENTS

xi CHAPTER 7

P r o j e c t i v e Differential G e o m e t r y of Hypersurfaces 7.1 7.2 7.3

7.4 7.5

Basic Equations of the Theory of Hypersurfaces . . . . . . . . . . Osculating Hyperquadrics of a Hypersurface .............. Invariant Normalizations of a Hypersurface . . . . . . . . . . . . . . . The Rigidity Problem in a Projective Space ............. The Geometry of a Surface in Three-Dimensional Projective Space ......................................... The Geometry of Hyperbands ............................

7.6 Notes ...........................................................

209 216 222 234 245 255 266

CHAPTER 8 Algebraization Problems in P r o j e c t i v e Differential G e o m e t r y 8.1 8.2 8.3 8.4 Notes

The First Generalization of Reiss’ Theorem . . . . . . . . . . . . . . . The Second Generalization of Reiss’ Theorem ............. Degenerate Monge’s Varieties ............................. Submanifolds with Degenerate Bisecant Varieties . . . . . . . . .

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

Bibliography Symbols Frequently Used

Index

271 277 279 285 295

297 333 335

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1

Chapter 1

Preliminaries 1.1

Vector Space

1. In what follows, the notion of a finite-dimensional vector space L" over the field of real or complex numbers will play an important role. We will not state here the basic axioms and properties of a vector space- they can be found in any textbook on linear algebra. Note only that a frame (or a basis) of an n-dimensional vector space L" is a system consisting of n linearly independent vectors e1,e2,. . . , e n . A transition from one frame R = { e l , e 2 , . . . , e n } to another frame R' = { e ; , e)z, . . ., eh} is determined by the relation ei

= a: ej , i, j = 1,. . . , n,

(1.1)

where (a:) is a non-singular square matrix. (In these formulas as well as everywhere in the sequel it is understood that summation is carried out over the indices which appear twice: once above and once below.) Thus, the family R ( L " ) of frames in the space L" depends on n2 parameters. The transformations of type (1.1) of frames form a group which is isomorphic to the general linear group GL(n) of non-singular square matrices ( a : ) of order n. As was the case for the family R ( L " ) of frames, this group depends on n2 parameters. Let us fix a frame Ro, and let R, be an arbitrary frame in the space L", where a is a set of parameters determining the location of the frame R, relative to the frame Ro. Denote by S, a transformation of type (1.1) which sends the frame Ra to the frame R,:

R, = S,Ro.

( 14

Equation (1.1) implies that the transformation S, is a differentiable function (in fact, linear) of parameters a. Let Ra+dabe a frame near the frame R,. Then, the transition from the frame R, to the frame Ra+dais determined by the transformation S,+daS;l. Since S,S;l = I , where I is the identity

1. PRELIMINARIES

2

transformation of the space L", and S, is a differentiable function of a , the transformation S a + d a S l l can be represented in the form:

=I

Sa+daSyl

+ S, + o ( d a ) ,

(1.3)

where o ( d a ) are infinitesimals of higher orders than da. The transformation S, defining the principal linear part of transformation (1.3) is called an infinitesimal transformation of a frame in the space 15". The derivational formulas or the equations of infinitesimal displacement of a frame are equations of the form:

d R , = S,R,.

(1-4)

Since the frame R, in the space L" consists of n linearly independent vectors e i , in the coordinate form, formula (1.4) can be written as follows:

dei = w i ej ,

( 1 -5)

where w{ are linear differential forms which depend on the parameters a (defining the location of the frame R,) and their differentials da. Let us find the explicit expressions for the forms w i in terms of the parameters a: of the group GL(n). For the vectors ei and e: of the frames R, and Ra+da,we have: ei = a{ ej" , e: = ( a da o(da))! ej",

+ +

where ej" are the vectors of the fixed frame Ro. It follows that e! = Zi e j ,

where (Zj) is the inverse matrix of the matrix ( a ! ) . The last two formulas imply e: = ei

+ da:Z$ej + oi(da).

(1.6)

Comparing formulas (1.5) and (1.6),we obtain: w{ =;id.:

= -a?dEi,

(1.7)

or in matrix notation w = a-'da.

(1.8)

The matrix form w = ( w i ) is called an invariant linear form of the general linear group GL(n). 2. Let us find the law of transformation of the coordinates of a vector under transformations of a frame in the space L". Suppose we have two frames R and R' whose vectors are connected by relations (1.1). An arbitrary vector t can be represented in the form of linear combinations of the vectors of these two frames:

1.1

Vector Space

3

=

I

- xi ei.

Using formulas (1. 1), we find from (1.9) that

=

3 Izj, Izi

= z(zj, 3

(1.10)

where (Z!) is the inverse matrix of the matrix (a!). In what follows, it will be more convenient for us to replace equations (1.10) by equivalent differential equations. We assume that the vector x is unchanged under transformations of aframe, i.e. we assume that dz = 0. If we differentiate the first equation of (1.9) and apply formulas (1.5), we obtain: 0 = dz'ei

+ x'dei

= (dz'

+ zjwf)ei.

The linear independence of the vectors ei implies that dz'

.

+ z'uj. . = 0.

(1.11)

These equations are the desired differential equations which are equivalent to equations (1.10). The latter equations can be recovered by integrating equations (1.11). Next, let us find the differential equations for coordinates of a covector 4. It follows from the definition of a covector that its contraction &xi with coordipates I' of an arbitrary vector x is constant, i.e. this contraction does not de end on a choice of a frame: I!

tix" = const. Differentiating this relation and using formulas (1.1 1), we find that d(izi +&dzi = (d& - (jui)xi = 0 Since this holds for any vector xi, it follows that

dt; - t j w ; = 0.

(1.12)

Similar equations can be derived for a tensor of any type. For example, let us consider a tensor 1 of the type (1, 2) with components t i k . It follows from the definition of such a tensor that its contraction with coordinates z j , yk and & of arbitrary vectors x , y and an arbitrary covector ( does not depend on a choice of a frame: t j k x j y k t j = const. Differentiating this relation and using formulas (1.11) and (1.12), we find the differential equations which the components t i k of the tensor t satisfy: dt'.j k - t i Ik wj1 - t i .j 1 w k1 + t!j k u Ii- 0.

(1.13)

1. PRELIMINARIES

4

By integrating equations (1.12) and (1.13), we can get the laws of transformation of the coordinates of a covector & and the tensor t f k under transformation (1.11) of a frame:

= ajej,

'tik

=a ~ a ~ ~ ~ t ~ m ,

(1.14)

To simplify the form of equations ( l . l l ) , (1.12) and (1.13) and similar equations, it is convenient to introduce a differential operator V defined by the following formulas:

VZ' = dz'

+dwj,

V& = d& - Ejwj.

(1.15)

Using this operator, we can write equations ( l . l l ) , (1.12) and (1.13) in the form:

vz' = 0,

v(i 1 0 ,

V t j k = 0.

(1.16)

In addition to the vectors and tensors which were considered above and which were invariant under transformations of a frame, we will encounter objects that get multiplied by some number under transformations of a frame. This number depends on a choice of a basis and on some other factors. Such objects are called relative vectors and relative tensors. Their coordinates satisfy equations which are slightly different from equations (1.16). For example, for a relative tensor of type (1, 2), these equations have the form:

V t j k = 0tjkl

(1.17)

where 0 is a linear differential form. The following law of transformation: (1.18)

corresponds to equations (1.17). The simplest tensor is the tensor of type (0, 0) or an absolute invariant, i.e. a quantity K which does not depend on a choice of a frame. For this quantity, equation (1.17) becomes

dK = 0.

(1.19)

A relative invariant is a quantity K which is multiplied by a factor under transformations of a frame. For this quantity, equation (1.17) becomes

dK = 0K

(1.20)

1.2

1.2

Differentiable Manifolds

5

Differentiable Manifolds

1. The second basic notion which will be needed is the notion of a differentiable manifold. We give here only the main points of the definition. For more detail, we refer the reader to other books (see, for example, [KN 631 or [D 711). A neighborhood of any point of a differentiable manifold M is homeomorphic to an open simply connected domain of the coordinate space R" (or C" if the manifold M is complex). This allows us to introduce coordinates in the neighborhood of any point of the manifold. The number n is the dimension of the manifold M . If neighborhoods of two points of the manifold A4 have a non-empty intersection, then the two coordinate systems defined in this intersection are connected by means of invertible differentiable functions. The differentiability class of these functions is called the class of the differentiable manifold. Coordinates defined in a neighborhood of a point of a differentiable manifold admit invertible transformations of the same class of differentiability. In what follows, we wiIl assume the differentiable manifolds under considerations to be of class Coo,and in the complex case we will assume them to be analytic. Consider a point z of an n-dimensional differentiable manifold M". In a neighborhood of the point z,we introduce coordinates in such a way that the point c itself has zero coordinates. Let zi = z i ( t ) be a smooth curve passing through the point x. We parameterize this curve in so that x'(0) = 0. The = tiare called the coordinates of the tangent vector ( to the quantities t=O curve under consideration, a t the point c. The parametric equations of the curve can be written as z i ( t ) = tit oi(t), where o'(t) are infinitesimals of orders higher than t. The set of tangent vectors to all curves passing through a point x E M" forms an n-dimensional vector space. This space is called the tangent space to the manifold M" at the point x and is denoted by T,(M"). The set of all tangent spaces of the manifold M" is called its tangent bundle and is denoted by T ( M " ) . An element of the tangent bundle is a pair (z,t),where z E M" and E T,(M"). This explains why the tangent bundle is also a differentiable manifold of dimension 2n, dim T ( M " ) = 2n. Next, we consider the set of all possible frames R, = { e i } in each tangent space. This set can be viewed as a fiber of a fibration R ( M " ) which is called the frame bundle over the manifold M". Since the family of frames at a fixed point 2: depends on n2parameters, the dimension of the fiber bundle R ( M " ) is equal to n2 n: dim R ( M " ) = n2 n . Let t be a vector of the space T,(M") : t E T,(M"). The decomposition of this vector relative to the basis { e i } has the form:

+

+

+

t = ua(€)e i,

(1.21)

where wi((l) are the coordinates of the vector ( with respect to the basis { e i } . These coordinates are linear forms constituting a cobasis (a dual basis) in the

1. PRELIMINARIES

6

space T,(M"). This cobasis is a basis in the dual space T,'(M"). An element of the dual space is a linear form over T,(Mn). I t follows from formula (1.21) that (1.22) The set of spaces T,'(M") forms the cotangent bundle T * ( M " ) over the manifold M". Since every tangent space T,(M") is an n-dimensional vector space, we can consider tensors of different types in this space. A tensor field t ( x ) is a function that assigns to each point x E M" the value of the tensor t at this point. We will assume that the function t ( x ) is differentiable as many times as we need. In each space T, ( M " ) , the frames { e i } admit transformations whose differentials can be written in the form (1.5). Since further on we will also consider displacements of the point x along the manifold M " , we will rewrite formulas (1.5) in the form: 6ei = I( ej ,

(1.23)

where 6 denotes differentiation under condition that the point x is fixed, i.e. 6 is the restriction of the operator of differentiation d to the fiber I t , ( M " ) of the frame bundle R ( M " ) , and the forms 7.3 are invariant forms of the general linear group GL(n) of frame transformations in the space T,(Mn). Parameters defining the location of a frame in the space T,(M") are called secondary parameters, in contrast to principal parameters which define the location of the point x in the manifold M". This is the reason why the symbol 6 is called the operator of diflerentiation with respect t o the secondary parameters and the 1-forms ~j are called the secondary forms. If a tensor field is given on the manifold M " , then the coordinates of this field must satisfy equations of type (1.13) at any point of this field. For example, for the field t f , ( x ) , these equations have the form: 6ti. 3 , - t;,7r;

,+ t;,aj = 0 .

- ti.3 1 72

(1.24)

If, in accordance with formulas (1.15), we denote the left-hand side of this equation by Vat;,, then this equation takes the form:

vat;,= 0 .

(1.25)

2. Let M and N be two manifolds of dimension m and n respectively, and let f : A4 + N be a differentiable mapping M into N . Consider a point a f M , its image b = f(a) E N under the mapping f and coordinate neighborhoods U, and ub of the points a and b. The mapping f defines a correspondence yu

= f"(x",

i = 1,...,m , u = l , ..., n ,

between coordinates of points x E U, and y E ub. A mapping f is differentiable of class r , f E C', if and only if the functions f" are differentiable scalar

1.2

Differentiable Manifolds

7

functions of the same class. If the functions f" are infinitely differentiable functions, then the mapping f belongs to the class C", and if the the functions f" are analytic functions, then f E C w . Consider the matrix

(a)

M = 8Y" 2'

having n rows and m columns. This matrix is called the Jacobi rnatriz of the mapping f . It is obvious that the rank r of this matrix satisfies the condition

5 min(m, n).

r

It is also obvious that the rank r depends on a point z E U,. If the rank reaches its maximal value at a point z,i.e. r = min(m, n ) , then a mapping f is said to be nondegenerate at the point z,and the point t itself is called a regular point of a mapping f. If r < min(m, n ) at a point z , then the point z is called a singular point of a mapping f . The following relations can exist between the dimensions m and n: a) m < n. In this case a mapping f is called injective. In a neighborhood of a regular point a , the image f ( M ) = V of a manifold M is an mdimensional submanifold of the manifold N , and the point b = f ( a ) is a regular point of the submanifold V . Moreover, the tangent subspace Tb(v)at a regular point b is an m-dimensional submanifold of the tangent subspace T b ( N ) whose dimension is equal to n. In particular, as we indicated earlier, if m = 1, the submanifold V is a curve in N , and if m = n - 1, the submanifold V is a hypersurface in N . b) m > n. In this case a mapping f is called surjective. In a neighborhood U, of a regular point a , this mapping defines a foliation whose leaves Fy are the complete preimages f-'(y) of the points y E ub, where b = f ( a ) . The dimension of a leaf is equal to m - n, and the dimension of the subspace tangent to the leaf Fy is also m - n. If dim N = 1, then we may assume that N c R, and the leaves F,, are the level hypersurfaces of the function y

defining the mapping M

+

= f(z1,. . . ,P)

R.

c) m = n. In this case, in a neighborhood of a regular point a , a mapping f is bijective. The tangent subspaces T , ( M ) and T b ( N ) to the manifolds M and N at the points a and b are of the same dimension, and the mapping f defines a nondegenerate linear map f. : T , ( M ) + T b ( N ) with the matrix M . Note also that if m < n , in a neighborhood of a regular point a the correspondence between the manifolds M and f ( M ) is bijective.

1. PRELIMINARIES

8

3. Let xi be coordinates in a neighborhood of a point x of the manifold M" and let f ( x ) be a function defined in this neighborhood. Then the differential of this function can be written in the form: (1.26)

The latter expression is a linear differential form in a coordinate neighborhood of the manifold M " . However, this form is a form of special type since its coefficients are partial derivatives of the function f ( x ) . A linear differential form of general type can be written in the form:

8 = aidx'.

(1.27)

Its coefficients ai = q ( x ) are coordinates of a differentiable covector field which is defined on the manifold Mn. The set of all linear forms on the manifold M" is denoted by A ' ( M " ) . For the linear forms, the operations of addition and multiplication by a function can be defined in a natural way. In addition, for two linear forms 81 and 82, the operation of exterior multiplication 81 A 82 can be defined. This operation is linear with respect to each factor and is anti-commutative: 8 2 A01 = -81 A 82. The product 81 A 62 is an exterior quadratac form. The exterior quadratic forms of general type are obtained by means of linear combinations of the exterior products of linear forms. The linear operations can be defined in a natural way in the set of exterior quadratic forms, and this set is a module over the ring of smooth functions on the manifold M " . This module is denoted by A2(Mn) (see, for example, [KN 631, pp. 5-7). The localization of this module over each coordinate neighborhood U c M" is a free module with (;) = generators. At each point the exterior quadratic forms form a over the field of real or complex numbers. vector space A' of dimension In a similar manner, one can define the exterior differential forms of degree p , p 5 n on the manifold M " , and these forms generate a module AP(M") over the same ring. The localization of this module over each neighborhood U C M" is a free module of dimension The multiplication of exterior forms of different degrees can be also defined. If 81 and 82 are exterior forms of degrees p and q , respectively, then their exterior product 81 A 82 is an exterior form of degree p q. This product satisfies the following property:

(g).

+

el A e2 = ( - y e 2 A e l .

(1.28)

By the skew-symmetry, the exterior forms of degree greater than n vanish. The exterior forms of different degrees form the G r a s s m a n n algebra on the manifold M":

A = A'+ A' + A 2 + .

..+A";

(1.29)

1.2

Diflerentiable Manifolds

9

here AP is the module of exterior forms of degree p . In particular, A* is the ring of differentiable functions on the manifold M " . Exterior forms of degree p are also called p-forms, and 1-forms are also called the P f a f i a n f o r m s . We now consider an exterior differential form of degree two on a manifold M " . In terms of the coordinates x i ,this form can be written as 6' = a i j d z i A d z j , i , j = 1 , . . , , n, where aij = - a j i , and dzi A dzj are the basis 2-forms. A skew-symmetric bilinear form is associated with the form 8. This bilinear form is defined by the formula:

e(€l

V)=aijpq,

where 6 and q are vector fields defined in T(A4"). If these two vector fields satisfy the equation

6'(6, 17) = 0, then we say that they are i n involution with respect to the exterior quadratic form 6.' The notion of the value of an exterior pform on a system consisting of p vector fields given on the manifold M" can be defined in a similar manner. Note further the following proposition of algebraic nature, which is called the C a r t a n lemma:

Lemma 1.1 ( C a r t a n ) Suppose the linearly independent 1-forms w ' , w 2 , .. . , wp and the 1-forms 6'1 , 0 2 , . . . ,8, are connected by the relation:

el A w1 + . . . + 0,

A U P = 0.

(1.30)

Then the f o r m s 8, are linearly expressed in t e r m s of the f o r m s wa as follows: 8 , = labWb,

(1.31)

where

Proof. Since the forms w " , a = 1 , .. . , p , are linearly independent forms 1 , . . . , n we complete in a vector space T*, by adding the forms wc, F = p wl,.,,,wP t o a b a s i s f o r T ' . Then

+

ea = iabLJb + iaEwE. Substituting this into relation (1.30), we obtain

1. PRELIMINARIES

10

which implies la(

= 0 and lab = lba. 1

In the algebra of differential forms, another operation-the exterior diflerentiation can be defined. For functions, i.e. exterior forms of degree zero, this operation coincides with ordinary differentiation, and for exterior forms of type (1.33)

8 = adzi1 A , . . A dxiP,

this operation is defined by means of the formula:

d8 = da A d z i l A . . . A dxiP

(1.34)

The operation of exterior differentiation is a linear operation; it is a linear mapping of the space Ap(Mn) into the space AP+'(M"):

d : AP -+ AP+'.

(1.35)

Using formula (1.34), the formula for differentiation of a product of two exterior forms can be proved. Namely, if the forms 81 and 82 have degrees p and q , respectively, then

d(O1 A 82) = d81 A 82

+ (-1)P81

A d82.

(1.36)

In addition, the following formula holds:

d(d0) = 0

(1.37)

This formula is called the Poincare' lemma. In particular, for a function f on M" we have d ( d f ) = 0. Conversely, if w is an 1-form given in a simply connected domain of a manifold Mn and such that dw = 0, then w = d f . A 1-form w satisfying the condition dw = 0 is called closed, and a form w satisfying the condition w = df is called eruct. Note also that in fact, the operation of exterior differentiation defined by formula (1.34) by means of coordinates, does not depend on the choice of coordinates on the manifold M n ,i.e. this operation is invariant: it commutes with the operation of coordinate change on the manifold M " . 4. As an example, we will apply the operation of exterior differentiation to derive the structure equations of the general linear group GL(n). In subsection 1,invariant forms for this group were determined for the frame bundle R ( L " ) of a vector space L" and were written in the form (1.7) or in the matrix form (1.8). Applying exterior differentiation to equations (1.8) and using equations (1.34), we obtain

dw = da-' A da.

(1.38)

From relation (1.8) we find that

da = aw,

(1.39)

1.2

Diferentiable Manifolds

11

and since aa-I = I, we have

da-' = -u-'dua-'

= -wa-'.

(1.40)

Substituting expressions (1.39) and (1.40) into equation (1.38), we arrive at the equation dw

= -W A W .

(1.41)

In coordinate form, this equation is written as

= - w i A wj" ,

dwf

or, more often, as dwi

= wj" A w i .

(1.42)

Equations (1.41) and (1.42) are called the structure equations or the MaurerCartan equations of the general linear group GL(n). 5 . Suppose that a system of linearly independent 1-forms %",a = p 1 , . . ., n , is given on a manifold M". At each point x of the manifold M", this system determines a linear subspace As of the space T,(M") via the equations

+

SO({)

= 0.

(1.43)

The dimension of this subspace is equal to p . A set of such pdimensional subspaces A= given at every point I of the manifold M" is called a p-dimensional dtstribution and is denoted by Ap(Mn). An integral manifold of a system of Pfaffian equations %a

=o

(1.44)

is a submanifold Vg of dimension q , q 5 p , which, at any of its points x, is tangent to the element A$ of the distribution Ap(Mn) defined by the system of forms 8". It is easy to prove that the system (1.44) always possesses one-dimensional integral manifolds. If the system has integral manifolds of maximal possible dimension p which form a foliation on the manifold Mn, we will say that the system is completely integrable. This means that through any point x E M " , there passes a unique pdimensional integral manifold VP of the system (1.44). A necessary and sufficient condition for a system (1.44) to be completely integrable is given by the Frobenius theorem (see [KN 631, vol. 2, p. 323).

Theorem 1.2 (Frobenius) The system (1.44) is completely integrable if and only if the exterior differentials of the forms 8" vanish by means of the equations of this system.

1. PRELIMINARIES

12 Analytically this can be written as follows: doa = Bb AO;,

(1.45)

where 6; are some new 1-forms. Note that the structure equations (1.42)of the general linear group GL(n), which we found earlier, are conditions of complete integrability for the equations (1.5) defining the infinitesimal displacement of a frame of the space L". Note also that if a system of forms ui is given, and it depends on p 5 n2 parameters and satisfies structure equations (1.42), then by Frobenius' theorem, this system uniquely (up to a transformation of the general linear group GL(n)) determines a pparameter family of frames RP in the space L". 6. If the system (1.44) is not completely integrable, then it could still possess integral manifolds of dimension q < p . We will say that the system of Pfaffian equations (1.44)is i n involution if at least one two-dimensional integral manifold V 2 passes through each one-dimensional integral manifold V' of this system, at least one three-dimensional integral manifold V 3 passes through each of its two-dimensional integral manifold V 2 ,etc., and finally, at least one integral manifold Vq of dimension q passes through each integral manifold Vq-' of dimension q - 1. Later on we will often apply the C a r t a n test for the system of Pfaffian equations (1.44)to be in involution. To formulate the Cartan test, first of all note that if Vg is an integral manifold of system (1.44),then on this manifold not only the system (1.44) vanishes but also the system

d6" = O .

(1.46)

A q-dimensional subspace A$ tangent to the integral manifold Vq is characterized by the fact that each of its vectors satisfies each equation of system (1.44), and each pair of its vectors is in involution relative to the exterior quadratic forms d P , i.e. the pair satisfies the system (1.46). These vectors are called the one-dimensional integral elements of system (1.44),and the subspaces of dimension k 5 q spanned by some or all of these vectors are called the k-dimensional integral elements of system (1.44). Let €1 be a one-dimensional integral element of system (1.44). A twodimensional integral element passing through the element & is determined by a vector € 2 which, in addition to the system of equations (1.43),together with €1 satisfy the system: d@'(€i, t 2 ) = 0.

(1.47)

If the vector €1 is fixed, the system (1.47)is a linear homogeneous system for finding €2. Denote by PI the rank of this system. Suppose that €2 is a solution of system (1.47).The vectors €1 and €2 determine a two-dimensional integral element E2 of system (1.44). To find a three-dimensional integral element of this system, we should consider the system:

1.2

Differentiable Manifolds

13

dea(ti,t3)= 0, dea(t2,€3)

= 0.

(1.48)

Each vector (3 satisfying equations (1.48), together with the vectors & and €2 determines a three-dimensional integral element E3. Denote by 7-3 the rank of . . , Eq. They system (1.48). Similarly we can construct integral elements E4,. are by the relation: €1

= El C E2 c E3 c . . . C Eq.

Denote by r b the rank of the system of type (1.48) defining a vector t k + l , which is in involution with the previously defined vectors [ I , . . . ,( k , and let s1 = r l , s2

= r2 - r l , . . . , sg-l = rq-l - r q - 2 .

Let sq be the dimension of the subspace defined by a system of type (1.48) for finding a vector En. The integers s1, s2,.. . , sq are called the characters of system (1.44), and the integer

Q = SI

+ 2 ~ +2 . . . + q s ,

is called its C a r t a n number. The characters of the Pfaffian system (1.44) are connected by the inequalities: s1

2

s2

2 . . . 2 sq.

(1.49)

The left-hand sides of equations (1.46) are exterior products of some linear forms from which q forms are linearly independent and are the basis forms of the integral manifold Vq. Let us call these 1-forms w n , a = 1,.. .,q . In addition, the equations (1.46) contains forms wu whose number is equal to s1 s2 . . . s q . Applying the procedure outlined in the proof of the Cartan lemma, one can express the forms w" as linear combinations of the forms wa. The number of independent coefficients in these linear combinations is called the arbitrariness of the general integral element and is denoted by the letter N . We can now formulate the Cartan test.

+ + +

Theorem 1.3 (Cartan's t e s t ) For a s y s t e m of P f a f i a n equations (1.44) t o be in involution, it i s necessary and s u f i c i e n i that the condition Q = N holds. Moreover, i t s q-dimensional integral manifold Vq depends o n S k functions of k variables where

Sk

is the last nonvanishing character in sequence (1.49).

Note also that if the system (1.44) of Pfaffian equations is not in involution, this does not mean that this system has no solution. The further investigation of this system is connected with its successive differential prolongations. Moreover, it can be proved that after a finite number of prolongations one obtains either a system which is in involution-and in this case there exists a solution of system (1.44)-or he arrives at a contradiction which proves that the system has no solution.

1. PRELIMINARIES

14

The reader can find a more detailed exposition of the theory of systems of Pfaffian equations in involution in the books [BCGGG 911, [Ca 451, [Fi 481, [Gr 831 and [GJ 871. Examples of application of Cartan's test can be found further on. 7. Let us find the structure equations of a differentiable manifold Mn.As we have already noted, if a function f ( x ) is given on the manifold M " , then in local coordinates x i , the differential of this function can be written in form of differentiation with respect to the coordinates xi (1.26). The operators form a basis of the tangent space T,(M"), called the natural basis. We view the differentials dx' as the coordinates of a tangent vector d = &dxi with respect to this basis. If we replace the natural basis {-&} by an arbitrary basis { e i } of the space T,(M"):

&

(1.50) where ( x i ) and (Zi) are mutually inverse matrices, then we can expand the vector d as

. .

d = ejzidx' = wjejl

(1.51)

where we used the notation: wj

= p ,d c i ,

. .

Z,J=1,

...,n.

(1.52)

The forms wj are called the base forms of the manifold M". Taking exterior derivatives of equations (1.52), we obtain

dw' = dZi A d x j .

(1.53)

Eliminating the differentials dxj by means of relations (1.52) from equations (1.53), we arrive at the equations:

dw' = dZi A xjkwj.

(1.54)

dw' = wj A w i ,

(1.55)

Equation (1.54) implies that

where the forms wj are not uniquely defined. In fact, subtracting (1.54) from (1.55), we find that

w' A (wj + XjkdZ;) = 0 . Applying the Cartan lemma to these equations, we obtain the equations

01

1.2

Differentiable Manifolds

wJ(

15

= -xSdd;

+xfkwk,

(1.56)

where x j , = x i j . Equations (1.55) are the first set of structure equations of the manifold M " . By the F'robenius theorem, it follows from equations (1.55) that the system of equations w' = 0 is completely integrable. The first integrals of this system are the coordinates zi of a point x of the manifold M". Let us find the second set of the structure equations of the manifold M " , which are satisfied by the forms w j . Exterior differentiation of equations (1.55) leads to the equations:

dwf = -dxS A d56

+ dXjk A wk + "jkw'

A w]".

(1.57)

The entries of the matrices ( x i ) and ( Z i ) are connected by the relation:

x;z; = sj. If we differentiate this relation, we find that

dx! = - x i x i d d ! . Substituting these expressions for dx; into equations (1.57) and using relations (1.55), we find that

dwj = Wj" A W ;

+ ( V X f k + XpIxskw') A

Wk,

(1.58)

where V x ; , are defined according to the rule (1.15). Define also the 1-forms:

w f k = Vzfk

+ X P l X i k W ' + xik{w',

(1.59)

where x f k l = x f f k . Using these equations, we can write equations (1.58) as

dwj = W ; A W ;

+ W;

Awk.

(1.60)

These equations form the second set of structure equations of the manifold M n . Using the same procedure which we just used to define the forms w i , w j and w j k on the differentiable manifold M " , and to find structure equations for these forms, we can define higher-order forms wj,,, . . . and find structure equations for them (see [Lap 661). However, in this book we will not need these higher-order forms and equations. As we already noted above, the forms wi are basis forms of the manifold M " . The forms w j are the fiber forms of the bundle R1(Mn)of frames of first order over M " , and the forms w f k together with the forms w f are the fiber forms of the bundle R 2 ( M " ) of frames of second order over M " . The fibers 72: and Rz of these two fibrations are defined on the manifolds R i ( M " ) and R2(M")by the equations w i = 0.

1. PRELIMINARIES

16

We denote by 6 the restriction of the differential d to the fibers Rk and 77.2 of the frame bundles under consideration. Let us also denote the restrictions of the forms uf and wfk to these bundles by T; = uj(6) and xfk = w i k ( 6 ) respectively. Then it follows from equations (1.60) that b ~= f ~ j Ak T : .

(1.61)

Equations (1.61) coincide with the structure equations (1.42) of the general linear group GL(n). Thus, the forms 7rj are invariant forms of the group GL(n) of admissible transformations of the first order frames {ei} associated with the point x of the manifold M " , and the fiber 77.; is diffeomorphic to this group. This fiber is an orbit of a point of a representation space of the group GL(n). This and relations (1.5) show that if u i = 0, the vectors ei composing a frame in the space T,(M") satisfy the equations bei

= <ej,

and the forms u' composing a coframe satisfy the equations 6J = - T ?Ju j .

(1.62)

Next, consider the forms rjk= ujk(6).Relations (1.59) imply that T3.k

= Vax;k,

rlj.

and thus irjk = It is not so difficult to show that the forms irfk satisfy the following structure equations 67rjk

= T j k A Tf

+

Tj

A Tfk

+

T:

A

Tjl

(see [Lap 661) and that these forms together with the forms T; are invariant forms of the group GL2(n) of admissible transformations of the second order frames associated with the point x E M " . The group GL2(n) is diffeomorphic to the fiber 77.2. 8 . In what follows we will use the notion of an affine connection in a frame bundle. The forms wf in equations (1.55) and (1.60) are invariantly defined in the frame bundle R2(Mn) of second order. An affine connection y on a manifold M" is defined in the frame bundle R 2 ( M " ) by means of a n invariant horizontal distribution A which is defined by a system of Pfaffian forms

r!

9(; = w;' - ' w k Jk

(1.63)

vanishing on A . The distribution A is invariant with respect to the group of affine transformations acting in R 1 ( M n ) . Using equations (1.63), we eliminate the forms LJ; from equations (1.55). As a result, we obtain

1.3

Projective Space

17

dw' = w j A 0;

+ Rjkwj A u k ,

(1.64)

where Rfk = I'bkl. The condition for the distribution A to be invariant leads to the following equations:

The Pfaffian forms 0 = (6;) with their values in the Lie algebra gl(n) of the group GL(n) are called the connection forms of the connection y . The quantities Rjkand R;klform tensors which are called the torsion tensor and the curvature tensor of the connection y respectively. Conversely, one can prove that if in the frame bundle R 2 ( M " ) ,the forms 0j are given, and these forms together with the forms wi satisfy equations (1.64) and (1.65), then the forms 6; define an affine connection y on M " , and the tensors Rjk and R;,, are the torsion and curvature tensors of this connection 7. As a rule, in our considerations the torsion-free affine connections will arise for which Rik = 0. For these connections the form w = (wf)can be chosen as a connection form. Under this assumption, the structure equations (1.64) and (1.65) can be written in the form: dw' = w j A w i , dw;

= w;

Au;

+ Rik,wkA w l .

(1.66)

A more detailed presentation of the foundations of the theory of affine connections can be found in the books [KN 631 and [Lich 551 (see also the papers [Lap 661 and [Lap 691).

1.3

Projective Space

1. We will assume that the reader is familiar with the notions of projective plane and three-dimensional projective space. These notions can be generalized for the multidimensional case in a natural way (see [D 641). Consider an ( n i1)-dimensional vector space L " t l . Denote by L"ntl = L"+'\{O} the set of all nonzero vectors of the space L " t l . We will consider collinear vectors of L"tl to be equivalent %nd define the n-dimensional projective space P" as the quotient of the set Lntl by this equivalence relation. This means that a point of Pn is a collection of nonzero collinear vectors XE of L"tl, i.e. a point of P" is a one-dimensional subspace of Lntl (minus the origin). A straight line of P" is a two-dimensional subspace of Lntl, etc. If in L " t l a basis defined by the vectors eo, el . .,e n , is given, then any vector c # 0 of L"tl can be decomposed relative to this basis: } .

z = zoeg

+ riel + . . .+ znen,

1. PRELIMINARIES

18

where the numbers xo,x',.. ., x" are the coordinates of the vector z relative to the basis { e i } . In the space I,"+', a set of collinear vectors corresponds to the point x of P",and the coordinates of this set are the numbers (XzO, Xz', . . . , Ax"). These numbers are called the homogeneous coordinates of the point z E P". Note that they are unique up to multiplicative factor. Linear transformations of the space L"tl give rise to corresponding projective transformations of the space P". Under these transformations, straight lines are transformed into straight lines, planes into planes etc. Since a point in P" is defined by homogeneous coordinates, transformations of the form y"

=px',

u=0,1,

...,n ,

define the identity transformation of the space P". Thus, the projective transformations can be written as pyu = u ~ z " ,u , v = 0,1,.. ., n ,

where det(ut) # 0. Therefore, the group of projective transformations of the space P" depends on ( n 1)2- 1 = n2 2n parameters. This group is denoted by P G L ( n ) . A projective frame in the space P" is a system consisting of n 1 points A,, u = 0 , 1 , . . . , n , and the unity point E which are in general position. In the space Ln+', to the points A, there correspond linearly independent vectors e,, and the vector e = Cz,o e , corresponds to the point E . These vectors are defined in L"+l up to a common factor. It follows that the set of projective frames { A , } depends on n2 2n parameters. We will always assume below that the unity point E is given along with the basis points A,, although we might not mention it on every occasion. We will perform the linear operations on points of a projective space P" via the corresponding vectors in the space L"+l. These operations will be invariant in P" if we multiply all corresponding vectors in L"+l by a common factor. In some instances, we will assume that a vectorial frame in L"+l is normalized by the condition:

+

+

+

+

eo h e 1 A . . . A e , = 1,

(1.67)

where the wedge denotes the exterior product. Condition (1.67) can be always achieved by multiplying all vectors of the frame by an appropriate factor. When such a normalization has been done, the vectors of a frame in L"+l corresponding to the point of a projective frame { A , } are uniquely determined. Thus, the group of projective transformations of the space P" is isomorphic to the special linear group SL(n 1) of transformations of L"+l. Sometimes we will write the normalization condition (1.67) in the form:

+

Ao A A1 A . . . A A, = 1.

(1.68)

1.3

Projective Space

19

The equations of infinitesimal displacement of a frame in P" have the same form (1.5) as they had in L":

dA, = wiAU,

(1.69)

but now the indices u and v take the values from 0 to n, and by condition (1.68), the forms ui in equations (1.69) are connected by the relation:

w:+w:+ . . . + w,"= o .

(1.70)

This condition shows that the number of linearly independent forms wi becomes equal t o the number of parameters on which the group PGL(n) of projective transformations of the space P" depends. The structure equations of the space P" have the same form as they had in the space L":

dw,U = w,"Aw;,

(1.71)

but now we have a new range for the indices u, v and w: u, v , w = 0 , l . . . ,n. It is well-known that a projective space P" is a differentiable manifold. Let us show that equations (1.71) are a particular case of the structure equations (1.56) and (1.60) of a differentiable manifold. To show this, first of all, we write equations (1.71) for v = 0 and u = i , where i = 1,. . . ,n, in the form:

dub=w{ABj, where 0; = wj - 6;~:. These equations differ from equations (1.55) only in notation. Next, taking exterior derivatives of the forms 0; and applying equations (1.71), we find that

de; = ejk AS:

+ (6;~;+ 6jw,O)A w,".

Comparing these equations with equations (1.60), we observe that they coincide if Wjk = 6;w;

+ 6'wO 1 k'

The latter relations prove that if a first order frame is fixed, the second order frames of a projective space P" depend on n parameters while on a general differentiable manifold they depend on n3 parameters. Note that the forms wiconstitute a basis in the cotangent space T,*(P") of a projective space P". The corresponding basis in the tangent space T,(Pn) is formed by the vectors vi which are directed along the lines AoAi (see [GH 791). Consider a hyperplane ( in P". The equations of this hyperplane can be written in the form: ( u ~ u = O , u = O , I ,...,n,

(1.72)

1. PRELIMINARIES

20

where the coefficients t,, are defined up to a constant factor. These coefficients can be viewed as homogeneous coordinates of the hyperplane t. They are called the tangential coordinates of the hyperplane t. This consideration shows that the collection of hyperplanes of a projective space P" is a new projective space of the same dimension n. This space is denoted by P"* and is called the dual space of P". Let us denote the left-hand side of equation (1.72) by (t,z). Then the condition of the incidence of a point z with coordinates zu and a hyperplane [ = ((,) has the form:

+

In the space P"*, let us choose a coframe consisting of n 1 hyperplanes mu which are connected with the points of the frame { A , } by the following condition:

( a U , A , )= 6,".

(1.74)

This coframe is called dual to the frame { A , } . Condition (1.74) means that the hyperplane a" contains all points A , , v # u, and that the condition of A,) = 1 holds. normalization (au, We write the equations of infinitesimal displacement of the tangential frame {a,} in the form:

dm,=G,"a',

u , v = O , 1 , ..., n.

(1.75)

Differentiating relations (1.74) and using equations (1.69) and (1.74), we arrive at the equation w," +G,"

= 0,

from which it follows that equations (1.75) take the form:

da, = -w,"a'.

(1.76)

The structure equations (1.71) are the conditions for complete integrability of both equations (1.69) of infinitesimal displacement of a point frame and equations (1.76) of infinitesimal displacement of a tangential frame. Thus, if the forms w,"depending on some number p, p 5 n2 n parameters, are given, then, up to a projective transformation of the space P", they define in P" a family of projective frames, and this family depends on p parameters. The location of this family of frames is completely determined by the location of a frame corresponding to initial values of parameters. Conversely, if in P" a family of projective frames which depends on p parameters is given, then the components w: of infinitesimal displacement of this family are unchanged under its projective transformation. Hence, the forms w,"are invariant forms with respect to transformations of the projective group.

+

1.3

Projective Space

21

2. As w a s already noted in Preface, a projective space can be used to represent all classical homogeneous spaces: affine, Euclidean, non-Euclidean, conformal and other spaces. To do this, one fixes certain invariant objects in a projective space P" and reduces the group of transformations of the space by requiring that they be invariant. We will now show how this is carried out for the basic homogeneous spaces. An afJine space A" is a projective space P" in which a hyperplane (Y is fixed. This hyperplane is called the ideal hyperplane or the hyperplane a t infini t y (or the i m p r o p e r hyperplane). The afJine t r a n s f o r m a t i o n s are those projective transformations that transform this hyperplane into itself. Straight lines of the space P" which intersect the ideal hyperplane at the same point are called parallel straight lines of the space A". Two-dimensional planes of P" intersecting the ideal hyperplane along the same straight line are called parallel 2-planes of the space A", etc. As a frame in the space A", it is natural to take a projective frame whose points A], . . . ,A, lie in the ideal hyperplane a. The equations of infinitesimal displacement of such a frame have the form:

dAo = w;Ao+ dAi =

~bAi, w l A , , i , j = l , . . . ,n.

(1.77)

Equations (1.77) show that in this case the forms up in equations (1.69) are equal to zero: up = 0. This and structure equations (1.71) imply that du: = 0. Thus, the form u: is a total differential: u: = dln 1x1. Substituting this value of the form w: into the first equation of (1.77), we find that

It follows that

(1.78)

If we set (1.79) then equation (1.78) can be written as dz = w b e i .

(1.80)

Differentiating the second equation of (1.79), we obtain

dei = q e j , where

(1 3 1 )

1. PRELIMINARIES

22

We may consider the point z as the vertex of an affine frame and the vectors ei as its basis vectors. Equations (1.80) and (1.81) are the equations of infinitesimal displacement of this affine frame ( 2 ,e i } . These equations contain n n2 linearly independent forms w6 and 0;. This corresponds to the fact that the group of affine transformations of the space A" depends on n n2 parameters. The forms wh determine a parallel displacement of the frame, and the forms e;' determine the isotropy transformations of this frame which keep the point z invariant. The structure equations of the space A" can be obtained from equations (1.71). In fact, we derive from those equations that

+

+

d w ~ = w ~ , A w 6 + w i A =u w~ i A 0 ; , d0j = dwi = wj" A u i = 0j" A0:.

As a result, the structure equations of the affine space A" have the form: d0j = 0j" A$:.

dub = w i A \ ; ,

(1.82)

These equations imply that the isotropy transformations form an invariant subgroup in the group of affine transformations of the space A", and this subgroup is isomorphic to the general linear group GL(n). In addition, equations (1.82) imply that the parallel displacements form a subgroup which is not an invariant subgroup. A Euclidean space En is obtained from an affine space A" if in the ideal hyperplane of the latter space a nondegenerate imaginary quadric Q of dimension n - 2 is fixed. The equations of this quadric Q can be written in the form: n

(1.83) The Euclidean transformations are those affine transformations that transform this quadric into itself. The quadric Q allows us to define the scalar product ( a , b ) of vectors a and b in the Euclidean space En. If we take vectors of a frame in such a way that (ei,

e j ) = 6;j

(1.84)

(here bij is the Kronecker symbol: 6ii = 1 and 6 j j = 0 if i # j),then the forms 8; from equations (1.81) are connected by the relations:

ei+e+o,

(1.85)

which are obtained by differentiating equations (1.84). The number of independent forms in equations (1.80) and (1.81) is now equal to n ;n(n - 1) = an(n+ 1). This number coincides with the.number of parameters on which the group of motions of space En depends. The structure equations of the space En still have the form (1.82).

+

1.9

23

Projective Space

A non-Euclidean space is a projective space P" in which a nondegenerate invariant hyper quadri c

Q ( X , X ) = guuzUzU = 0, u , v = 0 , 1 , . . ., n ,

(1.86)

is given. Suppose for definiteness that a non-Euclidean space is elliptic, i.e. the hyperquadric Q ( X ,X ) is positive definite. Then we may choose the points of a projective frame { A , } in such a way that they form an autopolar simplex with respect to this hyperquadric, and in addition, we normalize the vertices of this simplex. This means that we have (13 7 ) and the forms w: from equations (1.69) will satisfy u; +u,"

= 0.

(1.88)

The elliptic transformations are those projective transformations of the space

P" that preserve the hyperquadric Q. These transformations depend on i n ( n + 1) parameters, and the latter number coincides with the number of independent forms among the forms w,", The geometry of a conformal space C" can be realized as the geometry on a nondegenerate real hyperquadric Q of a projective space P"+l of dimension n+ 1, and the group of conformal transformations of the space C" is isomorphic to the subgroup of the group of projective transformations of the space P" that keep the hyperquadric Q invariant. 3. In what follows, we will often use a special construction called the projectivization. Let P" be a projective space of dimension n, and let Pm be a subspace of dimension m, where 0 5 m < n. We say that two points c and y of P" are an the relation Pm and will write this as zPmy if there exists the straight line cy intersecting the subspace P m . It is easy to check that the introduced relation is an equivalence relation. Thus, the points which are in the relation Pm are called Pm-equivalent. This equivalence relation divides all points of the space P" into the equivalence classes in such a way that all points of an (m+1)-plane P"+' containing the subspace Pmbelong t o one class. The equivalence relation introduced above allows us to factorize the space P" by this relation. The resulting factor space P"/Pm is called the projectivization of P" with the center Pmand is denoted by Pm(P"):

Pm(P")= P " / P . The projectivization Pm(P")is a projective space of dimension n - m - 1: Pm(P")= pn-m-1 . Let us take a basis in P" in such a way that its points A ; , i = 0, 1, .~. , m, belong to the center Pmof projectiviz_ation. Then the basis of the space Pn-m-lis ' formed by the points P m ( A a )= A,, a = m t 1 , . . . , n. Since the center Pm is unchanged under projectivization, the equations of

1. PRELIMINARIES

24

infinitesimal displacement of the frame {Ai,A,} of the space P" can be written in the form:

dAi = W! A j , dA, = wiAp

+w ~ A , .

Thus, in this family of frames we have wg = 0. Hence, the structure equations (1.71) of a projective space P" imply that

d w i = w;Y,A w t .

(1.89)

This allows us to consider the forms w t as the components of infinitesimal of the projectivization Fin-,-' - P, ( P " ) , so displacement of the frame that

{xa}

xa

On some occasions we will identify the points of the projectivization P,(P") with the points A, of the projective space P". Note that one can also consider the projectivization of a vector space Lm by its 0-dimensional subspace (0). The result of this projectivization is Pm-' = Lm\{O}. Actually, in the definition of projective space Pn itself (see subsection 1.3.1), we already used the projectivization, so that P" = L"+'\{O}. 4. We consider a point 1: in the space P" and associate with this point a family of frames such that A0 = I. If the point z is fixed, then this family depends on n ( n 2) - n = n ( n 1) parameters. Analytically, this family of frames is defined by the equations

+

+

w; = 0 , i = l , , . . , n ,

(1.90)

and the equations of infinitesimal displacement of this family of frames have the form:

+

6Ao = T ; A O , SAi = TPAO $ A j ,

(1.91)

where 6 is the operator of differentiation with the point z fixed, i.e. when w: = 0, and (1.92) The equations (1.91) are the equations of infinitesimal displacement of a frame of the stationary subgroup of the point Ao, and the forms in these equations are invariant forms of this subgroup. Since by equations (1.71) we have

1.3

Projective Space

25

then by the Frobenius theorem, the system (1.90) is completely integrable, and its independent first integrals are the nonhomogeneous coordinates of the point 2 = Ao. Equations (1.90) define a subbundle of the bundle of frames {A,,} of the space P". The base of this subbundle is the space P" itself, and its fiber over a point c is the family of frames with A0 = z. This subbundle can be denoted by A

where

A

: R ( P " ) + P"

(1.93)

is given by

A { A Q , A .~. ., , A , } = Ao. The forms w6, i = 1,.. . , n , which for brevity we will denote simply by u * , are the base forms of this subbundle. These forms are also called the horizontal forms for the subbundle (1.93). The forms A:, $ and 4 in equations (1.91) are its fiber forms. In a similar manner, we can find the stationary subgroup of a hyperplane ( of P". We associate a family of frames with this hyperplane in such a way that the points A 1 , . . . , A , belong to the hyperplane ( and the normalization condition ( ( , A o )= 1 holds. Then the hyperplane ( coincides with the basis hyperplane cyo from the coframe { a " } . By means of equations (1.76), the condition for this hyperplane to be fixed is

wi0 = O , i = l , ..., n,

(1.94)

and the equations of infinitesimal displacement in the family of frames associated with the hyperplane ao are 6a0

= -7rta0,

6ai

= -.;a0

.

.

- A2.ff'. 3

(1.95)

The 1-forms on the right-hand sides of these equations are invariant forms of the stationary subgroup of hyperplane ao = (. The system of equations (1.94) is completely integrable and determines a suLbundle A*

: R ( P " * ) + P"'

(1.96)

in the bundle of coframes, where A*{

aO,al , . . . ,a n } = a o .

The forms up are the base forms of this subbundle, and the forms TO", A; and R: are its fiber forms. We further consider an rn-dimensional subspace P" of a projective space P" and find its stationary subgroup. We place the points A i l i = 0, 1, . . . , m, in this

1. PRELIMINARIES

26

subspace P"'. If the subspace P"' is fixed, then the equations of infinitesimal displacement of frames associated with this subspace have the form:

SAj = $Aj,

6Aa = x L A ~+ i ~ t A p ,

where i, j = 0, 1, . . .,m and a,p = m frames is defined by

(1.97)

+ 1,.. ., n. It follows that this family of

wg = 0.

(1.98)

The symbol 6 in equations (1.97) is the operator of differentiation when the subspace P" is fixed (i.e. when w g = 0), and X: = u:(S) = w," . Since the

1

+

Wp=O

number of forms wg is equal to ( m 1)(n - m ) , the stationary subgroup of the subspace P" depends on n(n 2) - (rn l)(n - m ) parameters. The complete integrability of equations (1.98) can be verified directly. The independent first integrals of this system are nonhomogeneous coordinates of the subspace P" in the space P". Their number is ( m 1)(n - m ) , and this number coincides with the dimension of the manifold of m-dimensional subspaces of P". This manifold is called the Grassmannian of m-dimensional subspaces in P" and is denoted by G ( m ,n). The system of equations (1.98) is completely integrable and defines a subbundle of the set R ( P " ) of frames in P". The forms wg are base forms of this subbundle, and the forms 4 , 7; and X; are its fiber forms. The subspace P" is the intersection of the hyperplanes a p , p = m 1,.. . , n, of the tangential frame. It follows from equations (1.76) that the invariance condition for the subspace P" leads to the same equations (1.98), and the stationary subgroup of the subspace P" can be written as

+

+

+

+

6aP

=

-7r!pS, a' = -*id J

X

p

.

(1.99)

This implies that the Grassmannian G(m,n ) of m-dimensional subspaces in a projective space P" can be also defined as the Grassmannian G(n - m, n ) of ( n - m)-dimensional subspaces in the dual space P"*.

1.4

Some Algebraic Manifolds

1. We now consider some algebraic varieties in a projective space, which we will need in our considerations. First of all, we will study in more detail the Grassmannian G(m, n) of mdimensional subspaces in a projective space P". Consider a fixed frame {E,} in P" and denote the coordinates of a point X relative to this frame by xu. Thus, we have X = x"E,. Let P" be an m-dimensional subspace in P". Let us take m 1 linearly independent points Xi, i = 0 , 1 , . . ., m, in the subspace P". We will call them basis points of the P"'. Let us write the coordinates of the points X i relative to the frame {E,} in the form of a matrix:

+

1.4

27

Some Algebraic Manifolds

... (1.100)

...

. . .

Consider the minors p * O ' l - . ' m of order m . . . P"""""

-

+

I'

+ 1 of this matrix:

x'o"

2;

XT

2;'

... Xb" ... x p

1x2

x$

...

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

1.

(1.101)

Xkl

Since the matrix has m 1 rows and n + 1 columns, the total number of such minors is equal to ($:). If we change the basis in the subspace P m , the matrix (1.100) will aiso change but all its minors will be multiplied by the same factor, namely the determinant of the basis change matrix. Thus, these minors can be taken its homogeneous projective coordinates of a point in ;1.)These coordinates are the projective space I" of dimension N = (i: called the Grassmann coordinates of the P m c P". It is easy to see that these coordinates are skew-symmetric, and that not proportional sets of Grassmann coordinates correspond to different subspaces. . m-dimensional . . The Grassmann coordinates p ' o ' l . . . ' m are not independent-they satisfy the sequence of quadratic relations: . . . p w l . . .t m- 1 L p i o j l . . .jml= 0, (1.102)

which follows from equations (1.100) and (1.101) (see, for example, [HP 471). In formulas (1.102) (and many other formulas of this book), the square brackets enclosing some (or all) upper (or lower) indices denote the alternation with respect to t,he enclosed indices while the parentheses in the indices denote the symmetrization. For example,

&I

1(+j- t j , i ) 2

= 12(+j+ t j , i )

-

t[ijk] = L ( t i j k +tiki +tkij t j i k - t k j i - tikkj), 3' t ( i j k ) = L ( t i j k + tjki +tkij +tjik +t k j i + t i k j ) . 3!

Relations (1.102) define in the space P N an algebraic variety of dimension (m+ 1)(n- m), which is the number of linearly independent basis forms w s on the Grassmannian. We will denote this algebraic variety by O ( m ,n). There is a one-to-one correspondence between m-dimensional subspaces P m of P" and the points of the variety O(m,n). This correspondence defines the mapping (o : G(m, n ) -+ O(m, n ) , called the Grassmann mapping. As an example, we consider the Grassmannian G( 1,3) - the manifold of straight lines of the three-dimensional projective space P 3 . In this case the matrix (1.100) takes the form:

1. PRELIMINARIES

28

Its minors

are usually called the Plucker coordinates of the straight line 1 defined by the points X O and X I . Since (); = 6, the minors are homogeneous projective coordinates of a point in the space P5.It is easy to prove that these coordinates satisfy the single quadratic equation: p01p23 + p02p31 + p 0 3 p 1 2

= 0.

Therefore, the variety R(1,3) is a hyperquadric in P5, called the Pllicker hyperquadric. Let us study the structure of the Grassmannian G(m,n) and its image R ( m , n ) in the space P N , where N = (i: -; 1.)Let p and q be two mdimensional subspaces of P" having in common an (m- 1)dimensionalsubspace p m - 1 . These two subspaces generate a linear pencil Xp pp of m-dimensional subspaces. A straight line of the variety O(m,n) corresponds to this pencil. All subspaces of the pencil belong to the same subspace Pm+' of dimension m + 1, and a pair of subspaces Pm-' c Pm+l completely defines the pencil and therefore a straight line on O(m,n). Consider further an (n - m)-bundle of m-dimensional subspaces passing through a fixed subspace Pm-'. An (n-m)-dimensional plane generator (n-m of the variety O ( m ,n) corresponds to this bundle. Since the space P" contains the m(n - m 1)-dimensional family of subspaces Pm-l,the variety O(m,n) carries a family of (n - m)-dimensional plane generators ("-", and the latter family depends on m(n - m 1) parameters. Let Pm+' be a fixed ( m 1)-dimensional subspace in P". Consider all its m-dimensional subspaces P". They form a plane field of dimension m 1. An ( m 1)-dimensional plane generator qm+' of the variety O ( m ,n ) corresponds to this field. Since Pn contains the ( m 2 ) ( n - m - 1)-parameter family of subspaces P""', the variety O ( m ,n) carries an (rn 2)(n - m - 1)-parameter family of plane generators qm+'. If Pm-' c Pm+l, then the plane generators ("-*and qm+' of the variety O(m,n) corresponding to these subspaces intersect each other along a straight line. Otherwise, they do not have common points. Next, consider in P" a fixed subspace Pm. It contains an m-parameter family of subspaces Pm-'. Thus, an m-parameter family of generators ("-" passes through the point p E O(m,n) corresponding to the Pm.There is also an ( n - m - 1)-parameter family of subspaces Pm+lpassing through the same subspace Pm.Thus, an (n-m-1)-parameter family of generators qm+' passes through the point p E O(m,n). Moreover, any two generators (n-m and qm+l passing through the point p have a straight line as their intersection. It follows that all plane generators (n-m and qm+' passing through the point p E O(rn, n) are generators of a cone with its vertex at the point p , and this cone is located

+

+

+

+

+

+

+

+

1.4

Some Algebraic Manifolds

29

+

on the variety film, n). We will denote this cone by Cp(n- rn, m 1) and call it the Segre cone. The projectivization of the Segre cone with the center at a point p is the Segre variety S ( n - m - 1, m) which we will study later. In the space P", to the Segre cone C,(n - m, m 1) there corresponds the set of all m-dimensional subspaces intersecting a fixed subspace Pm along the subspace of dimension m - 1. It follows that the dimension of the Segre cone C,(n - m, m 1) is equal to n. 2. The so-called determinant submanifolds are interesting examples of submanifolds in a projective space. Consider a projective space P N of dimension N = ml m 1 in which projective coordinates are matrices (xg) with i = 0 , 1 , . . ., m ; a = 0 , 1 , . . . , I , and we suppose m 5 1. A determinant manifold is defined by the condition

+

+

+ +

1 5 rank (zp)5 r , r

5 m.

(1.103)

Consider first the extreme case r = 1. In this case, the matrix (x?) has the form of a simple dyad:

xg = P s i ,

(1.104)

where t a and si are homogeneous parameters which can be taken as coordinates of points in the spaces P' and Pm*. The determinant manifold defined by equation (1.104) is called the Segre variety and is denoted by S(m,l). This variety carries two families of plane generators si = Aci and t" = pea where ci and ca are constants. The generators of these two families are of dimensions 1 and m , respectively. The Segre variety is an embedding

P' x Prn* + PN

(1.105)

of the direct product of the spaces P' and Pm* into the space P N , and the dimension of the Segre variety is I m. Suppose now that in relation (1.103) the rank r = 2. In this case the entries of the matrix (xg) can be written in the form:

+

xp = A / t a s ; + p

I1

t a S Ii/ ,

(1.106)

i.e. the matrix (zg) is a linear combination of two simple dyads. Each of these dyads determines a point on the Segre variety S(m, I ) . If the parameters X and p vary, the point of the space P N with coordinates xg describes a straight line -the bisecant of the Segre variety S(m,l). Thus, if r = 2, equation (1.103) defines the bisecant variety for the Segre variety S(m, 1 ) . Similarly, for any r , the determinant manifold (1.103) is a family of ( r - 1)secant subspaces for the Segre variety S(m, I ) . For example, if m = 1 = 1, then N = 3, and the equations of the Segre variety S(1,l)can be written as

1. PRELIMINARIES

30

( 1.107) Eliminating the parameters ta and si from these equations, we arrive at the quadratic equation:

z:z: - z;z: = 0,

(1.108)

defining in the space P3 a ruled surface of second order which carries two families of rectilinear generators: s; = const and ta = const. This surface is an embedding of the direct product P1 x P1* into the space P3. Next, we consider another type of determinant manifold defined in a projective space P N of dimension N = $(m l ) ( m 2) - 1, where projective coordinates are symmetric matrices ( z ' J ) ,i, j = 0,1, . . ., m, by the equation:

+

rank (z'j) = P, P 5 m.

+

(1,109)

If r = 1,then each entry of a matrix (z'j) is the tensorial square of a vector t': zij

= tit].

( 1,110)

The parameters ti can be considered as homogeneous coordinates of a point in a projective space Pm. Thus, the manifold defined by equations (1.110) is a symmetric embedding of the Pm into the P N :

v:Pm+PN. The manifold (1.110) is called the Veronese variety and is denoted by V ( m ) . Its dimension is m. If P > 1, the determinant manifold (1.109) is the variety of (P - 1)-secant subspaces for the Veronese variety V(m). As an example of the Veronese variety, we consider the case m = 2. Then N = 5, and the variety V(2) defined by equation (1.110) for i , j = 0 , 1 , 2 , is a symmetric embedding of the two-dimensional projective plane into the space P5.The variety V(2) is a two-dimensional surface of fourth order in P5 (see, for example, [SR 851). Note some properties of the Veronese surface V ( 2 ) .To each straight line of the plane P 2 there corresponds a curve of second order on the Veronese surface V(2), and this surface carries a two-parameter family of second order curves. Through each point of the surface V(2) there passes a one-parameter family of such curves, and through any pair of points of the surface V(2) there passes a unique curve of this family. Two-dimensional planes in P5 containing these curves are called conisecant planes of the surface V(2). To the second order curve defined by the equation

aijtv =0

(1.111)

Notes

31

in the plane P2,there corresponds a fourth order curve on the Veronese surface V(2). This curve is the intersection of the Veronese surface V(2) with the hyperplane '3 = 0. a..x'3

(1.1 12)

If the curve (1.111) is composed of two straight lines, then the corresponding fourth order curve is decomposed into two conics. For curves of this type, we have det(aij) = 0, and the hyperplane (1.112) defining this curve is tangent to the Veronese surface V(2) at a point of intersection of these two conics. If the curve (1.111) is a double straight line, then aij = aiaj, and the hyperplane (1.112) is tangent to the Veronese surface V(2) along a double conic. If T = 2, the manifold defined in the space P5 by equation (1,109) is a hypercubic defined by the equation

; :1

zoo

xol

XO2

I

$ :;; = 0,

xij = xji,

( 1 . 1 13)

and called the cubic symmetroid. This hypercubic is a bisecant variety for the Veronese surface V(2). It carries two families of two-dimensional plane generators. One of these families consists of conisecant planes of the surface V(2), and the second consists of two-dimensional planes tangent to this surface. The Veronese surface V(2) is the manifold of singular points of the cubic symmetroid (1.1 13).

NOTES 1.2. For more detail on differentiable manifolds see, for example, the books [KN 631 or [D 711 and on the theory of systems of Pfaffian equations in involution the books [BCGGG 911, [Ca 451, [Fi 481, [GI 831 and [GJ 871. A more detailed presentation of the foundations of the theory of affine connections can be found in the books [KN 631 and [Lich 551 (see also the papers [Lap 661 and [ L ~ 691). P 1.3, For more detail on the notion of a multidimensional projective space see the book [D 641 by DieudonnC and the paper [GH 791 by Griffiths and Harris. 1.4 On Grassmann coordinates see, for example, the book [HP 471. On Veronese variety see the book [SR 851. The embedding (1.110) generating the Veronese variety was considered in many papers and books from different points of view (see [CDK 701, [EH 871, [GH 781, [GH 791, [J 891, [LP 711, [Nom 761, [NY 741, [Sas 911, [SegC 2la], [SegC 21b], [SegC 221 and [Sev 011).

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33

Chapter 2

The Foundations of Projective Differential Geometry of Submanifolds 2.1

Submanifolds in a Projective Space and Their Tangent Subspaces

1. Let M be a connected m-dimensional differentiable manifold, and let f be a nondegenerate differentiable mapping (an immersion) of M into a projective space Pn:

f:M-+Pn where m < n. The image V" = f ( M ) of this mapping is called an mdimensional submanifold in the projective space Pn. If I t , i = 1 , . . . , m, are differentiable coordinates on the manifold M , then the submanifold Vm can be given locally by the equations I" = I"(+), u = 0,1,. . ., 12, (2.1) where ~ " ( t ' are ) differentiable functions of the variables ti and the rank of the is equal to m. Since I" are homogeneous coordinates of a point matrix I of the space Pn,the functions I" admit multiplication by a common factor. The submanifold V" can be also given locally by a system consisting of n - m independent equations of the form:

(g)

+

P ( z 0 , c 1 ,... ,cn) = 0, cy = m 1 , .. ., n, (2.2) where F" are homogeneous differentiable functions. In a neighborhood of a is of rank n - m. Hence we may nonsingular point c,the Jacobi matrix

(s)

2. THEFOUNDATIONS

34

assume that equations (2.2) can be solved for the variables xa:

za = x y x o , 21,.. . ,z") = 0,

0=

m

+ 1 , .. .

,TI.

(2.3) Here the right-hand sides are homogeneous functions of first degree. Therefore, these right-hand sides as well as the right-hand sides of equations (2.1) contain m essential variables which determine the location of a point on the submanifold V"'. If we set x C " / z= o ti,we reduce equations (2.3) to the form (2.1). In Section 1.4 we considered some algebraic submanifolds in a projective space. Certainly, those are differentiable manifolds. Moreover, equations (1.98) defining the image n(m,n)of the Grassmannian G(m,n) in the space P N , where N = - 1, are of form (2.2), and equations (1.100) and (1.106), defining the Segre and Veronese varieties respectively, are of form (2.1). However, the parameters in equations (1.100) and (1.106) are homogeneous while the parameters in equations (2.1) are nonhomogeneous. But as we indicated above for equation (2.3), in a neighborhood of a nonsingular point, it is easy to change homogeneous parameters for nonhomogeneous ones. 2. Let Vm be a differentiable submanifold in the projective space P" and let x be its nonsingular point. In what follows we will always assume that a point x E Vm under consideration is nonsingular without specifying this additionally. Consider all smooth curves passing through a point z E Vm. The tangent lines to these curves at the point x, lie in an m-dimensional subspace T,(Vm) of the space Pn,called the tangent subspace to the submanifold I/"' a2 the point x. For brevity, we will also use the symbol Ti') for the subspace Ts (V"' 1. We associate a family of moving frames { A , , } , u = 0 , 1 , . . . , n, with each point x E Vm, and we assume that for all these frames the point A0 coincides with the point c , and the points A i , i = 1 , . . . ,m,lie in the tangent subspace The frames of this family are called first order frames. Since the differential d z = dAo of the point 3: belongs to the tangent subspace Ti'), its decomposition with respect to the vertices of the frame {A,,} can be written

(:z)

7"'). as:

dAo = w,OAo + wAAj. (2.4) Thus, in the space P", the submanifold Vm along with the family of first order frames is defined by the following system of Pfaffian equations: wo* = 0, ff = m S 1 , ...,n , (2.5) and the forms wi in equation (2.4) are linearly independent and form a cobasis For brevity, we denote these forms by wc: in the tangent subspace

Ti').

wt,= w'.

By the structure equations (1.67) of a projective space P" and by equations (2.5), the exterior differentials of the forms wi can be written as

2.1

Submanifolds in a Projective Space and Their Tangent Subspaces

35

This implies that the 1-forms

0; = ui3 - 6iu0 3 0

(2.6) are the base forms of the frame bundle R ' ( M ) of first order frames on the manifold M of parameters of the submanifold Vm. The forms w i are the basis forms of the manifold M as well as of the submanifold V". By relation (1.62), if the point z is fixed, the forms u isatisfy the differential equations: 6 W ' + w j ( T ; . -6;7r;)=o, (2.7) where, as in Chapter 1, the symbol 6 denotes the restriction of the differential d to the fiber RL of the frame bundle R 1 ( M ) ,and T: = u,U(6). If the point z is fixed on the submanifold V", then the forms u ivanish: us = 0. In this case, the tangent subspace Ti1)is also fixed. Hence the formswq also vanish. Thus, if the point z is fixed, then the admissible transformations of the moving frames are determined by the following derivational equations:

+

6Ao = a,OAo, 6Ai = TPAO < A j ,

+

+

6A, = TZAO T L A ~ r t A p .

(2.8)

The 1-forms a:, r:, kA,,

.

(2.59)

Since the tensor b>k is symmetric in the lower indices, the maximal number of linearly independent points B i j k does not exceed the number im(m+l)(m+2). Denote by m2 the maximal number of linearly independent points Bijk. Then it is obvious that this number satisfies the inequalities:

-

0

1

5 m2 5 -6m ( m + l ) ( m + 2 ) ,

m2

5 n - m - mt.

(2.60)

The number m2 is equal to the maximal number of linearly independent forms @$) defined by (2.56) and coincides with the dimension of the bundle (2.58).

-

The points & j k define a subspace of dimension m2 - 1 in the normal space N i 2 ) . We denote this subspace by ELz)and call it the second reduced normal i ~the z )space ) P” is of dimension m subspace. The subspace P ~ ~ m l ( fof ml m2 and is the linear span of the subspace Ti2)and the points Bijk = bTkAP1.By equation (2.53), the subspace P,&l (EL2))is the linear span of three-dimensional osculating planes of all curves of the submanifold Vm passing through the point x. For this reason, this subspace is called the third osculatzng subspace of the submanifold Vm at the point x and is denoted by TL3). We place the points Am+,,,] +1 , . . . , Amfml tm2into the third osculating subspace Ti3). Then the cubic forms i z = m ml 1 , .. . , m ml m2, defined by formulas (2.56), become linearly independent, and the forms @(a3j, a2 = m+ml + m z + l , . . . , n ,vanish. This implies the following equations:

+

+

+ +

b8.2 = 0, ZJ k

@i2 (3)Ib5jkwiwjwk,

+ + (2.61)

aaz ( 3 ) - O. -

(2.63)

Since the rank of the rectangular matrix ( b f j ) is equal to m l , it follows from the second of equations (2.62) that UP= I 1 = 0.

(2.64)

These equations show that the differentials of the points A i l , belonging to the second osculating subspace Ti2)of the submanifold V m , are contained in its third osculating subspace Ti3).

2 . THEFOUNDATIONS

46

T h e construction of osculating subspaces of the submanifold V" can be continued. In this construction, the osculating subspace Tiq)of order p of the submanifold V" a t a point x is the linear span of osculating planes of order q of all curves of the submanifold V" passing through the point x. While doing this construction, we can encounter two possibilities: 1. The osculating subspace of a certain order p coincides with the ambient space P". Then the sequence of osculating subspaces of the submanifold V" has the form:

x = A0 E Ti1) c Ti2)c . . . c T$') = P" In this sequence, the dimension of each preceding subspace is less than the dimension of the following one.

2. Starting from some order p , the dimension of osculating subspaces is stabilized, i.e. T?") = T i p ) . In the latter case we can prove the following theorem. Theorem 2.1 If starting from some order p , the dimension of osculating subspaces at every point x € V" is stabilized, then the subspace T i p )is the same for all points x of the submanifold V", and this submanifold lies i n this fixed osculating subspace Tip), Proof. If we make the specialization of moving frames similar to the specializations which we made earlier for the subspaces Ti2)and Ti3),it follows is defined by the points Ao, Ai, A i l , . . . , Aip--l. that the osculating subspace The differentials of these points have the form:

Ti')

+ + +

dAo = w,OAo wbAi, dAi = UPAO w i A j wil Aj,, dAi, = wl",Ao wI,Aj wi:Aj, wi:Aj2, ......................................................... dAip--l= wiO,-, AO + w { ~ - ~ .A. .~+w{:I:Aj,,-l + ~ i : - , A j , , .

+

+

+

(2.65)

+

Since, by hypothesis, we have Tipt') = T i p ) the , last term in the latter equation of (2.65) vanishes, i.e. we have

Hence, the subspace Tip)is fixed, and the submanifold V" entirely lies in this subspace. In what follows, we will always assume that the ambient space P" coincides with the fixed osculating subspace Tip),i.e. that we are in the situation of the first case indicated above.

2.4

2.4

Asymptotic and Conjugate Directions of Different Orders

47

Asymptotic and Conjugate Directions of Different Orders on a Submanifold

1. A curve on a two-dimensional surface V 2 of a Euclidean space E3 is called asymptotic if its osculating planes coincide with the tangent planes to the surface V 2 or are undetermined (see for example, [BI 211, p. 52 or [Bl 501, p. 65). This definition is projectively invariant and can be generalized to the case where we have a submanifold of any dimension m in a projective space P". Namely, a curve 1 on a submanifold Vm is said to be asymptotic if its two-dimensional osculating plane at any of its points t belongs to the tangent subspace Ti1)to the submanifold Vm at this point or is undetermined. If a curve 1 is given on the submanifold V m by a parametric equation 2 = t ( t ) ,then its osculating plane is determined by the points z ( t ) , z'(t) and t"(t). But since z = Ao, this plane can also be defined by the points Ao,dAo and d2Ao. Since for an asymptotic line the second differential of its point belongs to the tangent subspace Ti1),it follows from equation (2.28) that on this curve we have O(2) = w * ~ r A= , 0,

(2.66)

i.e. the second fundamental form of the submanifold Vm vanishes on 1. In coordinate form, this condition can be written as follows: . .

bGu*uJ= 0 .

(2.67)

10, and proved that if V 3 carries 00' straight lines, then it is foliated by w1 of quadrics. In [M 581 he studied submanifolds V" c P", which are foliated by projective spaces, found the minimal value for n and established conditions for these submanifolds to be Segre varieties or their projections. The p-ruled submanifolds V 3 c P" were also studied by Severi in [Sev 011 ( p = co), Sisam in [Si 301 ( p = 6,n = 4), Blaschke and Bol in the monograph [BB 381, p. 138 and p. 204 ( p = 3 , n = 7), and by Baldasari in [Bal 501, who studied algebraic p-ruled V 3 . 2.6. For classification of points of two-dimensional submanifolds V 2 in Riemannian spaces Vn, see the book [SS 381 by Schouten and Struik. The results on classification of points of three-dimensional submanifolds V 3 in the projective space P" are due to Polovtseva (see [P 881). The case ml = 3 was considered in detail in her three recent papers [P 901, [P 91a], and [P 91bI. In these papers she studied the submanifolds V 3 with a six-dimensional osculating subspace and with 0, 1 or 2 families of two-dimensional asymptotic directions.

73

Chapter 3

Submanifolds Carrying a Net of Conjugate Lines 3.1

Basic Equations and General Properties

In this chapter we will study special classes of tangentially nondegenerate submanifolds Vm of a projective space P" which have m linearly independent mutually conjugate directions a t each point. For any pair of vectors & and (j determined by these directions, the following relation holds:

We now consider a subbundle of moving frames whose vertices Ai lie on the straight lines determined by the vectors indicated above. Then the vectors k also has a diagonal form, i.e. b>k = 0 if at least one pair of the indices i ,j and k consists of different indices. Hence the fundamental forms of third order can be reduced to the form: (3.13) i

It is obvious that this consideration can be applied to the fundamental forms of any order k. This implies the following theorem:

76

3.

SUBMANIFOLDS CARRYING A

NET OF

CONJUGATE

LINES

Theorem 3.2 If a submanifold Vm carrying a net C2 i s referred to this net, then its fundamental forms of any order k can be reduced t o the canonical form of t y p e (3.13), 2.e. they are sums of kth powers of the basis forms w i . I n a similar manner we can consider conjugate nets C k , k 2 3, of higher orders k. For example, an independent system of m directions ti is called conjugate of third order if any three of these directions, two of which can coincide, but each pair of these direcare conjugate with respect to all the forms @Ti), tions is not conjugate with respect to the forms that

@ti).Analytically, this means

where a t least one pair of indices i, j and k consists of different indices, but @{i)(€i,€j)

#0

for at least one pair ( i , j ) ,i # j. For a net CB the cubic forms can be reduced to the form (3.13), but the quadratic forms have general form. Applying the same method used for the nets C2,we can prove that for nets c k all fundamental forms of order greater than o r equal t o k can be reduced l o the sums of lth powers of the basis forms w', where 1 2 k. We will consider below only conjugate nets Ca of second order.

@&

3.2

The Holonomicity of the Conjugate Net Cz

Let a net C (not necessarily conjugate) of curves be given on a submanifold Vm. Suppose that the points Aj of moving frames associated with a point z E Vm are placed on the tangents to the net curves passing through the point t. If the point t = A Dis fixed, then the straight lines AoAi are also fixed, and if w' = 0, then equations (3.3) and (3.4) hold. I t follows from these equations that the 1-forms ~f , i # j, can be expressed linearly in terms of the basis forms w k of a submanifold Vm, i.e.

(3.14) In the tangent subspace Ti1) of a submanifold Vm carrying a net of curves, the points Ao, A l , . . ., A;-1, Ai+l, . . . ,A, define a n ( m - 1)-dimensional subspace A,(.). The field of these subspaces forms a distribution Ai on V"', which is given by the equation wi = 0, where i is a fixed index.

Definition 3.3 A net C on Vm is said to be holonomic if each of the m distributions Ai is involutive.

3.2

T h e Holonomicity of the Conjugate N e t &

77

This means that the subspaces A i ( z ) are tangent to ( m- 1)-dimensional submanifolds Yrn-', and the lines of the net C belong to these submanifolds. Each pair of families of lines of a net C defines a two-dimensional distribution A i j . If a net C is holonomic, all these distributions are also holonomic, since the systems of equations (3.15) w k = 0, k # i, j, defining these distributions are completely integrable. Each pair of families of lines of a net C also defines a curvilinear two-web in the sense of Blaschke (see [Bl 551, pp. 96-98). If the distributions A'' are ' any holonomic, then this two-web is quadrilateral (see [Bl 551, pp. 99-loo), 1.e. quadruple of curves of this web forms a closed quadrangle (see Figure 3.1).

Figure 3.1 As was indicated above, a distribution A, on Vm is defined by the equation:

w' = 0, i is fixed.

(3.16)

If the net C is holonomic, each of equations (3.16) must be completely integrable. Applying exterior differentiation to the forms ui and using equations (3.14), we obtain

+

dw' = C w J A w f = wi A (wi- l j i w J ) j

ljkwj A w k .

(3.17)

j,k#i

By the Frobenius theorem (see Theorem 1.2), equation (3.16) is completely integrable if and only if the last term in equation (3.17) vanishes, i.e. if (3.18) lbk1= 0, j,k # i. For fixed i, this condition is the same as the involutivity of the distribution Ai. If condition (3.18) holds for all values of i, then all distributions Ai are involutive, and the net C is holonomic. Thus, condition (3.18) i s necessary and suficient f o r an arbitrary net C on a submanifold Vm to be holonomic. Suppose now that C is a conjugate net Cz. Then each distribution A i ( z ) is conjugate to the direction AoA,. The curves of a conjugate net Cz form nets

on the submanifolds vim-*. In general, these nets are not conjugate.

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

78

Definition 3.4 If a conjugate net C z is holonomic and each of the nets determined on the submanifolds vim-' by the net Cz is conjugate itself, then the conjugate net C z is called an m-conjugate system. Let us find necessary and sufficient conditions for a conjugate net CZ on Vm to be an m-conjugate system. Take, for example, a submanifold V;"-' which is defined by the equations w1 = 0, W" = 0 , a = m+ 1 , . . . , n , in the space P". On this submanifold we have: dA0 = w ~ A o + w " A , , a = 2 ,

...,m,

and

+

-

d2Ao/TL1)(VF-')= ltbwawbA1 b t p " W b A m . Since a net Cz is conjugate on the submanifold Vm, we have b", = 0 for a # b. Hence, the net defined by the net C z on the submanifold is conjugate if and only if the conditions lAb = 0, a # b, hold. It follows that the conjugate net Cz is an m-conjugate system on the submanifold V m if and only if the following conditions hold:

Vim-'

lj,=O,

i#j,k, j f k .

(3.19)

Therefore, we proved the following result.

Theorem 3.5 A conjugate net CZ on a submanifold V m is holonomic if and only if conditions (3.18) are satisfied. This net is an m-conjugate system if and only i f conditions (3.19) hold. By (3.19), if a conjugate net C2 is an m-conjugate system, the forms wJ, defined by equations (3.14) for an arbitrary conjugate net Cz, take the form:

J3.= la.3' .wi + [I 33. w j .

(3.20)

We will now prove the following property of rn-conjugate systems. Theorem 3.6 If a submanifold Vm carries an rn-conjagale sysiem, then all its submanifolds Ym-' also carry (m - l)-conjugate systems. Proof. Consider, for example, the submanifold Vy-' which is defined by the equations w1 = 0 , w" = 0 , a = m + l , . . . , n.

On this submanifold, equations (3.5) take the form: w r = 0, w," = bzwa, a = 2 ,..., m. If we take i = 1 and j = a , then from equations (3.20) we find that w; = lfiOW".

3.2

T h e Holonomicity of the Conjugate Net C2

79

This implies that the submanifold V r - ' has the following second fundamental forms: the second fundamental forms (3.6) of Vm, which on the submanifold VY-l are expressed by the formula m

and one additional form

which is also a sum of squares. Thus, the submanifold VT-' carries a conjugate net. On the submanifold V Y - ' , equations (3.20) take the form:

w; = l&WQ + l ; p b , which proves that this conjugate net is an ( m - 1)-conjugate system. It follows from Theorem 3.6 that if on Vm we consider the submanifolds yrI:q defined by the equations

- 0, . . ., = 0, m - q families of curves wzl

w8g

then the net consisting of defined on each of the submanifolds yyI( by the m-conjugate net C2 is also an (rn - q)-conjugate system. The properties of an rn-conjugate net C2 indicated above are connected with the structure of the matrix B. To establish this connection, we find the exterior differentials of equation (3.5). This gives the following quadratic equations: (3.21)

where

Abq = dby

+ bfw;

- bp(2wi - w : ) ,

and as we indicated above, there is no summation over the index i. Substituting the expressions of the forms w j from equations (3.14) into equations (3.21), we obtain

where

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

80

Since this relation must be satisfied identically, the alternated coefficients in w k A w j must vanish, i.e. we have

ba(l? 8 3 k - l iJ . )+ bcl! 3 s k - brlfj = 0 , k,j

# i.

(3.23)

Since the quantities b r define the points Bi (see (3.8)) in the subspace Ti2’, by contracting equations (3.23) with the points &, we find that

-

-rij)

&($k

-

-

+ Bjl:, - BkG = 0.

(3.24)

We can now prove the following result.

Theorem 3.7 Suppose that a submanifold Vm carries a conjugate net C2 and that one of the following two conditions is satisfied:

-

a) A n y three of the points Ba are linearly independent. b) The net C2 is holonomic on Vm and any two points linearly independent.

g; and g,, j # i,

are

Then the net C:! is an m-conjugate system. Proof. a) Since the points g,, gj and i i k for distinct i , j and k are linearly independent, equation (3.24) implies equation (3.19). b) The holonomicity of the net Cz implies the equations l i k = l i j . Hence equation (3.24) takes the form:

Since the points (3.19).

5, and k k

are linear independent, equation (3.25) again yields

I t is interesting to find submanifolds Vm that carry a nonholonomic conjugate net Cz. To this end we give the following definition.

Definition 3.8 A conjugate net C2 is said to be irreducible if each distribution Aaj is not involutive. Since the distribution Aij is defined by equations (3.15) and

dwk

211jlwi A d

(mod u k ) , k

# i, j ,

the conditions for a net Cz to be irreducihle are given by the following inequalities: l k13.

# I!.3 %

which must hold for any three different values i , j and k.

(3.25)

3.9

Classification of the Conjugate Nets C2

81

Theorem 3.9 Only submanifolds V"', whose osculating subspaces Ti2)are of dimension m + 1 or m + 2, can carry an irreducible net Cz.

Ti2)

Proof. In fact, the subspace is defined by the points Ao, A, and Bj. It follows from equations (3.24) that condition (3.25) can hold only if the dimento which the points gi = Bi/Ti2) belong, is sion of the normal subspace equal to o or I , i.e. i f m l = 1 or 2. W As we saw in Section 2.5, if ml = 1, a submanifold V"' carries a set of conjugate nets. On the other hand, if ml = 2, then the matrix B (see (3.10)) has the form:

fii'),

(3.26)

Thus the points g;lie on a straight line. If any two pairs of these points are distinct, then matrix (3.26) does not have proportional columns, and the submanifold Vm carries a unique conjugate net which is in general nonholonomic.

Classification of the Conjugate Nets

3.3

C2

1. Consider the points

g;

are the columns of the matrix B , the Since the coordinates of the points number of linearly independent points is equal to the rank of this matrix B , i.e. this number is equal to ml. These independent points generate the reduced normal subspace Gill, whose dimension is equal to ml - 1. We can now prove the result generalizing the Segre theorem which was proved in Section 2.5.

Theorem 3.10 Suppose that a submanifold Vm carries a net of conjugate lines and that at any point of Vm the following two conditions hold: a) ml

< m, and

b) Any subset of the points gj consisting of m - 1 points has rank ml and generates the subspace Si').

Then the submanifold V"' belongs t o its osculating subspace Ti') of dimension m

+

m1.

Proof. Consider equation (2.27) for j = i bi: = bi', equation (2.27) takes the form

bi'cyl'k = 0 , i

# k. Since b:;

# k.

= 0 if i

#L

and

(3.28)

82

3. SUBMANIFOLDS CARRYING A NET OF C O N J U G A T E LINES

By the condition b) of the theorem, there are ml linearly independent points among the points Bi, i # k. Thus, if i # k, then rank@:') = ml. This implies that the system (3.28) has only the trivial solution c::~ = 0. It follows that wrp1,' = 0.

(3.29)

Taking into account equations (3.29), we can write the equations of infinitesimal displacement of the moving frame in the following form:

+ +

dAo = W ~ A O d Ai, dAi = w;Ao W: Aj w:' A;,, dAj, = wf1Ao+ W ; ' A+ ~~:iAj,.

+

These equations show that the second osculating subspace Ti2) of the submanifold V" remains constant. Thus, the submanifold V" belongs to this osculating suhspace. The generalized Segre theorem was first proved by Bazylev (see [Ba SSa] and [Ba 55bl) and Akivis (see [A 61al). 2. Suppose now that ml = m. Then any subset of the points Bi consisting of m-1 points has rank m- 1. In this case dim Ti2)= 2m, and the submanifold V" has precisely m independent second fundamental forms Consider the subbundle of moving frames whose vertices Am+; belong to the osculating 2-plane of the curve Ci c C2 defined by the equations uj = 0, j # i. The matrix (3.10) becomes a square matrix, and for the frames from the subbundle indicated above, this matrix is diagonal. If we appropriately normalize the points Am+d,this matrix will be the identity matrix:

-

@;Ti.

1

0

... 0

1)

(3.30)

............ B = ( o0 0 .' . .

This implies that equations (3.11) take the following form: (3.31)

us' = 0, a1 = 2 m + 1,. . . , n .

(3.32)

The second fundamental forms of V" take the form:

am+; = (w'))", (2)

= 0, a1 = 2m

+ I , . . . ,n.

(3.33)

Submanifolds of this type were first considered by 8. Cartan in his paper [Ca 191. This is the reason that these submanifolds are called Cartan varieties. Proposition 3.11 A Cartan variety is an m-conjugate system.

3.3

Classification of the Conjugate Nets Cz

83

Proof. In fact, exterior differentiation of equations (3.31) leads to the following exterior quadratic equations:

If we apply Cartan's lemma to these equations, we get equations (3.20), which imply conditions (3.19). By Theorem 3.5, the submanifold Vm under consideration is an m-conjugate system. Note that the converse is not true, since for an m-conjugate system the number m l is not necessarily equal to m: ml 5 m. can be reduced to the form (3.13), the number m2 of Since the forms linearly independent forms (P$) does not exceed m: m2 5 m. This implies that

@GI

dim TL3) 5 3m. If ma = m, then we consider the subbundle of moving frames whose ver+ ~ to the osculating three-plane of the curve Ci C C2. By tices A z ~ belong to appropriately normalizing the points Azrn+i, we can reduce the forms the form: @m+i (3)

= (W')',

= 0,

a2

= 3m + I , . . . , n .

(3.34)

3. Suppose further that ml < m, but the set of points & has the property that among its subsets consisting of m- 1 points there are both subsets of rank rnl and subsets of rank less than m l . We will now prove the following result for submanifolds Vm carrying a conjugate net Cz satisfying the above condition.

Theorem 3.12 Suppose that a submanifold V m carries a conjugate net Cz and that a2 any point of Vm the following four conditions hold: a) ml

< m.

The subsets

- -

- -

- - . . , B,, B,+l,. . . , B1-1, B I + ~. .,. , B,

b) The subsets B1,. . . , Bl-1, B1+1,. . . , B,, B,+1,. . . , B , are of rank ml - 1 c)

B1,.

d) N o two points

B,

and

B,,

u

# v,

are of rank m l .

coinczde.

Then the submanifold V m is doubly foliated: Vm is an (m-q)-parameter family of the Cartan varieties Vq of dimension q , and Vm is a q-parameter family of submanifolds Vm-q of dimension m - q which belong t o their second osculating of dimension m ml - 2q. subspaces

Ti2)

+

Proof. By the condition b) of the theorem, the points B"1, . . . , B", are linearly independent, and they are not linearly dependent on the points 8",+1,. . . , B,

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

84

which determine a subspace L of dimension r n l - q - 1. Therefore, we can into a basis of the normal subspace fi:" by include the points 51,.. . , setting

zg

-

and place the points Am+g+l,. . ., Am+mlinto the subspace L . As a result of Am+l = B 1 , . . . , A m + q = B q ,

this specialization of moving frames, the matrix (3.10) is reduced to the form: 1 . . . 0

0

...

0

B = [ ................................. 0 . . . 1 0 ... 0

(3.35)

'.. 0 bm+g+l g+l ... b g + 9 + 1 ................................. 0 ... 0 pg+l t m l ... bgtmi 0

Moreover, the matrix

(by$+' .

...

bz+'?+l

~. . . . ~ . ... . ~ .. .. .. .

bm+ml

.,.

bz-kmi

has rank ml - q , and this rank is not decreased if we delete any column of this matrix. Suppose that a, b = 1 , . . . , q and u = q 1,. . . ,m. Then (3.24) for i = u , j = b and k = u takes the form:

+

B"a(I:u

- I:b)

-

gbl:u - B,I,Ub = 0.

From the form of matrix (3.35) it follows that the points are linearly independent. This implies

(3.36)

&,

B"b,

a

# b, and B",

l,Ub = 0.

(3.37)

Since dw" f l ~ , w " A w b

(mod w " ) ,

it follows from equation (3.37) that the system w" = 0 is completely integrable. This proves that the submanifold Vm under consideration is foliated by an ( m - q)-parameter family of submanifolds Vg of dimension q . By (3.35) and (3.37), the second fundamental forms of the submanifolds Vg have the form: @m+a (2)

= (wa)2,

'p$"

+

= 0,

q;)= 0 ,

(2)

+ +

- 0,

+

where a = 1 , . . . , q ; u = q l , . . . , m rnl and 1, ..., rn; u1 = m q a 1 = rn rnl+ 1,. . . ,n. Thus the dimension of the osculating space of each of the submanifolds V *is equal to 29, and these submanifolds are Cartan varieties. Next, we write system (3.24) for i = u , j = v , u # v and k = a:

+

3.4

Some Existence Theorems

-

a5

+

-

B,(l,U, - lzu) B,,l;" - gal:, = 0.

(3.38)

From the form of the matrix (3.35) and the condition d) of the theorem, it follows that the points &, E,,,u # v , and 5" are linearly independent. This and equation (3.38) yield 1:" = 0.

(3.39)

Since (mod w b ) ,

dw" E 1&wu A w"

it follows from equation (3.38) that the system w" = 0 is completely integrable. This proves that the submanifold Vm under consideration is foliated by a qparameter family of submanifolds Vm-q of dimension m - q . By (3.35) and (3.38), only the following second fundamental forms of submanifolds Vm-q do not vanish: GU1 (2)

b;'(w")',

-

u

=q

+ 1 , ...,m;ul = m +q+

1 , . . . , m + ml.

U

Since all these forms are sums of squares, each of the submanifolds Vm-q carries a conjugate net. But by condition c) of the theorem, the hypotheses of the generalized Segre theorem (Theorem 3.10) are satisfied for each of the submanifolds V"'-q. Thus, each of these submanifolds belongs to its second of dimension m ml - 2q. osculating subspaces Ti2)

+

3.4

Some Existence Theorems

1. First of all, we prove an existence theorem for submanifolds V"' carrying a conjugate system C z . Theorem 3.13 Submanifolds Vm carrying a conjugate system Cz exist, and the solution of the system defining such submanifolds depends on m(m - 1) arbitrary functions of two variables. Proof. As we saw earlier, the system of Pfaffian equations defining an mconjugate system Cz consists of equations (2.5), (3.5) and (3.20): WQ

= 0,

(3.40)

3 . SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

86

By (3.41), exterior differentiation of equations (3.40) leads t o identities. By (3.22) and (3.42), exterior differentiation of equations (3.41) gives the following exterior quadratic equations:

Abq A W' = 0,

(3.43)

where

Abtp = dbq

+ bp(wi - 2wi) + C(lijbr+ lLbt)wk + be,;. k#i

Finally, applying exterior differentiation to equation (3.42), we obtain:

A& Aw'

+ Alij Aw3 = 0,

(3.44)

where

+ w j ) + bSw& + lii

Alii = dlii + l{j(w: - 2wf

c

+

k#j

and

Alij = dl!j

+ lij(wi

- wf)

- wf

+ 1:

lfiwi,

Cljkjwk - c ljkjlFkwk, k#j

(3.45)

k#i,j

(3.46)

WJ

The system of equations (3.40)-(3.44) is closed with respect to the operation of exterior differentiation. We will apply the Cartan test (see Theorem 1.3 in Section 1.2) to investigate the consistency of this system. The number q of unknown functions A bf , and Alij in the exterior quadratic equations is: q = ( n - m)m 2m(m - 1) = m(n m - 2). The first character s1 of the system under consideration is equal to the number of independent exterior quadratic equations, i.e. s1 = (TI - m)m + m(m - 1) = m(n - 1 ) . Its second character sz = q - s1 = m(m - I ) , and the third and all subsequent characters are equal to 0: s3 = . . . = sm = 0. This implies that the Cartan number Q zz ~1 2 ~ = 2 ( n - m)m 3m(m - 1). Let us find the number N of parameters on which the most general integral element of the system of equations (3.40)-(3.44) depends. To find N , we apply the Cartan lemma to equations (3.43) and (3.44):

+

+

+

+

Abf = bgiw', . . . A[!. = /?..wk + I ? . . w J , 2% 3.ak 3 3 . A / ?. = 1'. . .wi + 1%.. .w3 , 33

133

383

The number N is equal to the number of independent coefficients in these equations, i.e. N = m(n - m) 3m(m - 1).

+

3.4

Some Existence Theorems

87

Since N = Q, the system of equations (3.40)-(3.44) is in involution, and an m-dimensional integral manifold V", defined by this system, depends on s2 = m(m - 1) arbitrary functions of two variables.

A submanifold V", carrying a conjugate net C2 which is not an m-conjugate system, can be more arbitrary. For example, a submanifold Vm of general type in a projective space Pm+' carries a conjugate net C2 which is not an mconjugate system (see Theorem 3.10). It is easy to see that this submanifold is defined by two arbitrary functions of m variables. 2. Consider now a submanifold Vm carrying a conjugate net C3 of third order and suppose that the osculating subspace Ti2'(Vm) has the maximal 1) = ;m(m 3) and m2 = m. For such a subpossible dimension m ;m(m manifoldVm we have: rnl = i m ( r n + l ) . In this case il = m + l , . . . , i m ( m + 3 ) , and the fundamental forms can be reduced to the form:

+

+

+

We use here a double index notation for numbering the forms damental forms 'Pi:, can be reduced to the form: (p(m+m1 ti)

(3)

=(43.

@ti).The fun(3.48)

To achieve this, the vertices A , of a moving frame must be chosen in such a way that Ai, E Ti2),A,, E Ti3)and A,, @ Ti3). Note that the quadratic forms (3.47) on the submanifold under consideration have the same structure which they had on the Veronese varieties (see Section 2.2). However, in contrast to the Veronese varieties, we now have m2 > 0. Theorem 3.14 The submanifolds Vm described above exist and depend on ;m(m - 1) arbitrary functions of m variables. Proof.

For the submanifolds V" under consideration the system (2.5) takes

the form:

,(id = 0,

+ +

where iz = m ml I , . .. , 2 m equations (2.18) become:

,l(i)

= ,I,

, ( 3i i )

JZ

+ ml; = 0,

= 0,

- 0, -

+ r n ~+ 1 , . . . , n . By (3.47),

a2 = 2m

,jij)

= ,j

(3.49)

j

,(U) k =0

(3.50)

where the indices denoted by different letters take distinct values. By (3.48)) equations (2.25) take the form: (3.51)

88

3 . SUBMANIFOLDS CARRYING A NET

+

OF

CONJUGATE LINES

+

where i z = m ml i. Moreover, the specialization of the moving frame indicated above implies the following Pfaffian equations: w!a

= 0 w?= z 0, w: 1

%

= 0.

(3.52)

Exterior differentiation of all Pfaffian equations (3.49)-(3.52) leads to the following exterior quadratic equations:

e;;,

:= w %A wia,2 = 0.

(3.60)

In equations (3.53)-(3.60) the forms 0 denote the exterior differentials of the corresponding equations of system (3.49)-(3.52). Note that the exterior differentiation of all equations (3.49) and the first two equations of (3.52) lead to identi ties. Each of (n - 2m - m1)m exterior quadratic equations (3.60) contains one form wi",' which does not enter into the other exterior quadratic equations. Thus we have for each of the quadratic equations (3.60): u1

= 1,

u2

=

.

I

.

= 0, = 0.

Next, consider the subsystem consisting of $m2(m+3) exterior quadratic equations (3.53), (3.54), (3.57), (3.58) and (3.59). This subsystem contains five groups of forms:

3.5

Laplace Transforms of Conjugate Nets and Their Generalizations

89

which do not enter into the other exterior quadratic equations. The number of these forms coincides with the number of exterior quadratic equations. Therefore, the characters of this subsystem are:

Finally, consider the subsystem consisting of m exterior quadratic equations (3.55)-(3.56) where the indices i and j, i # j, are fixed. The number of such subsystems is $m(m l ) , and each of these subsystems contains six group of forms:

+

which do not enter into the other exterior quadratic equations. Thus, we have the following values of characters for each of these subsystems: u1=m, u z = m - l ,

..., u m = l .

Since these subsystems are independent, the characters of the entire system are the sums of the corresponding characters of the subsystems: s1 s2 s3

= ( n - 2m - m l ) + f m 2 ( m + 3) = I m ( m - I)', =p ? ( m - l)(m - 2),

+ +m2(m- I),

................................................. s, = +m(m- 1).

Therefore, the Cartan number for the system (3.55)-(3.56) is:

1 1 Q = s ~ + ~ s z + .. .+ms, = (n-2m-ml)m+-m2(m+3)+-m(m+l)(m+2). 2

6

One can easily show that the number N of parameters, on which the most general integral element of the system under consideration depends, is equal to Q: N = Q, i.e. the Cartan criterion is satisfied. Thus, the system is in involution, and an m-dimensional integral manifold V m , defined by this system, depends on s, = i m ( m - 1) arbitrary functions of m variables. 1 This example was considered by 8. Cartan in Chapter 4 of his paper [Ca 191.

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

90

3.5

Laplace Transforms of Conjugate Nets and Their Generalizations

1. First we consider a two-dimensional surface V 2 in a three-dimensional projective space P 3 and a one-parameter family X of its lines. none of which is tangent to any of the asymptotic lines of V 2 . We associate a family of frames with the surface V 2 in such a way that the vertex A0 of a frame coincides with a point t E V 2 ,the vertex A1 lies on the line of X passing through the point t , and the vertex A2 belongs to the tangent plane T,(V2). Then we have

where (3.62)

= w1o, w 2 = w i and b l l # 0 since the line of X is not tangent to a n asymptotic direction. Let us find a point F on the tangent AoAl to the line of X which describes a two-dimensional surface, as the point A0 does. Since F = A1 zAo, then it follows from equations (3.61) and (3.62) that w1

+

The point F will describe a two-dimensional surface if and only if the 1-forms

+

8' = 1 ~ , w 1 (l&

+ z)w2

and wf = bllw'

+ b1zu2

(3.63)

are linearly dependent, i.e. the following condition holds:

It follows that (3.64)

Thus, on the straight line AoAl, there exists a unique point F which is different from A0 and which describes a two-dimensional surface V 2 when the point A0 describes the surface V 2 . The point F (as well as the point Ao) is called a focus of the straight line AoAl. The lines AoAl form a two-parameter family in the space P3. Usually such families are called congruences of straight lines (see [Fi 501). The surface F2 described by the point F is called the Laplace transform of the surface V 2 .

3.5

Laplace Transforms of Conjugate Nets and Their Generalizations

91

The equation uf = 0 defines a family 5 of lines on the surface V 2 , and the lines of this family are conjugate to the lines of the family A. In fact, the lines of X on V 2 are defined by the equation u2 = 0, and thus, by (2.76), the directions which are conjugate to the directions tangent to the lines of X are defined by the equation

r p ( ~ , G )= b1lw'G'

+ b12W1G2 = 0 .

Factoring out u 1 # 0, we arrive at the equation u: = 0. If uf = 0, the point A Ddescribes a line of 1on V 2 ,and this line is conjugate to a line of the family A. Moreover, the point F describes? line which is tangent to the straight line AoF and which lies on the surface V 2 .We say that the families X and X form a conjugate net on the surface V 2 . We further specialize the moving frame associated with a point A0 E V 2 by placing the vertex A2 on the tangent to the line of X passing through the point Ao. Then the form ufbecomes proportional to the form u l ,the coefficient b l z becomes 0, and equation (3.64) takes the form:

If we place the vertex A1 at the focus F of the straight line AoA1, then we obtain z = 0 and subsequently 1:2 = 0. This implies that equations (3.61) take the form: W:

The fundamental form Q, = squares: Q,

= bllwl, biju'wj

W:

= 1T1w'.

(3.65)

of the surface V 2 becomes the sum of

= bii(w')2

+

bzz(W2)',

and the form w: has the following expression: U;

= b22W 2 .

(3.66)

In the same way, the straight line AoAz tange_nt to the line of has a unique focus F' describing a two-dimensional surface ' V 2 . If we place the vertex A2 into the focus F', then the form w$ has the following expression: u;= 1;,w2.

'vz

(3.67)

The surfaces F2 and are two Laplace transforms of the initial surface V 2 . Let us_prove that the lines u2 = 0 and u1 = 0 are conjugat5on the surfaces and 'V'. We will prove this, for example, for the surface V 2 described by the vertex A l . By (3.65), we have for this surface

v2

3 . SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

92

p2 has the form

Thus, the equation of the tangent plane ( to the surface bll22

- 1;,x3 = 0,

(3.68)

where x o ,xl,x 2 and x 3 are the coordinates of a point in a frame { A o A ~ A ~ A ~ } . Let us find the second fundamental form of the surface f2.For this, we first apply exterior differentiation to equations (3.65). A simple calculation leads to the following equations: {d611

(dl11

+ b l l ( ~ , D- 2 ~ +: w:) + 1 1 2 1 ~ : ) A u1= 0 , + I : l ( ~ t - 2 ~ +: L J ~+) b l l u : } A w1 - A u 2 = 0 W:

The first of these equations can also be obtained from formulas (3.44) taken with m = 2 and n = 3. Applying Cartan's lemma, we find from the above equations that dbll+

dl;1

+

+

+ I:lw; + ~ q +) bllu:

=r d , =~ 1 1

+~ p 'l z w 2 ,

-w:

= p21ul

P22W2,

bll(w! - 2wi w;) l:l(w,O - 2 ~ :

+

(3.69)

where the quantities r and pij are related to the third order differential neighborhood of the surface V 2 and p 1 2 = p z l . Further, let us calculate the second differential of the point A1: &'A1

(u:w2

+

+ 611~,2))A2 + l?lug)As (mod Ta,(F2)).

+

+~l(dl:,

+wl(dbll

bllu;

Using equations (3.69), we can write the above relation as follows:

d2A1 5

+

( - P z ~ ( w ' ) ~- l f I ~ l ( ~-!2.:) pll(~')~)A2 (-bllul(w: - 2 4 ) r(u1)')A3 (mod T A , ( ~ ' ) ) .

+

Next, we find the the second fundamental form of the surface we obtain

v2.By (3.68),

Since this form does not contain the term with the product u1u2of the basis forms, the lines w 2 = 0 and-w1 = 0 on the surface V 2 ,which correspond to the lines of the families X and X of the szface V 2 , compose a conjugate net. The same property holds for the surface ' V 2 . Thus, under Laplace transform, a co_njugate_net of the surface V 2 corresponds to a conjugate net of the surfaces V 2 and ' V 2 . This is the main property of the Laplace transform. 2. We now turn to the multidimensional case. Let Vm be an rn-dimensional submanifold Vm of a projective space P", and let X be a one-dimensional foliation on V m such that the lines of X are not tangent to asymptotic directions at any of its points. As we did in the two-dimensional case, we associate with

3.5

Laplace Transforms of Conjugate Nets and Their Generalizations

93

a submanifold V" the bundle of frames of first order in such a way that the , i , i = 1 , . . . , m, point A0 coincides with the point z : A0 = t, the points A D A lie in the tangent subspace T,(Vm), and the line AoAl is tangent to the line of the foliation A. Then this line is defined by the equations w2

= . . . = u r n= 0.

After this specialization the forms w ? , a, b = 2, . . . , rn, become principal forms, i.e. they are expressed in terms of the basis forms w' and w b : w;

= iylo'

+ IYbW b .

(3.70)

We write equations (2.21) on the submanifold in question in the form wff = bfflw'

+ bybwb, w:

+ bzbwb.

= b:,w'

(3.71)

In this equation and in the rest of this subsection, we have the following ranges of indices: a , b , c = 2, . . . , rn and (Y = rn 1,. . . , n. Since the direction ADA1 is not asymptotic, at least one of the coefficients by, is different from 0. Let us find under what condition the straight line AoAl has points that describe submanifolds of dimension m, as the point A0 does'. Such points are called the foci of the straight line AoAl. Since

+

dAo = w,OAo +w'Al + w " A a , dA1= wPAo u : A ~ wYA, wPAa,

+

+

+

then, by (3.70) and (3.71), for a point F = A1 AoAl, we find that

d F = (w:

+

+ (dz + U P +

ZU')F +(ZTlw1 (I$

+

+ zAo lying on the straight line

Z(W;

-w:) - Z ~ W ' ) A O

+ zb,")wb)A,+ (by1w1+ byawa)Aa. Since the point F must describe a submanifold vmof dimension m, the forms 8" = I;,w1 + + z b , " ) ~ ' , w p = by1w' + byow" (3.72) ([Yb

must be expressed in terms of m - 1 linearly independent forms. Note that the form w' is one of those independent forms since at least one of the coefficients byl is different from 0. Placing the vertex A1 at the focus F describing the rn-dimensional manifold Vm, we obtain z, = 0. Assuming that the forms u 1 , w 3 , . . . ,urnare linearly independent on V", we find from (3.72) that

-

1Y2 = 0 , by2 = 0 .

(3.73)

The second from these equations means that on the submanifold V", there exists a one-parameter distribution (AoA2) which is conjugate to the distribution 'Note that it is possible to consider points on the straight lines AoAl that describe submanifolds of dimension less than m . For now, we do not consider this case.

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

94

{AoA1} tangent to the given foliation A. The integral lines of the distribution {AoAz} give rise to a foliation 5 on V", and this foliation is conjugate to the foliation A. Now it is clear that i f the straight line AoAl carries m - 1 foci, then at the point A0 E V", there are m - 1 directions conjugate to the direction AoA1, and these m - 1 directions define an ( m - 1)-dimensional direction which is conjugate to the direction AoA1. We will now prove the converse statement.

Theorem 3.15 Suppose that a submanifold V" of the space P" carries a onedimensional foliation A whose lines are not tangent to asymptotic directions and which has a complementary ( m - 1)-dimensional conjugate distribution. Then each straight line 1 which is tangent to a line of A, carries precisely m - I foci, if we count each of them as many times as its multiplicity. Proof. In fact, let us associate with a point z of the submanifold V" a family of frames of first order in such a way that the vertex A1 lies on the straight line I , and the points A z , . . . , A , span the ( m - 1)-dimensional direction which is conjugate to AoAl. Then the coefficients bya in formulas (3.71) must vanish, and equations (3.72) take the form: (3.74)

The condition defining the foci on the straight line AoAl can be written as det(l;,

+ 26:)

= 0.

(3.75)

This equation is an algebraic equation of degree m-I, and hence it has precisely m - 1 roots, if we count each root as many times as its multiplicity. Each of these roots defines a focus F = A1 rAo on the straight line AoA1. Thus, each straight line 1 tangent to a line of the foliation A c Vm carries m foci, since, besides the foci defined by equation (3.75), the point A0 also describes an m-dimensional submanifold. Such a family of straight lines 1 is called focal. In the general case, i.e. in the case when all the foci on the straight lines 1 are mutually distinct, a focal family places in correspondence to each of its focal submanifolds V"' m - 1 other submanifolds of the same kind as V". This correspondence is called a transformation of a submanifold Vm b y means of a focal family of rays. 3. The straight lines 1 of a focal family form a ruled submanifold Vm+l of dimension m 1 in the space P". This submanifold is characterized by the following theorem.

+

+

Theorem 3.16 A n m-parameter family of straight lines of a projective space P", n > m 1, is focal if and only if the tangent subspace Tn(Vm+') to the submanifold Vm+l formed b y the straight lines of this family depends on m parameters and is constant along a straight line of the family.

+

9.5 Laplace Transforms of Conjugate Nets and Their Generalizations

95

Proof. In fact, let us place the vertices A0 and A1 on the straight line 1 of the given family of straight lines. Then

(3.76)

where u = 2 , . . .,n . The forms wz and wy in formulas (3.74) are expressed in terms of m independent forms B p , p = 2, . . . ,m+ 1, determining a displacement of the straight line AoA1: w; = c ; p ~ p ,

wp

= cppep.

(3.77)

The foci F = A1 + zAo on the straight line AoAl can be found from the condition d F E AoAl. Since

dF

= (wy + zw;)A,

(mod Ao, A I ) ,

they are determined by the condition wp

+ zw; = 0 ,

which, by (3.77), takes the form (c;a

+

zc;a)ea

(3.78)

= 0.

This equation must determine those displacements of the straight line AoAl for which it describes one-parameter torses. Thus, equation (3.78) must have nontrivial solutions with respect to the forms 0'. The existence of such nontrivial solutions is guaranteed by the condition rank (cya

+

ZC&)

5 m - 1.

(3.79)

Suppose now that the family of straight lines AoAl is focal, i.e. each straight line AoAl has m foci (taking account of their multiplicity). This will be the case if and only if the system (3.79) has exactly m solutions, i.e. this system is reduced t o one algebraic equation of degree m. Therefore, among the rank determinants of order m of the matrix (cYa zcza), only one is essential. Assuming that this essential determinant is

+

detfc:,

+ zc&),

p , Q = 2 , . . ., m

+ 1,

and making an appropriate transformation of moving frames associated with the family of straight lines considered above, we easily arrive a t the following equations: C ~ ~ = O ca ,1P --O , a = m + 2

,..., n .

It follows that the tangent subspace TZ(Vm+')to the submanifold Vm+' at the points z E AoAl, spanned by the points Ao, A1, A,, is constant along the

3 . SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

96

straight line AoAl and depends on m parameters, in terms of the differentials of which the forms 8" are expressed. As was noted in Section 2.2, submanifolds of a projective space P" whose tangent subspace depends on a smaller number of parameters than a point, are called tangentially degenerate, and the number of parameters on which this tangent subspace actually depends is called the rank of this submanifold. Hence, the focal family of straight lines in P" considered above is a tangentially degenerate submanifold Vm+l of rank m. The general tangentially degenerate submanifolds will be considered in Chapter 4. Conversely, suppose that a submanifold Vm+' described by straight lines AoAl is tangentially degenerate of rank m. If we place the vertices Az, . . . ,Am+l in the tangent subspace common to all points of the straight line AoAl, then in equations (3.76), the index u takes the values 2 , . . . , m 1 . Thus, the foci will be defined from the single equation

+

det(eip

+ zc&)

(3.80)

= 0,

which defines precisely m foci on the straight line AoAl. This implies that the family of straight lines AoAl is focal. We will denote a tangentially degenerate submanifold Vm+' of rank m by V$+' (cf. Section 4.1). 4. Suppose that a focal family V:+l of straight lines has m mutually distinct foci. We may choose the vertex A0 in such a way that it does not coincide with any focus. Then the forms WE, p = 2 , . . . , m 1, are linearly In this independent and can be taken as basis forms on the family V;+'. basis, equations (3.77) can be written as

+

w P -- c p Pw o'I,

woa --& ) a -- 0 ,

p , q = 2 , . . ., m + 1 ; a = m + 2

, . . . , n.

(3.81)

(As above, we assume that the points A, are located in the (m+1)-dimensional tangent subspace of the submanifold V z + ' common to all points of the straight line AoAl.) This implies that equation (3.80) takes the form det(ci

+ zc;)

= 0.

(3.82)

This equation has precisely m mutually distinct roots. Thus the operator C = (c!) is diagonalizable, i.e we have c; = where cp # cq if p written as

#

qcp,

(3.83)

q . As a result, the first group of equations (3.81) can be

w;

= cpw;.

(3.84)

9.5

Laplace Transforms of Conjugate Nets and Their Generalizations

97

The forms ws define a net of torses on the focal family V Z t l , and this net is called the focal net. In general, the focal net is not holonomic since in general, the forms wg are not multiples of total differentials. By (3.83), equation (3.82) takes the form U ( C P

+

2)

= 0,

P

and the foci of the straight line AoAl can be written as

FP = A1

- CPAo.

Consider a second order neighborhood of a focal submanifold by the focus FP. First, we have

dFP

3

c ( c 9 - cP)wqA,

(mod Ao, A l ) ,

5;) described (3.85)

nfp

72

and thus the tangent subspace TFP to the submanifold is spanned by the points A o , A l , A , where q # p . Equation (3.84) implies that the corresponding tangent subspaces to the submanifolds described by the points FP and F9 have in common an (rn - 1)-dimensional subspace spanned by the points A01 A1 I A , , 7- # PI 9. To find the second differentials d2Fp of the points FP, we need some additional formulas. However, we will not derive them. Instead, we take them from Section 4.3. According to that section, the following relations hold on a focal family V;+l:

w; = b;wP, bp* = bFP,

(3.86)

m+l

(3.87) s=2 m+l

(3.88)

(The first two of the above equations correspond to equations (4.28) and (4.37), and the third follows from (4.40).) Differentiation of equation (3.85) gives

98

3. SUBMANIFOLDS CARRYING A NET OF CONJUGATE LINES

The coefficients in the points Ap and A , in the above expression are the second described by the point FP: fundamental forms of the submanifold

yz

In the above equations, the lower index ( p ) denotes, in contrast to Chapter 2, not the order of the corresponding form-all these forms are defined in a second order neighborhood-but that the corresponding form is a second fundamental form of the submanifold An analysis of these quadratic fundamental forms allows us to make the following conclusions:

yz.

yz,

1. On the submanifold the directions P A 0 (focal directions), tangent to the straight lines AoAl and defined by the system of equations w q =o,

nfP,

(3.89)

are conjugate to the (m- 1)-dimensional distribution defined by the equation U P = 0. This result matches Theorem 3.15 proved above.

2. If at least one of the coefficients l & , q , s # p , q # s, on a focal family V:+l is different from 0, then on the submanifold the net of lines corresponding to the focal net of the family V:+l is not conjugate. We focal. will still call this new net on

yz,

3. The focal net on the submanifold

yz is conjugate if and only if the fol-

lowing condition holds on VZ+l:

= 0, q , s

# P , 4 # s,

where the index p is fixed. Then the focal submanifolds Vm q # p , are called the quasi-laplace transforms of the conjugate net on the In general, the focal nets on the submanifolds are submanifold not conjugate since some of the coefficients Ct may be different from 0. Of course, a conjugate net Cz admits quasi-Laplace transforms along any of one-dimensional foliations composing this net. Focal submanifolds of all focal families tangent to these foliations are quasi-Laplace transforms of the original submanifold V(z.In general, the total number of all quasiis equal to m(m - 1). Laplace transforms of

0;

7 .;

7;

7;

75,

p = 2 , . . . , m-t 1, of a focal family V$+l are conjugate i f and only i f all coefficients l:d = 0 , q, s # p , q # s. Then a focal net of the family V:+' is holonomzc, and the conjugate nets

4 . The focal nets on all submanifolds

9.5

Laplace Tkansforms of Conjugate Nets and Their Generalizaiions

99

corresponding to this holonomic net on all focal submanifolds of V:+' are m-conjugate systems. In this case the focal nets on the submanifolds q # p , are called the Laplace transforms of the conjugate net C2 on the submanifold V" Since an m-conjugate system Cz on the submaniconsists one-dimensional foliations, the total number of its fold Laplace transforms is equal to m(m - 1).

yz, yz

0th

5 . Suppose now that a submanifold V" carries a conjugate net C z . I f the vertices Ai are placed on the tangents to the lines of this net, then, as we saw in Sections 3.1 and 3.2 (see equations (3.5) and (3.14)), the following equations hold on V":

Let us find the foci of the straight line AoA; when this straight line is displaced along the lines which are conjugate to the direction AoAi. The latter lines satisfy the equation w'

= 0,

which implies

(3.90)

If a point F = A;

+ zAo is a focus, then dFAAoAA;=O.

But by relations (3,90), we have

(I;,

dF

+ z6i)wkAh

(mod Ao, Al).

h,k#i

This implies that the foci of of the straight line AoA, are determined by the following system of equations: (/;k

+z6;)Wk

= 0.

(3.91)

This system has nontrivial solutions if and only if the quantity z satisfies the following equation: det(l:,

+ ~6,")= 0 .

(3.92)

Since in this equation h , k # i , it is an algebraic equation of degree m - 1 and has m- 1 roots z,, j # i , if we count each root as many times as its multiplicity. Each of these roots defines a focus

100

CARRYING A NET OF CONJUGATE LINES 3. SUBMANIFOLDS

A: = A; + ~ j A o on the straight line AoAi. This focus describes an rn-dimensional submanifold ( A : ) which is a quasi-Laplace transform of the submanifold Vm. If we substitute a root zj of equation (3.92) into the system (3.91), then the determinant of this system vanishes. This implies that the system has at least a one-dimensional solution yk

=

go.

(3.93)

I

This solution defines a one-parameter family of straight lines AoAi forming a torse whose edge of regression is described by the focus A : . Next consider the case when the torse defined by equation (3.93) corresponds to the j t h line of the conjugate net Cz passing through a point 2. This will be the case if the system (3.93) reduces to the form: wk=0, k#j.

Substituting these values of w k into (3.91), we obtain

lh. ,3 + 21. 6 :.- 0- . This implies 1; = 0, h , j # i , h

#j;

zj

= -l!j.

(3.94)

Thus, the focus A{ is expressed by

A: = Ai - 1:j Ao.

(3.95)

If the correspondence described above between torses of the families described by the straight lines AoAi and the lines of the conjugate net of the submanifold Vm holds for any i and j , then condition (3.94) is valid for all mutually distinct values i,j and h . This means (see Theorem 3.5) that the conjugate net on Vm is an m-conjugate system. Since the converse is obvious, we have proved the following result.

Theorem 3.17 Tomes of of the famalies, described b y the tangents to the lines of the conjugate net on the submanifold Vm, correspond to the lines o f this net if and only i f this net is an rn-conjugate system. The focal submanifolds of these congruences are Laplace transforms of the submanifold Vm. 6. As an example, we consider a submanifold V" of general type in a projective space Pm+'. As was indicated at the end of Section 3.2, in general such a submanifold carries a unique conjugate m-net which is not holonomic. Thus, such a submanifold admits quasi-Laplace transforms along any of the one-dimensional foliations composing the conjugate net, and these transforms

3.5

Laplace Transforms of Conjugate Nets and Their Generalizations

101

are not Laplace transforms. Moreover, the focal nets on focal families of straight lines realizing quasi-Laplace transforms do not correspond to the lines of the conjugate net of Vm. Submanifolds Vm, which are quasi-laplace transforms of V m , also carry conjugate nets but they do not correspond to the conjugate net of Vm. A similar situation occurs for a hypersurface Vm c Pm+l which carries a family of nonholonomic conjugate nets. As another example, we consider a tangentially degenerate submanifold V;+l C Pm+3. Focal submanifolds of such V;+l are of codimension three, and hence in general they do not carry nets of conjugate lines. This implies that a transformation by means of a focal family of rays is not reduced either to a quasi-Laplace transform or to a Laplace transform. The existence theorem for such focal families for rn = 3 and rn = 4 was proved by Akivis in his paper [A 62a]. 7. Consider a submanifold Vm carrying an rn-conjugate system Cz. With a point A0 E Vm, we associate a family of frames whose edges AoAi are tangent to the lines of the conjugate system. Denote by A : , j # i, the focus of the straight line AoAi corresponding to the displacement of the point A0 along the j t h line of the rn-conjugate system Cz. As was indicated above, the points A; describe rn-dimensional submanifolds carrying rn-conjugate systems. Each of those new rn-conjugate systems admits Laplace transforms which are also rnconju ate systems. Denote the points describing these new Laplace transforms by Ai, jf where the index k indicates a direction emanating from the point A i l and the index 1 indicates the number of the focus on the straight line emanating from the point A; in this direction. The submanifolds described by the points A{:, are called the second Laplace transforms of the conjugate system C 1 . It is not difficult to prove the following relations:

A{; = Ao, A!; = A ! , k

# i.

(3.96)

102 -see

3. SUBM.ANIFOLDS CARRYING A NET OF C O N J U G A T E LINES Figure 3.2.

Figure 3.2 This figure represents the case m = 3, and in this figure the line defined by the equations W J = 0, j # i, is denoted by l i .

/

Figure 3.3

3.6

Conic m-Conjugate Systems

103

The proof of the relations

= A i , A!!! = A:!,

(3.97)

where all indices i , j and k take mutually distinct values, is more complicated. These relations were proved by Smirnov in his paper [Sm 501. If m = 3, relations (3.97) are illustrated in Figures 3.3 and 3.4.

Figure 3.4

3.6

Conic rn-Conjugate Systems

1. We now consider a tangentially nondegenerate submanifold Vm c P" carrying a conjugate net CZ c Vm. With a point z E Vm we associate a family of moving frames { A " } , u = 0,1,. . ., n, whose vertex A0 coincides with the point z: A0 = z, and the edges AoAi, i = 1,.. . ,m ,are tangent to the lines of the conjugate net Cz passing through the point 2. The equations of the infinitesimal displacement of these moving frames can be written as dA, = wEA,, u , v = 0,1,.. .,n.

(3.98)

The specialization of the moving frames described above gives (see Section 3.1):

104

3 . SUBMANIFOLDS CARRYING

wa = 0,

CY

A

NET OF CONJUGATE LINES

= m+ 1 , ...,n)

(3.99)

i = l , ..., m,

wp=b9wi,

(3.100) (3.101)

(cf. equations (3.5) and (3.14)). In equations (3.99) and (3.100) and in other equations of this section there is no summation over the indices i, j , I, . . . unless the summation sign C is shown.

Definition 3.18 A submanifold Vm c Pn carrying a holonomic conjugate net Cz is said to be conic or a Peterson submanifold if the tangents to any line of each family ( C i ) of Cz taken at the points of intersection of this line with the ( m - 1)-dimensional submanifold conjugate to (Ci) form a cone. This definition implies that all foci of each of the straight lines AoAi, except A o , coincide, and each of these foci describes a curve. If we place the point Ai at the vertex of the cone associated with the family (Cj), we must have dAi A Aj z 0

(mod w ' ) .

(3.102)

Equations (3.98) and (3.102) imply that in equations (3.101) we have I:k = 0, I # i, i.e. .

.

= y;wi,

(3.103)

up = Z i i W i .

(3.104)

w;

and Since equations (3.104) imply Z:k = 0, R

#

i,j, i

# j , a conjugate

net on

Vm is an m-conjugate system. We will now prove the following result.

Theorem 3.19 If a submanifold Vm net C z is conic, then

c

P" carrying a holonomic conjugate

(i) The system defining a conic m-conjugate system is an involution, and an m-dimensional integral manifold V", defined by this system, depends on mn arbitrary functions of one variable. (ii) I n homogeneous projective coordinates the equations of a conic submanifold can be reduced t o the form:

+

+ +

pAo = L i ( u l ) Lz(u2) . . . Lm(um), where Li(ui) equations of the curves described by the foci A i .

(3.105)

3.6

105

Conic m-Conjugate Systems

Proof. (i) To prove the existence of conic m-conjugate systems, note that such a system is defined by Pfaffian equations (3.99), (3.100), (3.103) and (3.104). Exterior differentiation of these Pfaffian equations leads to the following exterior quadratic equations:

Ab? A wi = 0, Alii A wi = 0 , Al{i A wi = 0 , where

Abf = dbq

+ bq(w: - 2wj) +

c

lfibtwk

(3.106)

+ be.;,

k#i

c

+ l!,(W: - +W;) + bpwd + /filjkkWk, k#i,j Alii = dlii + 21ii(w,O - w:) + C l & w : + bqwi. A/ji = d(i

2Wi

k#i The system of equations (3.99), (3.100), (3.103), (3.104) and (3.106) is closed with respect to the operation of exterior differentiation. We will apply the Cartan test (see Theorem 1.3 in Section 1.2) to investigate,the consistency of this system. The number q of unknown functions Ab?, A& and Alii, and the number of the exterior quadratic equations are equal: p = s1 = m(m 1) m(n - m) m = mn. Thus s2 = 0 and Q = s1 2sz = mn. It is easy to check that the number N of parameters, on which the most general integral element of the system under consideration depends, is also equal to mn. Since N = Q, the system of equations (3.99), (3.100), (3.103), (3.104) and (3.106) is in involution, and an m-dimensional integral manifold Vm , defined by this system, depends on mn arbitrary functions of one variable. (ii) Note that equations (3.104) imply that dw,O = 0. Thus, the form w; is a total differential. It is convenient to write this form as

+

+

+

w: = -dln [pi.

(3.107)

We will integrate the system (3.98) where the components of infinitesimal displacement are defined by equations (3.99), (3.100), (3.103), (3.104) and (3.107). Since the point Ai describes a curve, this point can be written as

A* = piBi, where pi is a normalizing factor. The structure equations and equation (3.107) imply that dwi = (-dln IpI - w i ) A w k l and this means that each of the forms wi can be represented as w' = gidui.

Then

106

3. SUBMANIFOLDS CARRYING

A

NET OF CONJUGATE LINES

dAo = -dlnlplAo + ) p i B i d u ' ,

(3.108)

i

where p i = &pi (no summation over i), Exterior differentiation of the last equation gives C(dpi

+ pidln [ P I )A d u ' ~ =i 0.

i

Since the points B; are linearly independent, we find from the last equations that (dpj

+ jiidln IpI) A dui = 0

(no summation over i)

Multiplying this relation by p, we obtain d(pip) A dui = o (no summation over i).

This implies pip = vi(ui)S or

1 ' pip = -Cpi(U'). P Substituting this expression for p ; into equation (3.108), we arrive at

or (3.109)

where

Ci(u*) = p;(d)Bi(d) Integrating (3.109), we find that

+

+ . . . + Lm(um),

PAO= L(u') L2(u2) where

Lj =

I

Ci(ui)du'..

2. We will now define generalized conic systems. First, we suppose from the beginning that a net Cz on a submanifold Vm is an rn-conjugate system.

3.6

Conic rn-Conjugate Systems

107

With a point G E Vm we associate a bundle of moving frames { A " } , u = 0 , 1 , . . ., n , as described in the beginning of this section. Thus, we have equations (3.99), (3.100) and (3.101). In Section 3.4 we proved that for a submanifold Vm carrying an m-conjugate system, we have equations (3.19): Wf

= lf#< + I f j &

(3.110)

We showed in Section 3.4 that the exterior differentiation of equations (3.99) and (3.110) gives the following exterior quadratic equations:

Abp A w' = 0,

(3.1 11)

where

Abq = dbq -+ bp(w: - 2 4 ) + ~ ( l ~+ ,lfibi?)wk b ~

+ bfwz,

k#i

and

Alji A w i

+ Alij A w j = 0,

(3.112)

where

A / { j= dlij

+ /jj(U: -

Wi)

-Wp

+ I{j

c

k#j ljkjWk

k#j

-

c

k#Cj /jkj/fkuk,

k#:,j

and that by the Cartan lemma we have from equations (3.111) and (3.112):

Abq = bGiw',

(3.1 13)

and

Al3. 11 = [!..wi r1:

+ l 2'1! . . w j , All. '3 = [3..wi **3 + 13.'33. w j ,

(3.114)

respectively.

Definition 3.20 We will say that a submanifold Vm carrying an m-conjugate system is a generalized conic submanifold or a generalized Peterson submanifold if any two-dimensional submanifold K:, defined by equations w k = 0, k # i, j, is a conic system. The submanifolds defined by equations (3.105) are particular cases of generalized conic submanifolds.

Theorem 3.21 A generalized conic submanifold has degenerate Laplace transforms of dimension at most m - 1.

3. SUBMANIFOLDS CARRYING A NET OF C O N J U G A T E LINES

108

Proof. In fact, the Laplace transform ( A : ) of V" in the direction wi corresponding to the direction w j can be obtained as the Laplace transform of the surface l$ defined by the system w k = 0, L # i , j . Then, since the surface 6; is conic, it follows that the submanifold ( A ! ) does not depend on ui but can depend on all other u s , s # j , i.e. the dimension of (A:) is at most m - 1. We will now prove an existence theorem for generalized conic submanifolds.

Theorem 3.22 The system defining generalized conic submanifolds is in involution, and an m-dimensional integral manifold V", defined b y this system, depends on m(n + m - 2) arbitrary functions of one variable. Proof. As we saw in Section 3.5, the focus A! of the tangent AoAi correAi (see equations (3.95)). sponding to the line w j has the form A; = -lijAo By (3.98), (3.100), (3.110) and (3.114), its differential has the following form:

+

k# j

(3.115) Since w k = 0, L # i , j , on the submanifold 53, and since y3 is conic, the point A; describes the curve li with equations w k = wi = 0 . Expression (3.115) implies that the necessary and sufficient condition for this is 1.1.. $33

0.

(3.116)

These conditions and the equations (3.114) imply t h a t each term of the exterior quadratic equations (3.112) vanishes, i.e. we have A w' = 0,

A1ij A w j = 0.

(3.117)

Thus, generalized conic submanifolds are defined by Pfaffian equations (3.99), (3.100) and (3.110) and the exterior quadratic equations (3.111) and (3.117). The system is closed with respect to the operation of exterior differentiation. The number q of unknown functions Ab?, Ali, and Alii and the number of the exterior quadratic equations are equal: q = s1 = 2 m ( m - 1) m(n - m ) = m(n m - 2). Thus s~ = 0 and Q = s1 2sz = m ( n m - 2). It follows from equations (3.113), (3.114) and (3.116) that the number N of parameters, on which the most general integral element of the system depends, is also equal to m(n m - 2). Since N = Q , the system of equations (3.99), (3.100), (3.110), (3.111) and (3.117) is in involution, and an m-dimensional integral manifold V", defined by this system, depends on m(n m - 2) arbitrary functions of one variable. 3. We will now describe a procedure which can be used to construct a generalized conic submanifold.

+

+

+

+

+

+

3.6

Conic m-Conjugate Systems

109

First, note that if in the space P n t l , we fix two curves 11 and 12 and a hyperplane P", then the locus of the common points of P" with the straight lines joining arbitrary points of the curves 11 and 12 is a two-dimensional conic submanifold of general type. In fact, let N be a fixed point on the curve 12 (see Figure 3.5).

Figure 3.5 Projecting the line 11 from the point N onto the hyperplane P", we obtain a line n. Next take two points M I , M2 E 1 1 . Projecting the curve 11 from these two points onto the hyperplane P", we obtain the curves ml and m2. Moreover, the tangents t o the curves ml and m2 at the points of their intersection with the curve n are the projections of the tangent TN to the curve 12 a t the point N from the points M1 and M2. Therefore, the tangents to the curves ml and m2 have in common the point I< at which the tangent TN intersects the hyperplane P". Since the point I{ does not depend on the position of the points M I and Mz on the curve I ] , this point is the vertex of the cone passing through all curves m at the points of their intersection with a fixed curve n. In the general case, in a space Pntrn-', we consider m arbitrary curves l i , i = 1,. . . , m, and a fixed subspace P". Suppose that the curves 1, and the subspace P" are in general position. An arbitrary subspace Pm-l passing

3. SUBMANIFOLDS CARRYING

110

A

NET OF CONJUGATE LINES

through m points ti E 1; intersects P" at a point x which depends on m parameters. This point x describes a submanifold Vm. Let us prove that the submanifold V" is a generalized conic submanifold. In fact, consider an arbitrary two-dimensional submanifold obtained by fixing points on some m - 2 of m curves l i . Let Pm-' be a subspace defined by the m - 2 fixed points indicated above and Pn+m-2be a subspace defined by P" and Pm-'. Then the straight lines joining arbitrary points of the two at the points of a conic two-dimensional remaining curves intersect Pn+"-' submanifold W 2located in Pn+rn-2. If we project the submanifold W 2 onto P n from Pm-' as the center of projection, then we obtain a two-dimensional submanifold V 2 . Since the property of a submanifold to be conic is invariant under such a projection, the submanifold V 2 is conic. 4. Ryzhkov in [Ry 581 found the general form of parametric equations of a generalized conic submanifold. We will describe how these equations can be obtained and indicate their connection with the construction of the kind of submanifolds given above. For simplicity, we assume that the subspace P" is the coordinate subspace defined by vanishing of the first m - 1 homogeneous coordinates. Let m curves li be given in the space Pn+rn-lby quantities a i l , . . . , U ~ , ~ Um-l where a i l , . . . , are scalar functions of a single parameter ui and Ui = Ui(ui)is a point in the space Pn+rn-l. Then it is possible to prove that an equation of a generalized conic submanifold Vm can be written as follows:

A0 = det

all

a12

a21

a22

... ...

a1,rn-1

2)

a2,m-l

............................. am1 am2 . . . am,m-1 Urn

+

=~ I U I

a2U2

+ . . .+ a m u r n , (3.118)

where ai are the cofactors of the entries U;in this determinant. We now state Ryzhkov's result.

Theorem 3.23 The submanifold A0 = A o ( u l , . . . ,urn) defined b y equations (3.118) is a generalized conic submanifold whose conjugate system is form.ed b y the coordinate lanes, and the equations of any generalized conic submanifold can be reduced to the form (3.118). To establish the relationship between this result and the above construction of generalized conic submanifolds, we consider ail,ui2, . . . , ~ i , 1 ,~Ui- as coordinates of the point Ui describing the curve in the space Pn+"-l. Then A0 is the point of intersection of the coordinate subspace P" and the subspace Pm-l defined by m points taken on the curves Ui. Therefore, A0 = Ao(u', . . . , u r n ) is a submanifold which, according to the construction described above, is a generalized conic submanifold.

I ,

111

Notes NOTES

3.1. Submanifolds carrying conjugate nets were studied by 8. Cartan in [Ca 191, by Akivis in [A 61a], [A 62~1,[A 631 and by Bazylev in [Ba 531, [Ba 55a], [Ba 55b] and [Ba 671 (see also the papers [A 7Oa], [Ry 561 and [Ry 581 devoted t o conjugate systems). The papers [AR 641, [Ba 65.1 and [Lu 751 give surveys of the results on this subject. Akivis in [A 62c] investigated V" c P" carrying a conjugate net and admitting a field of axial (n - m)-planes (he called them the Voss normals) such that the osculating 2-planes to the Lines of the net a t a point z E V" meet the axial ( n - m)-plane passing through the point z along straight lines. He classified such submanifolds and proved existence theorems for some of them. 3.2. Theorem 3.7 was proved by Akivis in [A 6la]. A particular case of m-conjugate systems V" in P", n 2 2m, for which dim T(,2)(Vm)= 2m, was introduced by 8. Cartan in [Ca 191. 3.3. The generalized Segre theorem was first proved by Bazylev in [Ba 55a] and [Ba 55bl. Akivis gave a simpler proof of this theorem in [A 61a] and [A 631 and stated it in the form which we used in Theorem 3.10. Segre type theorems (giving conditions under which a submanifold V" is of class k) were proved by Villa and Muracchini when they studied quasiasymptotic curves and submanifolds (see [Vi 39e] and [M 51~1). Theorem 3.11 was proved by Akivis in [A 631. Submanifolds with a conjugate net, for which ml = m, were first considered by 8. Cartan in [Ca 191. Chern studied these submanifolds in detail in [C 471 and named them the Cartan varieties. In [C 471, Chern also considered Laplace transforms of Cartan varieties. 3.4. The example which we presented in this section was considered by 8. Cartan in Chapter 4 of his paper [Ca 191. 3.5. The theory of Laplace transforms is a very well developed chapter of differential geometry of conjugate nets on surfaces in a three-dimensional projective space P 3 . For the first time the geometric theory of Laplace transforms of two-dimensional surfaces in P3 was constructed by Darboux in [Da 721, vol. 2. This theory was also presented in the books [Tz 241, [Fi 371, [Fi 501, [Bo 5Oa], [Mi 581 and [Go 641. Note that in the book [Fi 501 the Laplace transforms of two-dimensional conjugate nets in an n-dimensional projective space were also studied. B. Segre in his book [SegB 711 presented Laplace transforms of two-dimensional conjugate nets in P" from the point of view of algebraic geometry. In the 194O's-l950's this theory was extended to the m-conjugate systems ( m > 2) in P". Koz'mina studied Laplace transforms of three-conjugate systems in P3 (see [Koz 471). Chern in [C 441 (see also [C 471) generalized Laplace transforms to mconjugate systems. However, he considered only Laplace transforms of m-conjugate systems that are Cartan varieties, i.e. the case when n 2 2m. Smirnov in [Sm 491 and [Sm 501 generalized Koz'mina's and Chern's results by constructing Laplace transforms of m-conjugate systems in P" where n 2 m. Four years later, Bell (who did not know the paper [Sm 501) in [Be1 541 independently arrived to the results similar t o those in [Sm 501. Korovin in [Kor 55.1 constructed a closed Laplace network consisting of 16 threeconjugate systems in P 3 . In [Kor 55b] Korovin extended the theory of Darboux invariants to three-conjugate systems V 3 C P' and introduced three-conjugate systems R (for them their asymptotic directions and asymptotic directions of their Laplace

112

3.SUBMANIFOLDS CARRYING A

NET OF

CONJUGATE

LINES

transforms correspond to each other). Geidelman in [Ge 571 generalized these results of Korovin and introduced rn-conjugate systems R in the set of rn-conjugate systems belonging to its second osculating space P" and possessing rnl = n - rn < rn linearly independent forms. Focal families of tangents to V 3 C P5 were first introduced by Akivis in [A 491 and [A 501 when he studied the T-pairs of complexes. Later Akivis gave the general theory of transformations of multidimensional submanifolds V"' C P" by means of a focal family of rays (see [A 61b] and [A 62al). Bazylev in [Ba 531, [Ba 55a], [Ba 55b], [Ba 611 and [Ba 641 constructed quasiLaplace transforms of the general and some special submanifolds V"' carrying conjugate nets. In the papers [V 64a] and [V 64b] Vangeldkre, who did not know the Akivis papers [A 491, [A 501, [A 61b], [A 62a] and the Bazylev papers [Ba 531, [Ba 55a], [Ba 55b], considered the submanifolds V3 c P5,whose two asymptotic cones at any point x E V 3 have four distinct asymptotic directions, and therefore such submanifolds V 3 carry a conjugate net. He generalized the notion of conjugacy of two families of lines on V 3 (one can find this generalized notion in [Ba 531) and constructed a particular case of quai-Laplace transforms of V 3 introduced earlier by Bazylev for general V"' C (see [Ba 531 and [Ba 55bl). 3.6. The results on conic (Peterson) submanifolds for n = 3 can be found in the paper [Bla 711 by Blank and for n 2 3 in the paper [Ry 581 by Ryzhkov (see also [Ry 561). However, in these papers the conic submanifolds were considered in an Euclidean and affine spaces, respectively. The presentation of conic submanifolds in a projective space P" appears in this book for the first time. Degen in [De 671 considered two-component conjugate systems on V"-' c P" with involutive distributions A, and A2. In particular, in [De 671 he introduced the Darboux tensors of A1 and Az, proved that if A1 defines the shadow submanifolds (along which V"-' and a hypercone of the same dimension as A2 are tangent), then one of the Darboux tensors vanishes, and considered the case when the shadow surfaces belong to some P" c P". Blank and Gormasheva in [BG 701 proved that a Peterson hypersurface V 3 in P4 can carry at most a six-parameter family of conic three-nets, and that if this is the case, then V 3is a hyperqnadric.

113

Chapter 4

Tangentially Degenerate Submanifolds 4.1

Basic Notions and Equations

1. We consider the Gauss mapping y of a submanifold Vm which maps a point z E Vm into its tangent subspace Ti1) (see Section 2 . 2 ) : y : V"'

+

G(m,n).

The basis forms on the Grassmann image y(Vm) of the submanifold V"' are the forms wf. These forms are expressed in terms of the basis forms of the submanifold Vm according to formulas (2.21): U8P

= bP.WJ 23 > b?. a3 = b?. 31'

(4.1)

As we noted in Section 2.2, if the submanifold Vm is tangentially nondegenerate, its tangent subspace depends on m parameters, and hence the forms wf cannot be expressed in terms of less than m linearly independent forms. Suppose now that the Grassmann mapping y of the submanifold V"' is degenerate, i.e. its Grassmann image y( Vm) depends on r parameters, where 0 5 r < m. Then we say that the submanifold V"' is tangentially degenerate and has rank r . We denote such submanifolds by Ifv"'. For a tangentially degenerate submanifold V,"' of rank r , the forms wf can be expressed in terms of precisely r linearly independent forms. Let us take the forms w'+l,. . . ,w"', where 1 = m - r , as these independent forms, and write the expressions of the forms w s in terms of the forms w q , q = 1 1,. . . , m, as follows:

+

w p = b i " , w q , q = l + l , . . . , m. Since the matrix ( b c ) is symmetric, this matrix takes the form:

(4.2)

4. TANGENTIALLY DEGENERATE SUBMANIFOLDS

114

and, by (4.2) and (4.3), the forms w p can be written as follows:

..., I,

wt=O,a=1, w:

(4.4)

= b&wq, p , q = I + 1, .. . , m ,

(4.5) where, as earlier, bFq = bFp. The 1-forms u p are basis forms of the Grassmann image r(Vpm)of the submanifold Vrm, and the quantities bFq form a tensor. By (4.4), the equations of infinitesimal displacement of the moving frame associated with the tangentially degenerate submanifold Vpm have the form:

+ +

+

dAo = w,OAo waAa wPAp, dAa = L J ~ A ~ uEAP, +w;Aa +u;Ap +up”A,, dAp = w~OAO w:A, wtAp. dA, = UZAO w;Aa

+

+ +

(44

+

Taking exterior derivatives of equations (4.4), we obtain the following exterior quadratic equations:

u{ Aw: = 0. Substituting expressions (4.5) into equations (4.7), we find that

b;,&

Auq

= 0.

(4.7)

(4.8)

Let us prove that, as was the case for the forms up”,the forms uC can be expressed in terms of the basis forms uq alone. Suppose that decompositions of the forms WE have the following form: w{

= c pa‘I W’I + < < W E ,

(4.9)

where the forms wc are some linearly independent forms. Substituting these expressions into equations (4.8), we find that

Since the exterior products from these relations that

u p A wq

and uc A wq are independent, it follows

(4.10)

and (4.1 1 )

4.1

Basic Notions and Equations

115

Since the forms wp" cannot be expressed in terms of less than r linearly independent forms, the rank of system (4.11) is equal to r , and the system has only the trivial solution: c$ = 0. Hence equations (4.9) take the form: WE

= c:,w9,

(4.12)

where the coefficients c$, are related to the coefficients b;, by (4.10). Note that under transformations of the points A,, the quantities C& are transformed as tensors. As to the index a, the quantities c:, do not form a tensor with respect to this index. Nevertheless, under transformations of the points A0 and A,, the quantities c;, along with the identity tensor 6; do transform as tensors. For this reason, the system of quantities c& is called a quasitensor. 2. We will now prove the following result.

Theorem 4.1 A tangentially degenerate submanifold Vrm of rank r is an rparameter family of I-dimensional planes (where 1 = m - r ) , and the tangent subspaces TA1)(Vm)remain constant along these 1-planes. Proof. On a submanifold Qrnconsider the system of equations: wp

= 0.

(4.13)

- b;w: - c;,w").

(4.14)

By (4.12), we have dwp = wq A (w:

By the Frobenius theorem, equations (4.14) imply that the system of equations (4.13) is completely integrable and defines a foliation of the submanifold Vrm into (rn - r)-dimensional submanifolds. Let us prove that all these submanifolds are planes. In fact, if wp = 0, then the first two of equations (4.6) take the form: (4.15)

This means that if wp = 0, then the 1-plane defined by the points A0 and A, remains constant. Thus the submanifolds defined on VTmby the system of equations (4.13) are planes of dimension 1. The same system of equations (4.13) implies that if wp = 0, then in addition to equations (4.15), we have the equation: (4.16)

Equations (4.15) and (4.16) mean that the tangent subspace T$')(V?) along the fixed 1-plane A0 A A1 A . . . A A1 remains constant. In what follows, we assume that if a submanifold Vrm contains a plane I dimensional submanifold, then the latter is extended to a complete 1-dimensional

116

4. TANGENTIALLY DEGENERATE SUBMANIFOLDS

plane. We will call this 1-plane an 1-dimensional generator of a submanifold Kmand will denote it by E'. We denote by up the first integrals of the completely integrable system of equations (4.13). These first integrals are parameters on which the generator E' and the tangent subspace Ti1)of a tangentially degenerate submanifold qm depend. The simplest particular case of a tangentially degenerate submanifold is a two-dimensional surface V; in the space P" formed by the tangents to a nonplanar curve. Such a surface is called a torse. This torse is a particular case of a tangentially degenerate submanifold VIm of rank one in P" which is also called a torse. Its generator is of dimension m-1, and its second osculating subspace is of dimension m + 1. The latter torse is a family of osculating planes of the mth order of a curve which does not belong to a projective space of dimension m. Note also the extreme values of the rank r of a submanifold Vm. If r = 0, then a submanifold Vm is an m-dimensional projective subspace, since in this case the matrix ( b ; ) is the zero matrix. On the other hand, if r = m, then it is obvious that the submanifold V" is nondegenerate. 3. Consider a correlative transformation K in the space P" which maps a point z E P" into a hyperplane ( E P", ( = K ( z ) , and preserves the incidence of points and hyperplanes. A correlation K maps a k-dimensional subspace Pk c P" into an ( n - k - 1)-dimensional subspace Pn-k-l C P". Consider a smooth curve C in the space P" and suppose that this curve does not belong to a hyperplane. A correlation K maps points of C into hyperplanes forming a one-parameter family. The hyperplanes of this family envelope a tangentially degenerate hypersurface of rank one with (n - 2)-dimensional generators. If the curve C lies in a subspace P 3 c P", then a correlation K maps points of C into hyperplanes which envelop a hypercone with an (n-s-1)-dimensional vertex. Further, let Vm be an arbitrary tangentially nondegenerate submanifold in the space P". A correlation K maps points of such V" into hyperplanes forming an m-parameter family. The hyperplanes of this family envelop a tangentially degenerate hypersurface Vg-'. The generators of this hypersurface are of dimension n - m - 1 and correspond to the tangent subspaces T,(1) (Vm). If the tangentially degenerate submanifold Vm belongs to a subspace P 3 c P", s > m, then the hypersurface VZ-' corresponding to Vm under a correlation K is a hypercone with an (n - s - 1)-dimensional vertex. Now let Vm be a tangentially degenerate submanifold of rank r . Then we can prove the following result.

Theorem 4.2 A correlation K maps a tangentially degenerate submanifold VF of rank r into a tangentially degenerate submanifold V:-'-', 1 = m - r , of the same rank T with ( n - m - 1)-dimensional plane generators.

4.2

Focal Images

117

Proof. Under a correlation K , to an 1-dimensional plane generator E' c Vrm there corresponds an ( n - 1 - 1)-dimensional plane En-'-', and to a tangent subspace T i l ) ( V T )= Em there corresponds an ( n - m - 1)-dimensional plane En-m-1 where En-m-1 c En-i-1 9 . Since both of these planes depend on r parameters, the planes En-"'-' are generators of the submanifold X(ym),and the planes En-'-' are its tangent subspaces. Thus, the submanifold K ( V F ) is a tangentially degenerate submanifold Vn-'-' of dimension n - 1 - 1 and of rank r. In particular, we note that, among tangentially degenerate submanifolds of rank r with 1-dimensional generators, where 1 = m - r , there are submanifolds for which 1 m = n - 1. A correlation K maps these tangentially degenerate submanifolds into submanifolds of the same dimension and with 1-dimensional plane generators. These tangentially degenerate submanifolds are called selfdual or autodual.

+

4.2

Focal Images

Let Vrm be a tangentially degenerate submanifold of rank r . By Theorem 4.1, such a submanifold carries an r-parameter family of &dimensional plane generators E', where 1 = m - r . Let z = zoAo + P A , be an arbitrary point of this generator. We will call this point focal if dx E E'. Since

dz/E' = (L'W:

+ zaw:)Ap,

the following conditions must hold for focal points

2:

zow: j-2%f: = 0.

(4.17)

Using relations (4.12), we can write these equations in the form:

(26;

+ z"cf:q)wq = 0.

(4.18)

These equations form a homogeneous system with respect to d .This system has a nontrivial solution if and only if its determinant vanishes: det(c06;

+ c"c&)

= 0.

(4.19)

The latter equation defines an algebraic variety of order r in the plane E'. To every point of this variety there corresponds a certain displacement of the plane E' which can be found from equations (4.18). In this displacement the family of generators E' has an envelope to which the point x belongs. The point L is called a focus of the plane E ' , and the algebraic variety defined by equations (4.19) is called the focus variety. Let us denote this variety by F Note that the point A0 does not belong to the variety F , since if za = 0, the left-hand side of equation (4.19) takes the form:

118

4. TANGENTIALLY DEGENERATE SUBMANIFOLDS

det(zoSi) = (do)'

# 0.

Theorem 4.3 The focus variety F is a set of singular points of a tangentially degenerate submanifold Vm, which are located on its generator E'. Proof. In fact, at these points the rank of the system of forms on the left-hand side of equation (4.17) is reduced. This implies a reduction of the dimension of the tangent subspace Ti')(F"').The points that do not belong to the focus variety F are regular points of the submanifold Vrm. The dimension of the tangent subspace Ti')(VF)at these regular points is not reduced. Since each generator E' of a tangentially degenerate submanifold Vp contains an ( I - 1)-dimensional variety F , the set of all these varieties is an (m- 1)dimensional submanifold of singular points of the tangentially degenerate submanifold F"'. For the two-dimensional tangentially degenerate submanifold V? of rank one (a torse) considered above, each of its generators E' has one singular point (a focus), and the tangent subspace of V 2 at this point degenerates into a straight line. The set of all singular points of such q2form the edge of regression of this tangentially degenerate submanifold. Consider one more example of a tangentially degenerate submanifold. In the space P", n 2 4,we take two smooth space curves C' and C2, which do not belong to the same three-dimensional space, and the set of all straight lines intersecting these two curves. These straight lines form a three-dimensional submanifold V 3 . It is easy to see that the submanifold V 3 is tangentially degenerate. In fact, the three-dimensional tangent subspace Ti1)(V3)to V 3 at a point x lying on a rectilinear generator El is defined by this generator E' and two straight lines tangent to the curves C' and C2 at the points F 1 and F 2 of their intersection with the line E l . Since this tangent subspace does not depend on the location of the point 2: on the generator E l , the submanifold under consideration is a tangentially degenerate submanifold V; of rank two. The points F' and F 2 are foci of the generator E ' , and the curves C' and C 2 are degenerate focus varieties. There are two cones through every generator E'. These cones are described by generators passing through the focus FI or the focus F2. On the submanifold V; these cones form two one-parameter families comprising a focal net of the submanifold V;. This example can be generalized by taking k space curves in the space P", where n 2 2k and k > 2, and considering a k-parameter family of (k- 1)-planes intersecting all these k curves. The question arises: do there exist in the space P" tangentially degenerate submanifolds without singularities ? Theorem 4.3 implies that from the complex point of view a tangentially degenerate submanifold Vrm does not have singularities if and only if it is an m-dimensional plane Pm, i.e. if r = 0 . From the real point of view a tangentially degenerate submanifold Vrm does not have real singularities if and only if its focal images in the plane generators are pure

4.8

Focal Images

119

imaginary, and this situation can occur only if the rank r is even. An example of such a submanifold will be considered in subsection 4.7.5. One can consider dual focal images of a submanifold Vrm. The tangent subspace Til)(VT)= Emof a tangentially degenerate submanifold Km of rank r depends on r parameters up, p = 1+1,. . . , m. Consider two tangent subspaces Em and 'Em corresponding to the values up and up duP of these parameters. Let = < m z m = 0, a = m 1,.. ., n, be a hyperplane passing through the tangent subspace Em. We say that this hyperplane is a focus hyperplane if it contains not only the tangent subspace E m but also the tangent subspace 'Em. Since the tangent subspace 'Emis determined by the points 'A0 = A0 dAo,'A, = A , dA, and 'A, = A, dA,, by formulas (4.6), the subspace 'Embelongs to the hyperplane if an only if the following conditions hold:


0, then the point x is called a parabolic point of the submanifold Vm. If all points of a submanifold Vm are parabolic, then the submanifold Vm is called parabolic. However, the second fundamental forms (4.73) are connected not so much to the metric structure of the submanifold Vm as to its projective structure, since these forms are preserved under projective transformations of the Riemannian submanifold Vm. If a Riemannian manifold V" admits a mapping f into a domain in n-dimensional projective space P", and if this mapping transforms the geodesics of V" into straight lines of P", then the second fundamental forms (4.73) coincide with the second fundamental forms of the submanifold --m V = f ( V m ) of the projective space which were defined in Chapter 2 by formula (2.32). We will call the mapping f : V" -+ P" introduced above a geodesic mapping. If we apply the notations of Chapter 2, then condition (4.74) defining the subspace FZ(Vm) can be written in the form: bP.2' = 0. $3

(4.75)

If we place the points A , , a = 1 , . , . , ~ ( x of ) ,our moving frames, associated with a point z = A0 of the submanifold 7= f ( V m ) , in the subspace ffFS(Vm), then from condition (4.75) we find that

b:j = 0 ,

(4.76)

and thus the matrices ( b ; ) of the second fundamental forms of the submanifold ? n ---m. V take the form (4.3). This proves that the submanifold V is tangentially degenerate, and its rank r is connected to the index u ( z ) of relative nullity of the submanifold Vm by the relation r = rn

- v(x).

I t follows that the index v ( x )of relative nullity of the submanifold Vm coincides with the dimension 1 of the plane generators of the submanifoldVm = f ( V m ) . Thus we have proved the following result.

Theorem 4.10 If a Riemannian manifold Vn admits a geodesic mapping f into the space P", and if a submanifold Vm c V" has the constant index u(x) of relative nullity, then the image = f ( V m ) of the submanifold Vm is a ) plane tangentially degenerate submanifold VTm of rank r = m - ~ ( x whose generators are of dimension I = .(XI.

vm

4. TANGENT~ALLY DEGENERATE SUBMANIFOLDS

134

Submanifolds Vm of a Riemannian space V" of a constant index v(z) = 1 of relative nullity are called I-parabolic submanifolds (cf. papers [Bori 821 and [Bori 851 by Borisenko). 2. We will now study complete 1-parabolic submanifolds in real simply connected Riemannian spaces V," of constant curvature c. If c = 0, then V: is the Euclidean space E". If c > 0, then is the elliptic space S". If c < 0, then V," is the hyperbolic space H". Each of these spaces admits a geodesic mapping into the space P", which is usually called the projective realization of the corresponding space V,". The Euclidean space E" is realized in the projective space P" from which a hyperplane E, has been removed (this hyperplane is called improper or the hyperplane at infinity), and the proper domain of the space E" can be identified with the open simply connected manifold Pn\E,. The elliptic space S" is realized in the entire projective space P", since the absolute of S" is an imaginary hyperquadric and its proper domain coincides with the entire space P". Finally, the hyperbolic space H" is realized in the part of the projective space P" lying within the convex hyperquadric that is the absolute of this space. This open simply connected domain is the proper domain of the hyperbolic space H". We denote by G the proper domain of the simply connected space V: in all these cases. Let Vm be a complete parabolic submanifold of a space V: of constant curvature. Suppose that Vm has a constant index v(z) = 1 of relative nullity. Let be the image of Vm in the domain G of the space P" in which the space V: is realized, and let Fmbe the natural extension of this image in the G. In this extension, 1-dimensional plane genspace P", so that 7= erators of the submanifold V are complemented by impyper elements from the complement P"\G. By Theorem 4.9,the submanifold Vm is a tangentially degenerate submanifold of rank r = n - 1. The focus variety F belonging to a plane generator E' of Vm is the set of all singularities of this generator. One of the important problems of multidimensional differential geometry is the finding of complete /-parabolic submanifolds Vm without singularities in spaces V: of constant curvature. Theorems 4.3 and 4.9 imply the following result.

K"

vm

VEQ

Theorem 4.11 Let Vm be a complete 1-parabolic submanifold of a simply connected space V: of constant curvature. Let = f(Vm) be the image of Vm in the proper domain G o f t h e space P" in which the space V," is realized, and let be the natural extension of this image in the space P". The submanifold Vm is regular if and only if the real parts IJe F of the focus varieties F belonging t o generators E' of the submanifold Vm lie outside of the proper domain G C P".

vm

vm

3. Let us examine the content of Theorem 4.11 for the different kinds of

4.7 Parabolic Submanifolds without Singularities

135

spaces V: of constant curvature. is the proper If c = 0, then V: is the Euclidean space En, and P"\E, domain of its projective realization. Thus a complete 1-parabolic submanifold Vm of the space E" is regular if and only if the real part Re F of thejocus variety F of each plane generator E' of the corresponding submanifold Vm c P" coincides with the intersection E' Em and constitutes a pfold ( I - 1)plane where 0 < p 5 r . If c > 0, then V: is the elliptic space S", and its proper domain coincides with the entire space P". Thus a complete 1-parabolic submanifold Vm of the space S" is regular if and only if thefocus variety F of each plane generator E' of the corresponding submanifold Vm is pure imaginary. If c < 0, then V: is the hyperbolic space H", and the proper domain of its realization lies inside of the absolute of this space. Thus a complete I-parabolic submanifold Vm of the space H" is regular if and only if the real part Re F of t_he focus variety F of each plane generator E' of the corresponding submanifold Vm lies outside of or on the absolute of this space. Parabolic surfaces of a three-dimensional space V," of constant curvature allow an especially simple description. In P 3 , to each parabolic surface V 2 there corresponds a torse, each rectilinear generator of which possesses a focus point. The locus of these focus points constitutes an edge of regression of the surface V 2 . If c = 0, then this edge of regression must belong to the improper plane E m , i.e. the edge of regression is a plane curve. But this is possible if and only if the edge of regression degenerates into a point. Therefore, a projective realization of a hyperbolic surface V 2 of a three-dimensional Euclidean space E3 is a cone with its vertex in the improper plane Em. Thus the surface V 2 itself is a cylinder. Hence in the space E3 there are no other regular parabolic surfaces except the cylinders. If c > 0, i.e. if we have the elliptic space S 3 , then there are no regular parabolic surfaces, since the edge of regression of the torse V 2 is always real. Finally, if c < 0, i.e. if we have the hyperbolic space H 3 , then there are regular parabolic surfaces, since the real edge of regression of the torse V 2 can be located outside of the absolute. Thus we have proved the following result.

n

Theorem 4.12 In the Euclidean space E 3 , only cylinders are regular parabolic surfaces. In the space S3, there are no regular parabolic surfaces at all, and in the space H 3 , regular parabolic surfaces exist and depend on of two arbitrary functions of one variable. The last statement follows from the fact that a torse in P 3 is completely defined by its edge of regression, i.e. by an arbitrary space curve, but these curves are defined by two arbitrary functions of one variable, as indicated in the theorem. 4. We will now consider some examples of regular parabolic submanifolds with a constant index of relative nullity in Riemannian spaces of constant curvature and dimension n > 3.

4. TANGENTIALLY DEGENERATESUBMANIFOLDS

136

One such example was pointed out by Sacksteder in his paper [Sac 601. I t is the hypersurface V 3 c E4, defined by the equation

+

x4 = x1 ~ 0 4 . ~ ) x2 sin(x3).

(4.77)

This hypersurface is everywhere regular in E4, and its index of relative nullity v(x) = 1. As a result, its rank T = 2. We will investigate this example from the standpoint of subsection 4.7.3. We introduce in the space E4 homogeneous coordinates (To,f1,c', T 3 ,Z4) such that xm = a = 1 , 2 , 3 , 4 , and extend this space to a projective space P4 by means of the improper hyperplane 2 = 0. Denote by p3 the natural extension z f the hypersurface V 3 in the space P4. The equations for the hypersurface V 3 can be represented in the parametric form:

5,

-0 x

-1 x

- s,

- -svsinu+tcosu,

-2 x

- svcosu+tsinu, -3 x - su, c4

(4.78)

=t.

These equations can be written in the form:

where A0 = (1,-vsinu,vcosu,u,O},

A1 = {O,cosu,sinu,O,l}

are points of the space P4. The straight lines A0 A A1 are the generators of the hypersurface V 3 . Differentiating the points A0 and A], we obtain

+

dAo = A z d ~ Asdv, dA1 = A Q ~ u , where A2 = (0,-vcosu,-vsinu,

l,O},

A3 = (0, -sinu,cosu,O,O}.

It can be easily verified that the points Ao, A1, A2 and A3 are linearly independent. As a consequence, the tangent hyperplane T, = A O A A ~ A A ~ A remains A~ constant along the straighte:1i A0 A A]. This hyperplane, like a rectilinear generator of the hypersurface V 3 , depends solely on the parameters u and v. We find the singularities (foci) of a generator A0 A A1 of the hypersurface V 3 C P4 in the same manner as for the general case in Section 4.2. A point x = ZoAo Z'Al is the focus of this generator if dx E A0 A A l , whence it follows that, for the focus,

-

+

-

xo(Azdu

+ A3dv) + Z1A3du = 0.

Since the points A2 and A3 are linearly independent, it follows that

4.7

137

Parabolic Submanifolds without Singularities

Zodu f l d u +Zodv

= 0, = 0.

(4.79)

This system should have a-nontrivial solution relative to du and dv, which defines a focal direction on V 3 . Consequently,

and (3’)’= 0. This means that the point A1 (the point at infinity of a rectilinear generator of the hypersurface V”) is the double focus of the line A0 A A l . It follows from (4.79) that the focal direction defined by the equation du = 0 corresponds to this focus. The torses on the hypersurface V 3 are therefore defined by the equation u = const. Each of these torses is a pencil of straight lines which is located in the 2-plane -3 1:

- u -0x , -4 x --- x1

(4.80)

cosu+F2sinu,

and whose center is the point A l . The point A1 describes the curve in Em defined by the equations -

xo = 0,

F3

= 0, (T’)2 + ( Z 2 ) 2 = (2”).

(4.81)

%

Besides the point A l , the plane (4.80) contains the point A3 = and is therefore tangent to curve (4.81). Thus, the hypersurface V 3 defined by equation (4.77) in the space E4, has no singularities in the proper domain of this space, since they have “retreated” to the improper hyperplane Em of this space. The example discussed above can easily be generalized. Let y be an arbitrary complete smooth curve in the improper plane E , of an Euclidean space E”. Suppose that this curve is described by the point A1 = A l ( u ) . We set A3 = and let T = ~ ( u be ) the smooth family of proper tangent 2-planes of the curve y. These 2-planes form a complete regular submanifold V$ of rank T = 2 with a constant index of relative nullity v(x) = 1. The proof of this assertion differs little from our above investigation of the structure of hypersurface (4.77) in E4. 5. In order to construct other examples, in Pn,n 2 4, we will consider a three-dimensional submanifold V$ of rank r = 2 with imaginary focus variety F . Equations (4.4), (4.12) and (4.5) defining this submanifold in P” take the form

%,

a = 4 ,..., n,

(4.82)

= b&wg, p , q = 2 , 3 ,

(4.83)

wt=wB=O,

W:

= <w:,

w;

while equation (4.19) defining the foci on the generator manifold, is written as

A0

A A1 of this sub-

4. TANGENTIALLY DEGENERATE SUBMANIFOLDS

138

Setting

f

= -A, we reduce this equation to the form X2

- (c;

+ c$)X + (c;c: - &;)

= 0.

Since the focus surface F is assumed to be imaginary, this equation has complex-conjugate roots X = c2 f icg, where c3 # 0. As a result, a real transformation converts the matrix A = (c:) to the form

Substituting these values for the components of the matrix A into equations (4.10), and taking into account that c3 # 0, we find that

bf2 + bg3 = 0. In view of this, the symmetric matrices B" can be written in the form

Equations (4.83) then assume the form (4.84) (4.85)

We will now find the osculating subspace Ti2) of our submanifold V 3 C Pn. Its tangent subspace is spanned by the points A o , A l , A z and A s . Since by (4.851,

Ti')

+

dA2 4 (b;wg b:w;)A, dA3 (bgw; - b;w:)A,

(mod Ti')), (mod Ti')),

the subspace Ti2)comprises the linear hull of the subspace B2 = b;A, and B3 = bgA,. Two cases are possible:

Ti1)and the points

(a) The points B2 and B3 are linearly independent. Then dimTil) = 5, and the dimension of the space n 2 5.

(b) The points Bz and B3 are linearly dependent. Then dimT,(') = 4, and n 2 4.

4. 7' Parabolic Submanifolds without Singularities

139

We will examine these two cases in turn. In case (a), we specialize the moving frames in P" in such fashion that A4 = B2 and A5 = B3. Then equations (4.85) take the form (4.86)

where A = 6 , . . . , n . Therefore the submanifold V 3 in case (a) is determined by the system of Pfaffian equations (4.82), (4.84) and (4.86). We will investigate the consistency of this system by means of the Cartan test (see Section 1.2). For this purpose we adjoin to the above Pfaffian equations the exterior quadratic equations obtained as the result of exterior differentiation of these Pfaffian equations. Exterior differentiation of the equations (4.82) leads to identities, by virtue of (4.84) and (4.86). Exterior differentiation of the equations (4.84) yields

( A c~ c ~ ( w+ ; w : ) ) A wi + ( A c 3 + ~ 3 ( -4 w ; ) ) A U ; = 0, - ( A c -~ c ~ ( w ; w:)) A U ; + ( A c + ~ c ~ ( w ;- w:)) ALL$ = 0,

(4.87)

where

+ +

+

A c= ~ dcz C Z ( W ; - w : ) - W ! ( ( ~ 2 -) ~( c ~ ) ~ ) w ; , Ac3 = d ~ 3 c ~ ( w ; w:) 2 ~ 2 ~ 3 ~ ; .

+

Exterior differentiation of the equations (4.86) gives (w,"

+ w t + - 2 4 ) A + - + + QUA) + + + + + + U:

C~U;

(LLJ;

U:

U;

A U:

= 0,

(u; - U: W $ ~ 3 1 4A)U ; - (w; W: C Z W ; - 2 4 ) A u: = 0, (u: - C ~ W ; 2 4 ) A w ~ (w: - u : -u; +u: C ~ U ; ) A w l = 0, (w: -wi - w i +w: c2wi) A w -~ (w: - C S W ~- 2 4 ) Au; = 0, ~2 A U +u," ~ A W ; = 0, w ~ X A U ; - wqX A"; = 0,

+

(4.88)

where A = 6 , . . ., n. The system (4.87)-(4.88) contains s1 = 2n-4 independent equations that include the following independent characteristic forms:

+ w:, - w:, + c2w; - 2w:,w; - w; + w; + c 3 4 , - c3w; - 2w;, wo" - w; - w; + wg + czw;,

A c ~A, c ~W,;

W;

wo" +w,4

w: wi,w,", A = 6

,...,n.

Their number is q = 2 n - 2. The second character of the system is therefore s2 = q - s1 = 2, and the Cartan number Q,= s1+ 2s2 = 2n. The number N of parameters on which the most general integral element depends is computed from the formula N = 29 - s1 = 2n. Since Q = N , by the Cartan test, the system of Pfaffian equations (4.82), (4.84) and (4.86) is in involution, and its general integral manifold depends on two arbitrary functions of two variables.

4 . TANGENTIALLY DEGENERATE SUBMANIFOLDS

140

In case (b), we have bg = b2ba and b$ = therefore be written in the form W; W$

b3ba.

Equations (4.85) can

+

ba(bzw: b 3 ~ ; ) , = b " ( b 3 ~ ; -a&).

Consequently,

+

dA2 E (b2wg b3wi)B (mod Ti')), dA3 5 (b3wg - b2w;)B (mod Ti')), where B = b*A,. We specialize our moving frame assuming A4 = B . Then equations (4.85) take the form W:

= bzWi

+b3~:, wz"

=o,

W;

= b 3 ~2 O- b2wo, 3

(4.89)

w;

=o,

(4.90)

where X = 5 , . . . , n. Exterior differentiation of the last two equations gives the following quadratic equations: w;Aw;=O,

w2Aw4x = O .

Since by (4.89), I-forms w: and w i are linearly independent, it follows from the last equations that w,x=o,

X=5,

..., n.

This means that the submanifold V 3 belongs to the four-dimensional space P 4 spanned by the points A ~ , A I , A ~and , A A4. ~ In case (b), the submanifold V 3 is thus a hypersurface in the space P4,being defined in this space by the system of equations (4.82) (with a = 4), (4.84) and (4.89). We will investigate the consistency of the last system. For this purpose we apply exterior differentiation to equations (4.89). As a result, we obtain the following quadratic equations:

+

+

(Ab2 - 2 ( b 2 ~ ; b 3 ~ ; )A) W : Ab3 A W ; = 0 , Ab3 A W: - (Abz - 2(b2~33- b 3 ~ : ) A ) W: = 0,

(4.91)

where

+

+

+

A62 = dbz bz(w; w:) (czbz - ~ 3 b 3 ) ~ : , Ab3 = db3 + b 3 ( ~ : W ; - W: w:) bz(w; - w:)

+

+

+ ( ~ 2 b +3 ~ s b z ) ~ ; .

The system of exterior equations (4.87) and (4.91) consists of s1 = 4 independent equations. They contain q = 6 characteristic forms. As a result, the character s2 = q - s1 = 2, and the Cartan number Q = s1 2s2 = 8. The number N of parameters, on which the most general integral element depends,

+

Notes

141

is also equal t o 8. Since Q = N , by the C a r t a n test, the system of Pfaffian equations (4.82), (4.84) a n d (4.89) is in involution, an d its general integral manifold depends on two arbitrary functions of two variables. Thus, three-dimensional parabolic submanifolds V? of rank 2 in P" that have n o real singularities exist, in both cases (a) and (b), and a general integral manifold, defining such submanifolds, depends o n two arbitrary functions of t w o variables. : of constant curvature be realized in a Now let a simply connected space V projective space P" and let G be its proper domain. If V: is a three-dimensional parabolic submanifold of rank 2 in P" th a t has no real singularities, then the intersection V 3 G is a submanifold having t h e sam e properties in V,.. Such submanifolds consequently also exist in V," , a n d a general integral manifold of t h e system, defining such submanifolds, depends o n two arbitrary functions of two variables.

n

NOTES 4.1-4.4. Tangentially degenerate submanifolds V" of rank r < n were considered by 8. Cartan in [Ca 161 in connection with his study of metric deformation of hypersurfaces, and in [Ca 191 in connection with his study of manifolds of constant curvature; by Yanenko in [Ya 531 in connection with his study of metric deformation of submanifolds of arbitrary class; by Akivis in [A 571 and [A 62b]; by Savelyev in [Sav 571 and [Sav 601; and by Ryzhkov in [Ry 601 (see also the survey paper [AR 641). Brauner in [Br 381 studied such submanifolds in Euclidean n-space. Most of the results presented in these sections are due to Akivis (see [A 571). 4.5. The results presented in this section are due to Akivis (see [A 62bl). 4.6. The results presented in this section are due to Safaryan (see [Sa 701). 4.7. The notion of the index of relative nullity was introduced by Chern and Kuiper in [CK 521 (see also [KN 631, vol. 2, p. 348). Complete parabolic submanifolds in an Euclidean n-dimensional space were studied by Borisenko in [Bori 821 and [Bori 851. In [Bori 921, he used the notion of parabolicity t o formulate and prove a theorem on the unique determination of V" c E" from its Grassmann image. Akivis recognized that the problem of finding singularities on complete parabolic submanifolds in a Riemannian space V," of constant curvature and of distinguishing those submanifolds that have no singularities, is related not so much to the metric as to the projective structure of the spaces V,". In this section we follow Akivis' paper [A 87b] written on this subject. The hypersurface defined by equations (4.78) was studied by Sacksteder in [Sac 601.

143

Chapter 5

Submanifolds with Asymptotic and Conjugate Distributions 5.1

Distributions on Submanifolds of a Projective Space

Consider a submanifold V" in a projective space P". If a subspace A: of dimension p is given in each tangent subspace Til'(Vm), we say that a distribution AP is defined on V'" (cf. Section 3.2 where distributions were already discussed). As always, we assume that the subspaces A: depend differentiably on the point x. With this distribution we associate a bundle of first order moving frames, which is a subbundle of the bundle of first order moving frames associated with the submanifold Vm: the points Ab, b = 1,. . . , p , of moving frames of this subbundle are in the subspace A: passing through the point x = A*. If at a point z a vector ( E A: is given, then relative to the above moving frame associated with the point z we have the following equations: w"(()=O,

u = p + 1 , . . . , m.

Thus, the equations wu = 0 are the equations of the subspace A$ at the point X.

Since the subspace A$ is constant if the point x is fixed, and since the point x is fixed if the equations w' = 0, i = 1,.. . , m, hold, then under this condition the equations of infinitesimal displacements of the points A0 and A* of the subspace A: have the form:

6Ao = $Ao,

6Ab = ?r;Ao + r i A c .

(5.1)

144 5. SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

Hence the forms w l , u = p + 1 , . . . , m, vanish if w i = 0, i = 1, . . . , m. Therefore the forms w: are linear combinations of the forms w ' : - I ua i W i .

wu a

(5.2)

Equations (5.2) are the equations of the distribution AP in the chosen subbundle of frames of first order. A distribution AP is said to be integrable or holonomic (cf. subsection 1.2.4) if there exists a family of submanifolds VP of dimension p such that a) At any point x E VP the tangent subspace Ti(Vp) = A:, and b) Through any point x E V" there passes a unique submanifold VP. The submanifolds VP are called integral submanifolds of the distribution Ap. An integrable distribution AP defines a foliation of the submanifold V" into an ( m - p)-parameter family of submanifolds Vp. Let us find the conditions of integrability of a distribution AP. By the Frobenius theorem, this condition is that the exterior differentials dw" of the forms w u must vanish modulo these forms, i.e. the following relation must hold: 0

dwU

(mod w u ) .

(5.3)

Since dw" = W : A w U

+

W'

A W:

+

W"

Aw;,

then, by (5.2), we have: dw" = w: A w "

+ w" A w ; + lZbwa A w * + I&wa

A 0'.

(5.4)

Thus, by (5.3), for a distribution AP to be integrable, the term in equations (5.4) which does not contain the forms w" must vanish, i.e. we must have:

lG*] = 0.

(5.5)

We have arrived at the following result.

Theorem 5.1 A distribution AP is integrable if and only if condition (5.5) holds. The integral curves of a distribution AP are the curves that are tangent to the tangent subspaces A{ c AP at each of points x E V". All such curves satisfy the equation wu

= 0.

If a distribution AP is integrable, then its integral curves belong to its integral submanifolds VP. Note that integral curves of a distribution exist independently on the fact whether the distribution AP is integrable or not.

5.2

5.2

Asymptotic Distributions on Submanifolds

145

Asymptotic Distributions on Submanifolds

1. In Chapter 2 the asymptotic directions on a submanifold V" were defined as directions annihilating the second fundamental form @ ( 2 ) :

= bGw'wj A,,

Since @(Z) system of equations:

@(Z)(E)

= 0.

the asymptotic direction [ satisfies the following

b g [ j = 0.

(5.6)

Suppose that the tangent subspace Til)(V") has asubspace AP, c T i l ) ( V m ) with each of its directions asymptotic. We will call such subspace Ag asymptotic. If there are asymptotic subspaces A$ at any point c E Vm, then we say that the submanifold V" carries an asymptotic distribution AP of p dimensions. Integral curves of an asymptotic distribution are asymptotic lines of the submanifold V" . The osculating 2-planes of these curves belong to the tangent subspace Til)(Vm). We associate a bundle of frames to a submanifold V" carrying an asymptotic distribution in the same way as in Section 5.1. Then for any direction [ c A: we have

= btb:bEaEb

@(Z)([)

(5.7)

O.

These equations give the equations:

and B,,

-

btb = 0 .

(5.8)

This implies that B,, are the only nonvanishing points among the points &j = bP.2, defining the normal subspace Nil)(Vm). The number of Y these nonvanishing points is p(m - p )

1 1 + -(m - p ) ( m - p + 1) = -(m - p ) ( m+ p + 1). 2 2

It follows that for submanifolds V" carrying an asymptotic distribution AP, the dimension m ml of the osculating subspace Ti2)satisfies the inequality:

+

rn

+ ml 5 m + -21( m - p ) ( m+ p +

1).

By (5.8), equations (2.21) take the form:

w t = b:,w",

w," = bE,wa

+ bEvwu.

(5.9)

Taking exterior derivatives of the first group of these equations and using equations (5.9) and (5.21, we obtain the following exterior quadratic equations:

146 5. SUBMANIFOLDS WITH ASYMPTOTICA N D CONJUGATE DISTRIBUTIONS

+

+

+

- l ~ , , b ~ , w " }A w U

{Vb:u (--l:bb&, 1FubGb lrub:w)wb +(-libbEC 1rcb:,)wb A w c = 0,

+

(5.10)

where

V b z u = dbzu

+ bzuw: - bzvw: - b&w: + bt,wF.

It follows that the last term in (5.10) must vanish. Thus,

b:,,(lyc

+ b&1:,

- &) - bz,,l:b

= 0.

(5.11)

Equations (5.11) will play an important role in our further considerations. E N i l ) , we find that Contracting these equations with the points

A",

gah,(lrc - /:b)

+

N

- gcu/:b Bbul,",

= 0.

(5.12)

Equations (5.11) and (5.12) are similar to equations (3.24) and (4.43). Following Lumiste (see [Lu 59]), we will prove the following result.

Theorem 5.2 Let a submanifold Vm of a projective space P" carry an asymptotic distribution APs p > 1,_and the bundle of moving frames be chosen as above. If the points B,, and Bb, are linearly independent f o r a n y fixed a and b, a # b, then the asymptotic distribution AP o n Vm is integrable, and its integral manifolds are p-planes. Proof. Setting

c = a in relations (5.12), we obtain

ga,(l:,

- 2l:b)

By the linear independence of the points that

lra- 21:b

-

+ Bb,/:,

= 0.

(5.13)

gauand gb,, equations (5.13) imply

= 0,

= 0.

(5.14)

Interchanging a and b in the first of these equations, we find that l:b

- 21y* = 0.

(5.15)

Thus, it follows from equations (5.14) and (5.15) that /:b

= 0.

(5.16)

By Theorem 5.1, it follows from equation (5.16) that the distribution AP is integrable. Further, by (5.16), we have

5.2 Asymptotic Distributions on Submanifolds

147

This and equations (5.9) give the following equations:

dAo

w:Ao

+ w'Ab

(mod w " ) , dAb

wfAo + w ~ A ,(mod w").

(5.17)

Equations (5.17) mean that if w' = 0, the point A0 describes a pdimensional plane. Thus, the submanifold V" is foliated by an ( m - p)-parameter family of pdimensional planes. 2. In the second osculating subspace Ti2'(Vrn), we now consider the subspace Ex defined by the points Ao,Ai and B,,, where B,, = b&A,. Let us prove that this subspace is invariantly connected with the asymptotic subspace A$. In fact, it follows from equations (5.10) that if the point t is fixed (i.e. wi = 0), we have

Vsb;,, = 6b;,

+ b:,,:n;

- b&r; - b&rt

+ bfU$

= 0,

(5.18)

where, as above, the symbol 6 denotes differentiation with respect to secondary parameters, i. e. 6 is the restriction of the differential d to a fiber of the frame bundle associated with the submanifold V". We set T," = w,"(6). For the points B,, = bz,,x,, we get from (5.18) and the obvious equation V s A , = 0, we have:

-

-

-

- + bzuVaA,-

VsB,, = Vabz,A,

Conditions (5.19) mean that the points

g,,

= 0.

(5.19)

determine an invariant subspace

zcin the kormal subspace N i l ) . Together with the tangent subspace Ti'),the subspace Ex determines an invariant subspace Ex in the second osculating subspace Ti2)(Vrn). It is easy to see that the dimension of the subspace Ex does not exceed m p(n - p ) . The hypotheses of Theorem 5Z: arcequivalent to the condition that the (m-p-1)-dimensional subspaces E, C Ez determined by the points B,,, where a is fixed, are mutually in general position in E x . Since dim E, = m p - 1, for this co_nditions to be satisfied, it is necessary that the dimension of the subspace Ex be not less than 2 ( m - p - 1) 1. In particular, if the dimension of the subspace Exis equal to p ( m - p ) - 1, the hypotheses of Theorem 5.2 will be automatically satisfied. If p = m - 1 , the subspaces E, are reduced to the points Barn. If these points lie on a straight line but mutually are linearly independent, then the hypotheses of Theorem 5.2 will hold. Thus, if a submanifold V" carries an ( m- 1)-dimensional distribution A"-' of asymptotic directions, the subspaces Ex are of dimension greater than or equal t o m 2, and the subspaces E, are mutually distinct, then the submanifold V" is foliated by a one-parameter family of ( m - 1)-dimensional planes. Let us establish a geometric meaning for the subspace Ex for a submanifold V" which is foliated by an (rn - p)-parameter family of pplanes A:. Take an arbitrary point y E A$:

+

-

-

+

+

-

-

+

148 5 . SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

+

(5.20)

Y = YOAO y"Aa.

If we differentiate the point y and projectivize the result by Ag, then by (5.2), (5.9) and (5.16), we find that

dy/A$ = ((~'6:

+ Y"l:,)Au +

YaBav)uU.

(5.21)

Thus, the tangent subspace to the submanifold V" a t a point y E EP is determined by the subspace A$ and the points Cv(y) =(YOS,U + ~ " l : , ) A u+ y a B a v .

(5.22)

We can see from this that the linear span of all such tangent subspaces is defined by the points Ao, A, ,A , and B,, , and thus coincides with the subspace E x . Moreover, in fact the subspace Ex depends not on the point x but on the plane generator A$ of the submanifold Vm. Therefore, the subspace Ex depends on the same m - p parameters on which the generator A; depends.

Submanifolds with a Complete System of Asymptotic Distributions

5.3

1. Suppose that the submanifold Vm carries a system of asymptotic distributions AP1, . . . , APk. This system is called complete if any pair of subspaces ASA and A 2 , A, p = 1 , .. ., k,X # p , are in general position, i.e. ASAn A 2 = x, and p l . . . + p k = m. For simplicity we consider the case k = 2. We will now prove the following result.

+

Theorem 5.3 If a submanifold Vm carries a complete two-component system of asymptotic distributions AP and Am-pJ then this submanifold is contained in its second osculating subspace Ti2)of dimension m + m l where ml 5 p ( m - p ) . Proof. For the submanifold Vm in question, the second fundamental tensor

(bj)has

the form:

Thus the second osculating subspace TL2)(Vm) is determined by the points Ao, Ai and B,, = bE,A,. Therefore, the dimension m ml of this subspace is not greater than m p ( m - p ) . We choose a moving frame in the same way as in Section 2.3, by taking the points Am+l,. . . ,Am+m, in the subspace Ti2).Then

+

+

b,": = 0 , a1 = m

+ ml + 1 , .

..,TI

5.9

Submanifolds wath a Complete System of Asymptotic Distributions 149

As in Section 2.3, we can prove that, as a result of the choice of moving frames indicated above, the forms wE1 are expressed in terms of the basis forms (2.41): UP' 1 1 = cP1 S i kW E ,

(5.23)

where the coefficients c:k are connected with the second fundamental tensor by equation (2.43). Since we now have

b"bb = 0, bt: = 0, equations (2.43) take the form: (5.24) b2u c::b = 0, b2u cig; = 0. Since the rank of the ml x p ( m - p ) matrix (b"d,) is equal to m l , equations (5.24) imply CPl

sib

= 0,

cP1

WP'

= 0.

SlV

= 0.

By (5.23), these equations give 11

(5.25)

Using equations (5.25), we obtain

+

+

dAi, = wf1Ao ~ f , A j ~ i : A j , . It follows that the second osculating subspace defined by the points Ao, Ai and Ail is constant, and the submanifold Vm lies completely in this subspace. Theorem 5.1 also implies the following result.

Theorem 5.4 Let a submanifold Vm carry a complete two-component system of asymptotic distributions AP and Am-p, where p > 1 and m - p > 1. I f in addition the following inequality holds: ml

2 2m a x h m -PI,

and each of the systems of points, {&, & U } where a and b are fixed and a # b and nu,&,} where u and v are fixed and u # v , are linearly independent, then the submanifold V" is doubly foliated by families of plane generators of dimensions p and m - p . The hypotheses of Theorem 5.4 are automatically satisfied if ml = p ( m - p ) and p > 1, m - p > 1. Thus a submanifold Vm which carries a complete twocomponent system of asymptotic distributions and whose second osculating subspace T i 2 )is of maximum dimension, is foliated by two families of plane generators of dimensions p and m - p . An example of a submanifold V m , which has a double foliation by families of plane generators of dimensions p and m - p and for which ml = p(m - p ) , is

AND CONJUGATE DISTRIBUTIONS 150 5 . SUBMANIFOLDS WITH ASYMPTOTIC

the Segre variety S ( p , m - p ) . This variety is embedded into a projective space P" of dimension n = m+p(m-p), which coincides with the second osculating subspace Ti2) of S ( p , m - p ) (see Section 2.2, cf. [Bert 241). If ml < p(m - p ) , the submanifolds in question can be obtained by projecting a Segre variety S ( p , m - p ) into a projective space P" of dimension n < m p(m - p ) (cf. [M 581). We will now prove that these examples exhaust all those submanifolds Vm , which are doubly foliated by families of plane generators. In fact, let us write the equations of such a submanifold in the parametric form:

+

x=t(A",A"),

a = I ,...,p , u = p + l , ..., m,

where A", A" are coordinates on Vm chosen in such a way that if A" = const, the point x describes a pdimensional plane generator A:, and if Aa = const, this point describes a q-dimensional plane generator A:, where q = m - p . If we denote by A0 and A, the basis points of the generator A:, and by Bo and B, the basis points of the generator A! of the submanifold Vm, then the equation of this submanifold can be written in the form x = A0

+ AUAa,

where the points A0 and A , depend on Xu, or in the form 2

= Bo + A,&,

where the points Bo and B, depend on A". But the last two equations imply that the equation of the submanifold Vm has the form 3:

= c + C,A"

+ CJ" + ca,xaxu,

(5.26)

where C, C,, C, and C,,, are fixed points of the space P". Introduce homogeneous coordinates on the generators A: and A: by setting Aa = $ and A" = c 5 0 7 respectively. Then equation (5.26) takes the form

where the indices 2 and 4 take the values from 0 to p and from 0 to q , respectively. If the points CGGare linearly independent, then the submanifold Vm is the Segre variety S(p,q) = PP x Pg imbedded into a projective space P" of dimension n = p q p q = m + p(m - p ) . On the other hand, if the points Cia are linearly dependent and the maximum number of linearly independent among them is equal to n 1, where m < n < m p(m - p ) , then the submanifold Vm is a projection of the Segre variety S(p, q ) onto a projective space P". Multicomponent systems of asymptotic distributions can be investigated in a manner similar to our considerations for complete two-component systems of asymptotic distributions, (see [Lu 591). In particular, Theorem 5.3 can be generalized as follows.

+ +

+

+

5.4

9-Dimensional Submanifolds Carrying a Net of Asymptotic Lines

151

Theorem 5.5 If a submanifold Vm carries a complete k-component system of asymptotic distributions A P I , . . .,A P k , then this submanifold is contained in its osculating subspace TJk)of order k , whose dimension satisfies the following inequality: E

dimTJk) i ~ P A , P A . .PA,, ~ . C=l

where summation is taken ouer the combinations ( A l , . . .,A,) 1 , 2 ,...,K , i.e.

5.4

of numbers

Three-Dimensional Submanifolds Carrying a Net of Asymptotic Lines

1. Consider a three-dimensional submanifold V 3in a projective space P" which carries a complete system of one-dimensional asymptotic distributions. Since such distributions are always integrable, their integral curves form a net of asymptotic lines on V 3 . Such submanifolds were briefly considered by Cartan in [Ca 181. In this section we will study this kind of submanifold in more detail. We place the vertices Aj of a moving frame associated with a point t E V 3 onto the tangents to the asymptotic lines. Then the second fundamental forms @&, of V 3 take the form:

@T2) = by2w'w2 + bg3w2w3 + b : l w 3 ~ ' .

(5.27)

Equations (5.27) imply that the number ml of linearly independent forms does not exceed three. We will consider the case rnl = 3, since other cases were already considered in Section 2.5. So, let rnl = 3. We can choose the points A,, (Y = 4 , . . . , n , so that the forms take the form:

@Tz,

a4 (2)

- 2w2w3,

a5 (2)

- 2w3w1,

q2,= 2w1w2,

a?;) = 0,

(5.28)

@F2)

where a1 = 7 , . . . , n. The forms and @;z) constitute a basis of the bundle of second fundamental forms associated with the point t E V 3 . Relations (5.28) imply that equations (2.41) and (2.42) are reduced to the form: w;=o, w 51- - w 3, w 61 - w 2, w;1

w; = w 3 , w;=o, w;=w1,

= 0,

a1

w;=w2, w;=w1, w;=o,

= 7 , . ..,n.

(5.29)

(5.30)

152 5.

SUBMANIFOLDS WITH

ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

Exterior differentiation of these equations, and application of Cartan's lemma to the obtained exterior quadratic equations, lead to equation (2.50):

where the quantities cyl\ are connected with the second fundamental tensor b:f. by equations (2.52), which (by the structure of the tensor b;; following from (5.29)) lead here to the following equations: 41

- ' 5011 2

- c63

011

def - p

c;; = c;; = c;; = c;; = c;

(5.32)

1 ,

= ;c;

= 0.

(5.33)

If in equations (5.31) we replace the quantities ct\ by their values (5.32)and (5,33),then equations (5.31)take the form: W i I

= c"lw1,

= C Q ' W 2 , w;' = crn'w3,

Wf'

(5.34)

and, by (5.28)and (5.34),the third fundamental forms @Ti) defined by equations (2.55)take the form: @a"

(3)

- 6ca1w1wZw3.

(5.35)

Two cases are possible: all the coefficients c'l are equal to zero or at least one of these coefficients is different from zero. 2. In the first case, when cal = 0, equations (5.34)give: UP' = 0. I1

(5.36)

By (5.36),we have dAi, = wf1Ao

+

WflAk

+w ~ " A . I1

31

This implies that the sax-dimensional second osculating subspace T5")(V3)is constant, and the submanifold V 3 lies i n this subspace. The submanifolds V 3 of this type are defined by the system of Pfaffian equations (2.5)and (5.29). Let us investigate this system. Exterior differentiation of equations (2.5) leads to identities, and exterior differentiation of equations (5.29) gives the following exterior quadratic equations:

+ +

(wy - w g ) A w 2 (wf - u : ) A u 3 = 0, ( w ~ - w ~ ) A w ~ + ( w ~ - w ~= O) , A w ~ ( w i - w t ) A w1 (w; - w:) A u 2 = 0,

(5.37)

5.4

3-Dimensional Submanifolds Carrying

+ +

a

Net of Asymptotic Lines

(w: - w z ) A w l + 2 w ; A w 2 (w; + w z -wE - w z ) (wi + w i -w: - w : ) A w l (wi - w ; ) A w 2 2 4 2 ~ A; w l (w: + u ; - U: Aw’ (u: (u;- LJ:) A w l w i -w: A w 2 2w; 2w: A w l (w: - w g ) A w 2 ( w i + w i - w: - w z ) (w: w i - w t -w,“) A w l 2 4 A w 2 (ui- w : )

+ +

+

+ (wi +

WE)

+ +

-ui)

+

+ -WE) +

+

153

A w 3 = 0, Aw3 = 0,

A w 3 = 0, Aw3 = 0, A w 3 = 0, A w 3 = 0.

(5.38)

In addition to the basis forms w i ,quadratic equations (5.37) and (5.38) contain 3 - w::!, i # j, w i w i - WE the following independent 1-forms: wf, w? w j , w i w i - w: - w ; , and w: w i - w: - wg. Their number is q = 15. The system consists of s1 = 9 independent exterior quadratic equations. Thus the successive characters of this system are s2 = 6 and s3 = 0. The Cartan number Q = ~1 2 ~ 2 3 ~ = 3 21. Applying Cartan’s lemma to equations (5.37), we find that

+

+

+

+

+

w: - w; w: - w; w ; - w;

= 21iZw2 + 21:3w3, = 21;3w2 21i3w3,

+ + 21;1wl, W! +2111~1, w; - wg = 21&w’ + 21:2w2, w; - w: = 2q2w1 + 21:2w2. = 21;3w3 = 2&w3

(5.39)

Replacing the forms wf - wi$ in quadratic equations (5.38) by their values taken from equations (5.39), and applying Cartan’s lemma to the quadratic equations obtained after this replacement, we arrive at the following equations:

= /i2w1+ aw2 + a2w3, + w t - w: - w i = 21i3w1 + 2a2w2 + 2a3w3, u; = li3w1 + a3w2 + Aw3, w i = li3w2 + bw3 + b2w1, + - - = 2 1 3 2 , ~+~ 2b2w3 + 2b3W1, ~7 = 1:,w2 + b3w3 + Bu’, w: = + + c2w2, + w i - wg - WE = 2 / : 2 ~ +3 2c2w1 + 2c3w2, w; = + cBwl+ wg w;

W:

W;

W:

W;

(5.40)

1+3

U:

1gZw3

cw2.

Equations (5.39) and (5.40) prove that the number N of parameters on which the most general three-dimensional integral element depends, is equal to 21: N = 21. Since N = Q, the system is in involution, and its general integral manifold V 3 c P6 depends on six arbitrary functions of two variables. 3. Suppose now that at least one of the coefficients ca’ is different from zero. Then there is only one independent form among the third fundamental and dimTi3)(V3) = 7. If we take the point A7 of our moving forms @$)(V3), frame in this subspace T i 3 ) ( V 3 )we , obtain:

154 5 . SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

a7(3)-0, - 6w1w2w3, aaa (3) -

02

= 8,...,n.

(5.41)

In the case under consideration, equations (5.34) take the form: w4 7 = W 1 ] w;

= w 2 , w: = w 3 ,

(5.42)

= 4,5,6.

(5.43)

= 0,

il

Let us prove that in this case the subspace TL3)(V3)is constant, and thus the submanifold V 3 lies in this 7-dimensional subspace. In fact, exterior differentiation of equations (5.43) gives the following exterior quadratic equations: wi A U : ~= 0 ,

which yield WL2

= 0.

(5.44)

Therefore, the subspace Ti3’(V3)defined by the paints Ao, Ai, A3+i and A7 i s constant] and the submanifold V 3 lies an this subspace. This result matches Theorem 5.4 stated above. 4. The submanifold V 3 in question is defined in the space P7 by Pfaffian equations (2.5), (5.29), (5.30), where 01 = 7, and (5.42). By (5.42), exterior differentiation of equations (5.30) leads to identities, and exterior differentiation of equations (5.29) give exterior quadratic equations (5.37) and (5.38). If we find exterior derivatives of equations (5.42) and apply (5.39) and (5.40), we arrive at the following three exterior quadratic equations:

0 = w:

+ w; + w; - 2w; - w;.

(5.46)

Equations (5.45) yield the following equations:

Zi3= 1z2, z;~ = 1i3,1t2 = 1f1 and

(5.47)

5.4

$-Dimensional Submanifolds Carrying a Net of Asymptotic Lines

155

We can now establish a geometric meaning for conditions (5.47). By (5.40), we have the following relations:

dw’ = (1& - 1i3)w2A w3 (mod wl), dw2 (li3- 1 3 w 3 A w 1 (mod w 2 ) , dw3 =- (2211 - 2 1 22)w1A w 2 (mod w3).

=

Thus, conditions (5.47) mean that the two-dimensional distributions defined by three asymptotic directions on a submanifold V 3 are integrable. Therefore, an asymptotic net on a submanifold V 3 i s holonomic. 5 . Let us find under what conditions the asymptotic lines of a submanifold V 3 in question are straight lines. First of all, note that equations (5.37) and (5.38), and therefore equations (5.39) and (5.40), are valid even in case the submanifold V 3 is not contained in a space P 6 . Next, we find the differentials of the points A1,A2 and A3 located on the tangents to the asymptotic lines of the submanifold V 3 passing through a point z = Ao. Applying equations (5.39), (5.40) and (5.47), we find that

dA1 = w:Ao

+ (wl1 - c3w2 - b2w3)A1 +w1(cA2 + BA3) + w 2 &

+w3.&,

(5.49)

d A z = w ; A o +(w; - U ~ W ~ - C ~ U ’ ) A ~ + W ~ ( U A ~ $ C A ~ ) + (5.50) W~.&+U~A”~, and

dA3 = w;Ao

+ (wx - b3w1 - a 2 w ’ ) A 3 + ~ ~ ( b A 1+AA2) +w1X5 +w2A”4, (5.51)

where

-

A4 2 5

A6

+ I&Ai + a d z + = A5 + /:,A2 + b3A3 + b2A1 , = A6 4+CQAI +CzAz. = A4

(5.52)

Consider these equations together with the equation

d A o = w,OAo

+ w l A 1 + w ’ A ~+ w3A3.

(5.53)

Then from equations (5.49) and (5.53) we find that

d(Ao A A l )

(u:

+ w:)Ao A A1 +

A0

A (cA2

+ BA3)w’

(mod w 2 , w 3 ) .

It follows that the first family of asymptotic lines of a submanifold V 3 ,defined by the equations w 2 = w3 = 0, are straight lines if and only if the conditions

c=B=O

(5.54)

156 5.

SUBMANIFOLDS WITH

ASYMPTOTIC AND

CONJUGATE

DISTRIBUTIONS

are satisfied. Similarly, we can prove that the second and third families of asymptotic lines of the submanifold V 3 are straight lines if and only if the conditions a=C=O

(5.55)

b=A=O

(5.56)

and are satisfied, respectively. We now return to the general case. Consider the nondevelopable ruled surface A12 generated by the straight lines A0 AA1 and defined by the equations w1 = w3 = 0. Since by (5.49) and (5.53) we have

dAo dA1

= wgAo +wlAl (mod w 1 , w 3 ) , wYAo + (w: - c3w2)A2+ w2A6

(mod w’,w3),

the point 2 6 belongs to the 3-plane A0 AAl A A z A i 6 = El2 which contains the tangent 3-planes to the surface A12 a t the points of the straight line Ao-AA1. In a similar manner, the point A5 belongs to the 3-plane A0 A A1 A A2 A As = El3 which contains the tangent 3-planes to the ruled surface A13 defined by the equations w1 = w 2 = 0 at the points of the straight line A0 A A l . Similarlyequations (5.50) and (5.53) imply that ifw2 = w 1 = 0 , the straight line A0 AA2 describes the ruled surface A23 whose tangent planes at the points of this straight line belong to the 3-plane A0 A A2 A A3 A A4 = E23, and if w 2 = w3 = 0 , this straight line A0 A A2 describes the ruled surface A21 whose tangent planes at the points of this straight line belong to the 3-plane A0 A A2 A A1 A A6 = Ezl. Finally, by (5.51) and (5.53), the tangent planes to the ruled surfaces A31 and A32, described by the straight line A0 A A3 when w3 = w2 = 0 and w3 = w 1 = 0 respectively, belong to the 3-planes E31 = A0 A A3 A A1 A 2 5 and E32 = A0 A A3 A A2 A A4 respectively. Moreover, it is clear that El2 = E21, i.e. the tangent 3-planes to the surfaces A12 and A21 coincide. Similarly, we have E23 = E32 and E31 = E13.- From equation (5.52) it follows that the points Ao,Al,Az,A3,A4,& and A6 are linearlx independ_ent. Thus we-can specialize our moving frames by taking A4 = A4,A5 = AS and A6 = As. Then the point A4 belongs to the plane E23, which contains first order differential neighborhoods of the straight lines A0 A A2 and A0 A A3 of the surfaces A23 and A32; the point A5 belongs to the plane E31, which contains first order differential neighborhoods of the straight lines A0 AA3 and A0 AA1 of the surfaces A31 and A13; and the point A6 belongs to the plane El2, which contains first order differential neighborhoods of the straight lines A0 A A1 and A0 A A2 of the surfaces A12 and Azl. Having made the choice of moving frames indicated above, we obtain 12 11

- 1322 - li3= 0 ,

= a3 = bz = b3 = ~2 = ~3 = 0 .

By (5.57), equations (5.39) and (5.40) take the form:

(5.57)

5.4

3-Dimensional Submanifolds Carrying a N e t of Asymptotic Lines

w i = aw2, w l = bw3, w: = c w l , w; = A w 3 , W : = Bwl, W : = C w 2 ,

- w; = 21;3w3, - w j = 21ilW1,

w; - w: = 21;3w2, w; - w: = 21ilW3, w; - w; = 21y2w2, w; - w: = 21f2W',

w:

w;

+ w: - wo" - wq4 = 21i3W', + w; - wo" - w; = 21i1W2, + w; - w: - w: = 21:2w3. Let us further prove that the coefficients 1i3,1i1 and lyz w; w; w:

157

(5.58)

(5.59)

(5.60)

in the right-hand sides of equations (5.59) and (5.60) can be reduced to 0. By formulas (5.58)(5.60), the differentials of the points A4, A s and A6 satisfy the following relations:

It follows that if we replace the point A7 of our moving frame by the point

-

A7

= A7 - 21:,A4 - 2 1 & A ~- 21f2A6,

then the coefficients 1i3,1& and 172 will be reduced to 0, i.e. in addition to relations (5.57), we obtain the following relations:

ii3 = 1:' =

= 0.

(5.61)

After this specialization of moving frames, equations (5.59) and (5.60) take the form (5.62)

w: = w;+w; - w ; , w; = w; + w ; - w;, w: = w ; + w ; - w : ,

(5.63)

respectively. In addition, by (5.61), equations (5.46) and (5.48) imply w; = w ; + w ; +w:

- 2w:.

(5.64)

158 5 . SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

6. The remaining coefficients in equations (5.58) cannot be reduced to 0 since, as we will show in subsection 5.4.7, they are relative invariants of the submanifold V 3 in question. As we proved above, the vanishing of these coefficients is related to the rectilinearity of asymptotic lines of the submanifold v3.

Consider the case where all these coefficients are equal to 0, i.e. where the submanifold V 3 carries three families of rectilinear asymptotic lines. In this case the ruled surfaces A12 and A21, which we considered in subsection 5.4.5, are two families of rectilinear generators of the same quadric located in the 3-plane Elz, the surfaces A23 and A32 are two families of rectilinear generators of the same quadric located in the 3-plane E23, and the surfaces A31 and A13 are two families of rectilinear generators of the same quadric located in the 3-plane &I. Thus, the submanifold V 3 in question is foliated by three families of quadrics, and its parametric equation can be written in the form: x = A0 +u'Ai +u2A2 +u3A3+u2u3A4 +u3u1A5+u'u2A6 +u1u2u3A7. (5.65)

Introduce homogeneous coordinates (so,d ) ,(yo, yl) and ( z o ,2') so that u1 = 2' 0 0 1 u2 = and u3 = $, and homogeneous coordinates w i j k ,i , j , k = 0 , 1 , in the space P7.Then equation (5.65) can be written in the form: (5.66)

This equation proves that the submanifold (5.65) is the Segre variety S(1, 1, 1) which is an imbedding of the direct product of three projective straight lines into the space P7. We will now return to the general case and establish another geometric meaning of the relative invariants a , b, c , A , B and C. Consider one of the two-dimensional surfaces by which the submanifold V 3 is foliated. Let us take, for example, the surface V z , defined by the equation u3 = 0, and find the second fundamental forms of this surface. By equation (5.28), one of the nonvanishing second fundamental forms of this surface is the form

a&)= 2w1w2.

(5.67)

Besides this form, another nonvanishing second fundamental form of this surface is the form

q2) = w 1w 13 + w

2w2, 3

which by (5.58) takes the form:

a)&) = B(W')2

+ U(W"2.

(5.68)

Therefore, the relative invariants B and u are the coefficients of the second nonvanishing second fundamental form of the surface V:. Since each of the

5.4

PDimensional Submanifolds Carrying a Net of Asymptotic Lines

159

surfaces Vz has only two nonvanishing second fundamental forms, the second osculating subspaces of these surfaces are four-dimensional, and each of these surfaces carries a conjugate net. It is easy to prove that this conjugate net is real. Similarly, the relative invariants C and b are the coefficients of the second nonvanishing second fundamental form of the surface V:, defined by the equation w1 = 0, and the relative invariants A and c are the coefficients of the second nonvanishing second fundamental form of the surface V . , defined by the equation w 2 = 0. Our previous considerations can be easily complemented by the following result.

Theorem 5.6 If the asymptotic lines of a submanifold V 3 are asymptoiic lines on all two-dimensional surfaces by which the submanifold V 3 is foliated, then these asymptotic lines are rectilinear, and the two-dimensional surfaces are quadrics.

Proof. Consider, for example, the surface V: defined by the equation w 3 = 0. This surface carries asymptotic lines if and only if its second fundamen-

tal form (5.68) is proportional to the form (5.67), i.e. if B = Q = 0. Similarly, surfaces V;”and V. carry asymptotic lines if and only if the conditions C = b = 0 and A = c = 0 hold, respectively. But as we proved earlier, the simultaneous vanishing of all these coefficients leads to the rectilinearity of all asymptotic lines of a submanifold V 3 ,and consequently, V 3 is foliated by three families of quadrics. 7. Let us again return to the general case. Exterior differentiation of equations (5.58) and (5.62)-(5.64) gives the following exterior quadratic equations:

0: A w l 0: Aw’

+ Aa A w 2 + 0;A w3 = 0 ,

+ A b A w 3 + fl& A w l = 0 , !2: A w 3 + AcA w1 +fig A w2 = 0 , 0;A w l + 0: A w 2 + AA A w 3 = 0 , 0; + 0;A w3 + AB A w1 = 0, 0: A w 3 + 0:Aw’ + A C A w 2 = 0 , 0;A w l + 20; A w2 + 20; A w3 = 0, fl; A w 2 + 20; A w 3 + 20; Aw’ = 0, 0;A w3 + 2 3 g A w ’ + 2 0 ; Aw’ = 0, where

2Cl: A w 2 20: A w 3 2 0 ; A w1 2 0 ; A w3 2n; A w1 20: A w 2

+ 0;A w 3 = 0 , + 0 ? r \ w 1 = 0,

+ 0; A w 2 = 0 , + 0: A w Z = 0 , + 0;A w 3 = 0, + 0;A w l = 0 ,

(5.69)

160 5. SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

+ + + + + +

+ + + + + +

Au = da a(wg - 2w: wi), Ab = db b(w1 - 2 4 w i ) , Ac = dc c(w," - 2wi w g ) , A A = dA A(w; - 2 4 w g ) , A B = dB B(wt - 2 w i w i ) , AC = dC C ( w t - 2w; w ; ) , 0: = w: - baw2 - CAw3, !2: = w: - cbw3 - ABw', 0;= U : - ~ C U '- BCw2, Note that by (5.69), exterior differentiation of equation (5.64) gives an identity. Applying Cartan's lemma to equations (5.69), we obtain the following Pfaffian equations:

Aa

= =rwl+ =qw'+

w2-w; wj-w;

AA = Ab = w:-w;

= p 2 +

wg-wy

=TO'+

(5.70)

AB = Ac = u:-wy

=qw3+

wi-w;

= p 3 +

AC =

0:

w: w:

n;

+ abw2 + ACw3, = rw3 + caw' + CBw2, = pw'

= qw2 + bcw3 + BAw' ,

=

a;=

2rw1+ ng = 2qw'+

2rw2

+2qw3, 2pw3,

(5.71)

(5.72)

2p2.

It follows from equations (5.69) and (5.70)-(5.72) that we have: q = 18, ~1 = 15,~2 = 3 , S z = 0, Q = ~1

+ 2 ~ +2 3 ~ 3 21, N = 18.

Since Q # N , the system is not in involution, and we need to find its prolongation. Note that equations (5.70) imply that the quantities a , b, c , A , B and C are relative invariants. 8. We will now prove that some of the coefficients in expansions (5.70)(5.72) can be reduced to 0. To prove this, we apply (5.58), (5.62), and (5.64) to find the differentials of the points Aq, A5 and As. As a result, we find that

5.4

%Dimensional Submanifolds Carrying a Net of Asymptotic Lines

+ + + + + +

+ + + + + +

161

+ ( w i - @)A2 + ( w i - rw1)A3 + u1(A7 + P A ] +Q A+~rA3), + (u!- rwZ)A3+ ( w i - p 2 ) A 1 + d ( A 7 + pA1+ qAz + PA^), + (u: - p 3 ) A 1+ - pw3)Az + w3(A7 + pA1 + qAz + rA3).

dA4 = W ~ A O(abw' ACw3)A1 + ~ t A 4 Cw2A5 bw3& dA5 = w;Ao ( b w 3 BAw')Az +u;A5 Ao3A6 cw1A4 dA6 = w:Ao (caw' CBu2)A3 +&:As Bwl-44 awZA5

(id;

It follows that if we replace the vertex A7 of our moving frame by the point

-

A7 = A7

+ PAI+ QAZ+ rA3,

we reduce the coefficients p, g and r to 0, i.e. we obtain: p = q = r = 0.

(5.73)

Note that the above choice of the vertex A7 of moving frames is consistent with our previous choice made in subsection 5.4.5, since there we used a linear combination of this point with the points As+;, while here we use its linear combination with the points A;. Let us write the expression of dA4 using equations (5.73) and replacing the forms w: and ws by their values taken from equations (5.70):

+

dA4 = W:AO ( a b d +(w:

Consider the point

-

+ ACw3)A1+ (w: + ylw2 + z1w3)Az

+ t 1 w 2 + ylu3)A3+ ~ t A +4 Cu2A5+ bw3As + w1A7.

x4= A4 - ylAo. Its differential has the form:

+ Y ~ ( w :- wgO) - &)A0 + ( a b d + ACw3 - ylw')A1 + r1u3)Az+ (u;+ t1u2)A3+ w t i 4 + Cu2A5+ bw3A6 + w1A7 If we change our moving frames by replacing the vertex A4 by the point A4, then dA4

= (w,"

+(u:

we reduce the coefficient yL to 0. In a similar m_nner, by replacing the vertices As and A6 by the points A5 = A5 - yzAo and A6 = A6 - y3A0 respectively, we reduce o the coefficients y z and y3. to 0. As a result of this choice of moving frame, we obtain: y1 = yz = y3 = 0 .

(5.74)

9. As we showed earlier, the system of Pfaffian equations (5.58)) (5.59), (5.62) and (5.64) is not in involution. However, this does not mean that this system has no solution. It followsfrom the general theory (see Section 1.5) that the further investigation of this system is connected with its successive differential prolongations. As a straightforward calculation shows, in our case even after double prolongation of the system (5.58), (5.59), (5.62) and (5.64) the prolonged system is still not in involution. Therefore, we consider some particular

162 5.

SUBMANIFOLDS WITH

ASYMPTOTIC AND

CONJUGATE

DISTRIBUTIONS

cases of the system (5.58), (5.59), (5.62) and (5.64) which are characterized by the vanishing of some of the coefficients Q, b, c, A , 3 and C . In subsection 5.4.7 we already considered the case when all these coefficients vanish. Consider now the following case:

B=c=O,

(5.75)

C=a=O.

First of all note that since B = a = 0, the considerations in subsection 5.4.6 prove that the submanifold V 3 in question is foliated b y a family of quadrics V3". Next, by (5.70), equations (5.75) imply a1

= a3 = pz = p3 = 0 ,

21

= 23 = z2 = 2 3 = 0 .

(5.76)

By (5.75) and (5.76), four equations of system (5.70) are identically satisfied. By (5.75), (5.76), (5.73), and (5.74), the remaining eight equations of system (5.70), and equations (5.71) and (5.72) take the following form:

- w; = 0,

w:

- W ! = 0, w ; -wl" = 0, w i - w; = 0, w:

W:

- w ! = 0,

w3 ,

+

AA = Z ~ W ' p1w3, Ab = n2w3 ~ ~ w l ,

-U;

+

- W:

= z2w3,

wg=o,

wg=o,

W:

=o,

U;

=q

w: - w :

W$

= 0, w! - w: = ( z l

(5.77)

(5.78)

+

22)w3.

(5.79)

Exterior differentiation of equations (5.77)-(5.79) gives the following quadratic equations:

fl; A w 1 - 2 w i Aw3 = 0, fl; A w ' - 2 ~ :A w 2 + A z A~ w 3 = 0,

+

A w 2 Ap, A w 3 = 0, A C Y ~ A W ~ $ A X= O~, A W ~ 0: Aw' Ax2 A w 3 - 2w,O Au' = 0, A w 2 - 2w: A w 3 = 0, 0; ~~3 - 2 4 ~~2 = 0, A21

+

n:

Aw3 - 2

4

(5.80)

= 0,

and (5.82) where

5.4

%Dimensional Submanifolds Carrying a N e t of Asymptotic Lines

+ + + +

163

A z= ~ dzl ~ 1 ( 2 ~ 2: 4 ) - ~ A w ; , n;2 -= w;2 - w:, Ax2 = dx2 x : ~ ( ~ w , O - 2 4 ) - 2bw:, a7 - w7 - W E , ACUZ = d a z C U ~ ( ~ W; 3 ~ : w;) 4Aw30, 0: = W ; - w:. Ap1 = dp1 p 1 ( 2 ~ :- 3 ~ : w : ) 4 b 4 .

+ +

+

+

Application of Cartan's lemma to equations (5.80)-(5.82) gives

= -pwl Azl = s1w2 A/31 = t l w 2 2 4 = -pw3 2w:

- s1w3, Amp = p2w3 + qzw',

+ tlw3, + u1w3,

+

Ax2 = q2w3 r2w1, 2 ~= : -p2- r2w3,

(5.83)

and w+ - w:

w;- w w; - w :

= pw;,

5"-

(5.84)

- P 1

= pJ3.

Let us prove that by an appropriate change of the vertex A7 of our moving frames, we can reduce the quantity p to 0. We write the expression of dA7 using relations (5.77), (5.79), (5.83), and (5.84):

dA7 = w;Ao

+ (-h1w3 + pwl)Al +

+ ~+ :wzA5 4 ?I

Consider the point

dA7 =

x7

+ &)A2 + pw 3A3

( - $ T ~ W ~

+ (w: + w 3 ) A 6+ w77A7.

= A7 - pAo. Its differential has the form:

+

(w; - d p p(w; - w:))Ao - $s1w3A1 - $3w3A2 + w : A ~ U ; A ~ (u: z ~ w ~ ) w;X7. A ~

+

+ +

+

If we change our moving frames by replacing the vertex A7 by the point then we reduce the quantity p to 0:

p=o.

x7,

(5.85)

By (5.85), equations (5.83) and (5.84) take the form: 2w:

= -s1w3,

A z ~= s1w2 Ap1 = t l W 2 2w:

= 0.

w:-w:=o,

+ tlw3, + u1w3,

+

Aa2 = p2w3 q z w l , Ax2 = q2w3 T ~ w ' , 2~4.~8 = - T ~ w ~ ,

w:-w~=o,

+

(5.86)

w;-w,o=o.

(5.87)

Exterior differentiation of equations (5.87) gives the following three exterior quadratic equations:

164 5 . SUBMANIFOLDS WITH ASYMPTOTIC AND CONJUGATE DISTRIBUTIONS

w 07 A u 1 = 0 ,

w;Awz=O,

w;Au3=0.

(5.88)

It follows from equations (5.88) that w ; = 0.

(5.89)

Exterior differentiation of equation (5.89) leads to an identity. Exterior differentiation of equations (5.86) gives the following quadratic equations: As1 A w 3 = 0 , A p z A w3 As1 A w 2 At1 A w 3 = 0 , A q z A w 3 At1 A d Aul A w 3 = 0 , Arz A w 3

+

+

+ A q z A w1 = 0 , + Arz A w l = 0 ,

(5.90)

= 0,

where As1 At1

dsl

1

+ Si(3W; -LJ: - 2 4 ) - ~ z ~ w , O ,

= d t l + t i ( 3 w ; - 3 4 ) + 4ZlWg - 2 P l w ; , A U I = d u i U 1 ( 3 W ; - 4 4 + w;) Apz Aqz Arz

+

+

+ + + + + +

+ +

+lO@lwg 4A(bw: A w i ) 2 A ( s l u 2 r2w1), = dpz pz(3w; - 4 ~ 3 3 w : ) +10a& 4b(bw: Aw;) 2b(slw2 r z w l ) , = dqz q2(3W; - 3 4 ) 4C& - ~ C U Z W ~ , = drz rz(3w; - w : - 2 4 ) - zzwy.

+

+ +

+

Application of Cartan's lemma to equations (5.90) gives: As1 =

Atl = S1w2+ Aul = T1w2+

+ +

S1w3, A p z = Pzw3 Q z w l , T 1 w 3 , A q 2 = QZw3 R 2 w 1 , U1w3, Arz = R2w3.

(5.91)

Exterior equations (5.90) are independent and they contain six unknown 1-forms A s l , A t l , A u l , A p z , Aq2 and Arz. Thus, we have: q

=6,

~1

6 , sz = 0 , Q = ~1

+ 2.92 = 6 .

Equations (5.91) prove that the number N of parameters, on which the most general three-dimensional integral element depends, is equal to 6 : N = 6 . Since N = Q ,the system is i n involution, and its general integral manifold V 3 c P7,carrying three one-dimensional asymptotic distributions with t w o of t h e m being distributions of straight lines, depends on six arbitrary functions of one variable. This result implies that the general submanifolds V 3 carrying a complete system of one-dimensional asymptotic distributions, are not necessarily reduced to submanifolds with all asymptotic distributions consisting of straight lines.

5.5

5.5

Submanifolds with a Complete System of Conjugate Distributions

165

Submanifolds with a Complete System of Conjugate Distributions

1. In Chapter 3 we considered a submanifold Vm having m independent one-dimensional conjugate directions at each of its points. In this section we consider submanifolds carrying systems of conjugate directions of arbitrary dimension. Consider a complete system of distributions AP1, . . . , APk on Vm. This system is called a conjugate system of distributionsif any two directions E A!* and 77 E A?, X # p , at any point x E Vm are conjugate with respect to the system of the second fundamental forms of V"', i. e.


lji.

(6.68)

In this formula and in other formulas of this section there is no summation over the indices i and j. Since ci # cj if i # j , it follows that b$ = 0,

l;j

= 0 if i # j.

(6.69)

6.4

Submanifolds with Flat Normal Connection

189

The first of these relations proves that the submanifold V" carries a conjugate net (see Section 3.1). Moreover, by means of relations (6.69), equation (6.7) of the focus cone 0 takes the form: m

+ €&)

n:(€Olii

= 0,

(6.70)

i=l

i.e. the focus cone 0 decomposes into m bundles of hyperplanes whose centers are spanned by the (m - 1)-plane I, and one of the points

Let us write equations (6.64) and (6.65) taking into account relations (6.69): (6.72)

Contract first of these equations with the point Ap and add the obtained sum to the second of equations (6.72) multiplied by Ao. This gives:

&; Bj j

= cLj Bi, ,

(6.73)

I t follows from equation (6.73) that, if the points B;, are mutually distinct, then we have:

dOi= 0, i # j .

(6.74)

This equation implies that equation (6.6) of the focus variety F has the form: m ...

n(y0

+ yQ&)

= 0.

(6.75)

i=l

This proves that the focus variety F decomposes into m planes of dimension n - m - 1. Since, by our assumption, the variety F does not have multiple components, all these (n - m - 1)-planes are distinct. Moreover, the equation

defining the focal directions in the congruence of first normals n, (see Section 4.3), proves that to each of these (n - m - 1)-planes there corresponds the focal displacement along the curve Ci defined by the equations uj = 0 , j # i. In this case, the congruence of first normals n, is said to be conjugate to the submanifold Vm, and the corresponding normalization of V"' is also called conjugate. All these considerations prove the following theorem.

Theorem 6.4 Suppose that a submanifold Vm has the harmonic normalitaiion and the flat normal connection y,, . If the focus variety F of each of its first normals n, does not have multiple components, then the submanifold Vm carries a net of conjugate lines, and the focus cone @ of each of its second normals

190

6 . NORMALIZED SUBMANIFOLDS IN A PROJECTIVE SPACE

I, decomposes into m bundles of hyperplanes. If in addition, the points B;;] which together with 1, define the centers of these bundles, are mutually distinct, then the focus variety F decomposes into m planes of dimension n - m - 1, and the congruence of first normals n, is conjugate to the submanifold Vm. Moreover, the pseudocongruence of second normals I, is foliated by m families of torses corresponding to the lines of the conjugate net of Vm. H The last statement of Theorem 6.4 follows from formulas (6.4) and (6.69). 2. Consider now a submanifold Vm which is normalized by means of first and second normals n, and 1, and which carries a conjugate net C. If the straight lines AoAi are tangent to the conjugate lines passing through the point Ao, then, as was established in Chapter 3, on Vm we have: w a = 0, wr

and

= b:ui

. . w'3 = /jkWk.

(6.76) (6.77)

(see (3.5) and (3.14)). Since the submanifold Vm is normalized by means of first and second normals, we have:

(see (6.3) and (6.4)). Let us find a geometric meaning for the condition: li, = 0, i # j , which arose in the previous subsection (see (6.69)). Since

dAi = wPAo by (6.76), (6.79) and (6.80), we have

+ wiAj + w r A , ,

(6.79)

(6.80)

+

(6.81) dAi - w:Aj = (1;;Ao bzAcr)w;. It follows that if the point A. moves in any direction conjugate to the direction AoAi, i.e. wi = 0, the point A; describes a curve which is tangent to the subspace Til)(Vm). We will call such a point A; the strong pseudofocus of the straight line AoAi. (For the general definition of a pseudofocus see [Cas 501 and [Ba 65bl.) Thus all the points Ai are strong pseudofoci of the straight lines AoA;. Now it is not so difficult to prove the following result.

Theorem 6.5 I f Q submanifold Vm of a space P" carries a conjugate net and is normalized in such a way that its congruence of first normals n, is conjugate to V" and the points of antersection of the second normals 1, with the tangents to the conjugate lines are strong pseudofoci of these tangent lines, then the normal connection yn induced on Vm b y this normalatation is flat,

6.4

191

Submanifolds with Flat Normal Connection

Proof. If we take a subbundle of moving frames associated with Vm as indicated above, then the conditions of the theorem can be written as

b; = 0, ckj = 0, I,, = 0, i # j. (6.82) But these relations imply that the tensors of normal curvature, R;jj and R t j j , defined by formulas (6.48) and (6.49) vanish.

3. Let us prove the existence of the conjugate-harmonic normalization of a submanifold Vm carrying a conjugate net C and admitting the flat normal connection 7". By (6.69) and (6.74), equations (6.3) and (6.4) take the form:

. . = c;jwt, W p = l'+J'. (6.83) Exterior differentiation of (6.83) by means of (6.41), (6.42), (6.76) and (6.77) leads to the following conditions: W t

. . (2ij - C k j ) $ k ljj(l$k - l i j )

where i # j , k , j conditions:

#

- ( c t k - c k i ) / i j = 0,

(6.84)

+ ljjrik - lkklk.r l - 0,

(6.85)

k . Let us recall that equations (6.76) also imply the

+

bg(l$k - $) bjajlik - b&l& = 0. (6.86) Relations (6.84), (6.85) and (6.86) form a homogeneous system of equations with respect to the quantities / $ k . In general, this system has only the trivial solution:

(6.87) Geometrically, this solution signifies that the net C on Vm is holonomic and that the lines of this net also form a conjugate net on any submanifold defined by the equation wi= 0. The system of equations defining the conjugate-harmonic normalization of a submanifold Vm consists of equations (6.83) and quadratic equations that are obtained by exterior differentiation of (6.83). By (6.87), these quadratic equations have the form: I:k=o,

i#j,k, j # k .

+

(6.88) (VC$ - w:) A w i = 0, (Vlii b&;) A W' = 0. Applying Cartan's test to the system of equations (6.87) and (6.88), we find that q = s1 = m(n - m l ) , s2 = . . . = s, = 0, N = Q = m(n - m 1). Therefore, the system, defining the indicated above conjugate-harmonic normalization of a submanifold V"', is in involution, and a general integral manifold, defined by this system, depends on m(n - m 1) arbitrary functions of

+

+

+

one variable.

6. NORMALIZED SUBMANIFOLDS IN A P R O J E C T I V E SPACE

192

6.5

Intrinsic Normalization of Submanifolds

1. So far we have assigned the first and second normals, n2 and I , , of a submanifold Vm by the points Ao, A , and Ai of a moving frame. However, this is not always convenient. On many occasions, the first normal is assigned by the point A0 and the points

Y, = A ,

+ &Ail

(6.89)

+ .~iAo.

(6.90)

and the second normal by the points

Zi = A ,

As earlier, in this case, the normals n, and I , must be invariantly connected with the point 3: = Ao. Thus, if a point 2 is fixed, i.e. w' = 0, the points Y, must satisfy the conditions:

+

6Y, = U;AO uCY~,

(6.91)

and the points Zi the conditions: 6Zi = 4 Z j .

(6.92)

Differentiating equations (6.89) and (6.90) by means of the operator 6 (i.e. by setting ui = 0) and using the equations of infinitesimal displacement of projective frames and equations (2.5) and (6.2), we find that

6Zi = 4 Z j

+ (6%; + zi.0"

-~

j +4r:)Ao.

(6.94)

Comparing these equations with equations (6.91) and (6.92), we find that the conditions for the normals n, and I , to be invariant have the forms

and

+ .:= 0

6%;+ q.0" -

respectively, or

VayL and

+

=0

(6.95)

+ .:= 0.

(6.96)

v&zi

T;

Note that if yk = 0 and z; = 0, the last two equations are reduced to the form ~b = 0 and TP = 0. This corresponds to equations (6.3) and (6.4) and

6.5 Intrinsic Normalization of Submanifolds

193

implies that the normal n, is defined by the points A0 and A , and the normal 1, by the points Ai. The quantities yb satisfying equations (6.95) are called the normalizing objects o f t h e first kind and the quantities zi satisfying equations (6.96) are called the normalizing objects of the second kind. In Section 6.1 we assumed these quantities to be equal to zero and obtained a normalization of a submanifold V"' which however was not intrinsically connected with the geometry of a submanifold V"'. The problem of construction of an invariant normalization that is intrinsically connected with the structure of a submanifold V"' is more difficult. To construct such a normalization, we must express the normalizing objects yk and zi in terms of the fundamental objects generated by the submanifold V"', i.e. the objects which are obtained by differential prolongations of the initial equation wa = 0 of the submanifold V"' and those equations that are connected with a specific structure of this submanifold. As we noted earlier, to define an affine connection -yt in the tangent bundle, in addition to the first and second normals, n, and I,, we additionally have to find an ( n - m - 1)-dimensional subnormal m, c n, which is also invariantly connected with the point z = Ao. The latter subnormal can be given by the points: 2, = Y,

+ taAo.

(6.97)

The condition for this subnormal to be invariant has the form: SZa = $zP.

(6.98)

Differentiating equations (6.97) by means of the operator 6, we obtain

62, = 7rgz-p

+ (62, + 2a,:

- zpr{

+ 9.Ly

+ 7ri)Ao,

(6.99)

where yk is the normalizing object of the first kind satisfying equations (6.95). Comparing these equations with equations (6.98), we obtain the condition for the subnormal m, to be invariant in the form: (6.100) Note that if yb = 0 and z , = 0, these equations are reduced to the form = 0. This corresponds to equation (6.5) and proves that the subspace m, is spanned by the points A,. Let us prove that if the geometric object :y defining the first normal n, is constructed, then, by its prolongation, we can easily construct an object z , determining the subnormal m,. It follows from equations (6.95), which the object yk satisfies, that 7r:

(6.101)

194

6. NORMALIZED SUBMANIFOLDS

IN A PROJECTIVE SPACE

Exterior differentiation of these equations leads to the exterior quadratic equations:

{ V y ’a3. - 6j(w:

+ y t w i ) + y t b f j w g + y i b f j w t } A W’

=0.

(6.102)

Applying Cartan’s lemma to these quadratic equations and setting w’ = 0, we find that

V,YLj -:,(a;

+ y t 7 r i ) + bfj(Yt7r$ +

= 0.

(6.103)

Using equations (6.95) and the fact that V a b f j = 0, we eliminate the forms 7rL from equations (6.103). This gives:

Va(& - b f j y $ Z ) - $(T:

+ Y ~ T , ”=) 0.

(6.104)

Contracting these equations with respect to the indices i and j , we obtain the equation:

Va(d/- bPklY@Ya) 1 k - m(?r:+

= O’

(6.105)

Comparing these equations with equations (6.100), we see that the geometric object zd, defining the location of the subnormal m,, can be determined by the formula:

One can easily prove that the subnormal m z , defined by means of this geometric object z,, is the polar of ihe point A0 with respect to the focvs svrface F of the first normal n,. 2 . As an example of construction of the invariant normalization of a submanifold Vm which is intrinsically connected with its geometry, we will perform such a construction for a submanifold V’”belonging to its osculating subspace Ti2)of dimension m ml,where m1 5 i m ( m 1). On such a manifold, the following equations hold:

+

+

w a = 0, I

- 6% I,W 1 , b?. $1= bu. Jl

(6.107) (6.108)

(see (2.5) and (2.21)), and all second fundamental forms @?2)

= b$w’Wj

(6.109)

are linearly independent. The coefficients of these forms satisfy the following system of equations:

6.5 Intrinsic Normalization of Submanifolds

195

Vb; = b 5 k w k ,

(6.110)

where the coefficients bGk are symmetric in all lower indices (see formulas (2.26)), and as was indicated in Chapter 2, these coefficients do not form a tensor. We assume that a relative invariant (6.111) Z = Z(b;) # 0 can be constructed from the components of the tensor b 4 , and that this invariant is a homogeneous algebraic polynomial of degree p in b; . Consider the tensors (6.112)

By Euler's theorem on homogeneous functions, we have (6.113)

Using relation (6.112)' we write the latter equation in the form: b'jbu. = p .

(6.114)

'3

On the other hand, relation (6.112) implies that d l n l = bydb;.

(6.115)

Substituting expressions db; from (6.110) into (6.115), we obtain dln I = 2bgbrk8f - bZbGt9;

+ bzbGkwk,

(6.116)

wt

where 0; = wj" - 6Bw; 3 and O t = - 6iw;. But the relative invariant I satisfies an equation of the type

dr = q e

+ I&)),

which can be written as dlnl =6

+ zkWk.

(6.117)

Comparing formulas (6.116) and (6.117), we see that the form 0 must be a linear combination of the forms 0: and 0;. But since the form 0 does not have free indices, it is of the form:

1.e.

196

6. NORMALIZED S U B M A N I F O L D S I N A PROJECTIVE SPACE

dln I = AOi - PO,"

+I p k .

(6.118)

Comparing formulas (6.118) and (6.116), we find that 2bz brk = A$,

b z b$ = p6:

(6.119)

and

Ik = b'jb?. a $3,.

(6.120)

Contracting the first of equations (6.119) with respect to j and k and the second one with respect to cr and B and taking into account formula (6.114), we find the values of A and p : (6.121) By (6.121), formula (6.118) becomes: (6.122) Since the quantities b a form a tensor, they satisfy a system of differential equations of the form: Vbz = b2k,wk.

(6.123)

If we take exterior derivatives of system (2.26), apply the Cartan lemma and set w' = 0, we obtain (6.124)

uCp =

bgsb - 6;~:.

(6.125)

In the same way, we find from equations (6.123) that Vdbzk = -3b&j'uL{,

(6.126)

where

-

, kl i - Pb ki l r p - 6 L ~ p .

Consider the contraction b g , b r . Equations (6.124) and (6.123) imply that (6.127)

6.5 Intrinsic Normalization of Submanifolds

197

(6.128)

The quantities W$ form a tensor, and therefore

V6WP:f = 0.

(6.129)

The number of equations in system (6.127) is the same as the number of forms upY. This system can be solved with respect to the forms UP, if the tensor W';

is nonsingular. Assuming that this is the case, we introduce the tensor such that WY'$? Pi E l = 676!. c 3 Contracting relation (6.127) with the tensor

@ :

(6.130)

Ez,we obtain (6.131)

. -

bG&kWP,' = -A:€.

(6.132)

Then relations (6.131), (6.132) and (6.125) imply

Contracting the constructed quantities A,S, with the tensor ,b: new object

1; = AyP. :b

we obtain a

(6.134)

Applying (6.119) and (6.121), we can find from equations (6.133) and (6.126) that this new object satisfies the following system of differential equations:

Val: = --T:P ' m

+ bLkri.

(6.135)

In order to construct the normalizing objects y; and zi satisfying equations (6.95) and (6.96), we must separate the forms in formula (6.135). For this we need to construct one more object of the same type as I f . It is possible to prove that such an object can be constructed, and it has the following form: (6.136)

Applying (6.126), (6.123) and (6.124), we find that the object h i satisfy the equations

6. NORMALIZED SUBMANIFOLDS

198

IN A PROJECTIVE SPACE

" ' ~ b+ m ( m + 2)b27rj0] .

Vah'k,= 2 m ( n - m ) [ - p ( 2nn- m )

(6.137)

Eliminate now the forms 7rg from differential equations (6.135) and (6.137). As a result, we obtain (6.138)

+

Suppose that n > m 1, i.e. the submanifold Vm under consideration is not a hypersurface. Then we can set

'1

- p z ( n --m - 1) [p(m+ 2 ) G - 2(n - m)hb .

a -

pa

(6.139)

Relations (6.138) and (6.139) imply Vapi,

+ 7r;

= 0.

(6.140)

Comparing equations (6.140) and (6.95), we find that the constructed quantities p', form a normalizing object of the first kind. To construct a normalizing object of the second kind, we eliminate the forms T ; from equations (6.140) and (6.135). As a result, we obtain (6.141)

If we contract equation (6.141) with the tensor :bG, then we find that (6.142)

Introduce the notation: (6.143)

Then equations (6.142) and (6.143) imply

Var; + 7rp = 0.

(6.144)

Comparing equations (6.144) and (6.96), we find that the constructed quantities ri form a normalizing object of the second kind. Formulas (6.134) and (6.136) imply that if n > rn 1, the normalizing objects of the first and second kinds can be expressed in terms of the tensors b; and bGk connected with a third order differential neighborhood of the submanifold Vm. Thus, if n > m 1, there exists an invariant normalization of a submanifold V" C P" defined by a third order differential neighborhood of

+

+

Vm

6.6 Normalization of Submanifolds Carrying a Conjugate Net of Lines

199

+

As we will see in Chapter 7, if n = m 1, i.e. for a hypersurface, the construction of invariant normalization of V"' c Pm+' requires the use of objects connected with a fourth order differential neighborhood of the submanifold V" . As was noted in subsection 6.5.1,if a family of first normals n, is found for a submanifold V" c P", there is a standard procedure to construct a family of subnormals m, c n, but this requires the objects connected with a fourth order differential neighborhood of the submanifold Vm. In our construction of invariant normalization for a submanifold Vm c P", we mostly followed the paper [Os 661 by Ostianu. The assumption for existence of a nonvanishing invariant I , which is necessary for construction of the inverse tensor b:, was already substantiated in the papers of Weise (see [We 381). The possibility to construct the inverse tensor for the tensor Wi:, which is necessary in our construction, was proved in the paper [Os 661 mentioned above.

6.6

Normalization of Submanifolds Carrying a Conjugate Net of Lines

1. We will construct now an invariant normalization of a submanifold V" carrying a conjugate net. Suppose that m > 2 and that a conjugate net on Vm is not conic (see Section 3.6). On such a submanifold the following equations hold: wa

wi3 = li.w'+ ia 1jjwj

= 0,

+

C

(6.145)

/jkuk,

ijt j

(6.147)

k#i,j

(cf. (2.5), (3.5) and (3.14)). Exterior differentiation of equations (6.145) leads to an identity by means of relations (6.146). The exterior differentials of equations (6.146) have the form (3.21), and by relations (3.23), they can be written in the form:

Vbq A w' = 0;

(6.148)

in these equations as well as in (6.146) and (6.147) there is no summation in i, j,. . . unless there is the summation sign, and

Vbq = dbq + btw; - bp(2w: - w:).

(6.149)

Differentiating equations (6.147), we obtain an exterior quadratic equation of the type

200

6. NORMALIZED SUBMANIFOLDS IN A P R O J E C T I V E S P A C E

xA/jkAWk+fl;

=o,

(6.150)

k

where the quadratic form 0; is a linear combination of exterior products of the basis forms uiand

Al'. = Vl!. + bTwL, 3.3 13 Altj = Vl?. -wf, EJ Al:k = Vl:,,, i # j # k

# i.

In these expressions, the operator V is constructed in the same manner as in Chapter 2 with the additional restriction following from the fact that on a submanifold Vm carrying a conjugate net, the group of admissible transformations of moving frames is reduced by the equations = 0, i # j. Because of this, we have: Vli. = d / < .+ /? . (u:+ ui - 2&), 1.3 a3 13 Vl!. = dl?. + l!.(w: - w i ) , ?3 17 y Vljk = dljk 1ik(w8 + w i ' - u '1 - u:).

i~i

+

It follows from equations (6.148) and (6.150) that

Let us denote by li the expression: li

c

1 =-

m - 1 ,3 f a ,

(6.155)

Then, using relations (6.153), we find that the quantities li satisfy the equations:

Val; -P,.

= 0.

(6.156)

Comparing equations (6.156) and (6.96), we see that the latter equations are satisfied if we set Z' I - -1. a ,

(6.157)

i.e. the quantities -4 form a normalizing object of the second kind. This implies that the points Zi = Ai - liAo define the second normal I , intrinsically connected with the geometry of a submanifold V"' carrying a conjugate net.

6.6 Normalization of Submanifolds Carrying a Conjugate Net of Lines

201

We shall separate the construction of a normalizing object of the first kind into a few steps. 1. Construct the object:

-

13. r3

= 13. - l i ,

(6.158)

$3

which by (6.156) and (6.153) satisfies the equation

Vb1ij

= 0.

(6.159)

-

2. Consider the mean square of the quantities l j j :

(6.160) which satisfies the equation

-

V6li = 0

(6.161)

We will assume that li # 0 since otherwise l j . = Ii and as will be proved *? below, if m > 2, then this implies that a conjugate net on a submanifold Vm is conic. 3. Introduce the following quantities: (6.162) By relations (6.151), (6.152) and (6.161), these quantitiessatisfy the equations:

Introduce also the object:

-1"'

=

c-""'; 3 31'

i #i By equation (6.163), this object satisfies the equations:

(6.164)

6 . NORMALIZED SUBMANIFOLDS IN A PROJECTIVE SPACE

202

4. Assume now that the conditions of the generalized Segre theorem hold on a submanifold Vm (see Section 3.3), i.e. we have ml < m, and any subsystem of points B, = b y A , , consisting of m- 1 points, is of rank ml. For the matrix B = ( b q ) of coordinates of the points & this assumption

-

-

means that its rank ml-and it will not be reduced if we delete any column. The matrix B = ( b y ) satisfies the same conditions as the matrix

B. The matrix

in equations (6.165) is the Gramm matrix for the nonsingular matrix obtained from the matriz? by deleting the ith column. Thus, this matrix has the inverse matrix ( b k p ) . By (6.163)) the entries of the matrices tions: "4- 2 6-4 Sbi , T Oo 6 X i p 2pip7r;

+

RP)and pip)satisfy the equa-

-_

+ b-7s i IT^a -0 - , - $&T; - b;p?r'. = 0.

p + b-07 i T-,

(6.167)

zp) @i,)

These formulas prove that the matrices and are the matrices of components of mutually inverse symmetric tensors that are connected only with transformations of the points A,. 5. Consider the system of the following quantities:

-1; = -.bLplp'. -.

(6.168)

By (6.167) and (6.165), these quantities satisfy the equations:

-+

Val:,

T:,

= 0.

(6.169)

Comparing these equations with equations (6.147) defining the normalized object of the first kind we see that the latter equations are satisfied if we set .

y:,

-

= I:,.

(6.170)

+

Thus, the points Ya = A , FiAi and the point A0 determine the first normal n, intrinsically connected with a submanifold Vm. We will now clarify the geometric meaning of the first normal we just constructed. To do this, we find the osculating two-plane E,? of the curve Ci of the conjugate net C on the manifold Vm. This curve is defined by the equations:

6.6 Normalization of Submanifolds Carrying a Conjugate Net of Lines

wJ = 0, j

# i.

203

(6.171)

For the curve Ci we have:

+

dAo = w,OAo w ’ A ~ , d2Ao ( X l i , A j b:A,)(wi)2 j#i

+

(mod Ao,Ai).

(6.172)

By setting

we conclude that the osculating two-plane E: of the curve Ci is defined by the points Ao, Ai and L;. Let us change the normalization of the point Li by setting (6.173)

Consider the first normal defined by the points A0 and Y, whose form is given by relations (6.89). Eliminatingthe points A , from relations (6.173) and (6.91), we find that zi

=

x(qi

-X:Y’&)Aj +%qY,.

(6.174)

j#i

ti

The quantities -@$; give a deviation of the osculating two-plane E; from the normal defined by the points Ya.The quantity

*,3 3#*

is the mean square of these deviations. We will try to find the quantities y: in such a way that this mean square is a minimum. To do this, we find

e:

At the point where the minimumoccurs, all these derivatives must vanish. This implies that

-

p j

= T;Pyip.

4and using (6.168), we find that -= (6.175)

Solving these equations with respect to -.

y‘, = w,#j

-,

13a ’

6 . NORMALIZED SUBMANIFOLDS IN

204

A PROJECTIVE SPACE

Thus, the first normal we have constructed minimizes the mean square S of above mentioned deviations. We will next clarify the geometric meaning of the second normal we have constructed in this section. Consider a family of lines (6.171) of the conjugate net C on the submanifold Vm. Consider also the straight line AOAi tangent to the curve Ci and passing through the point Ao, and consider the displacements of this straight line along the integral curves of the equation w' = 0 where i is fixed. For such displacements we have:

(6.176)

We define the foci of the straight line AoAi as the points

satisfying the condition:

dX AAo AAi

0

:

(6.177)

for some integral curve of the equation w g = 0 passing through the point A0 (cf. Section 3.5). Since

dX

= (lik + z6i)wkAj

(mod Ao, Ad),

(6.178)

it follows from condition (6.177) that (6.179)

This system has a nontrivial solution defining a focal displacement of the straight line AoAi if and only if det(lfk

+ 16i) = 0.

(6.180)

Equation (6.180) is an algebraic equation of degree m - 1 with respect to I. This equation can be written in the form: (6.181)

By the Vikta theorem, we have (6.182)

where

11,

. . . , 1,-1

are roots of equation (6.180). Thus the point

Notes

205

1 FI. -- A I. - __

m - 1 . .

/;,A0

(6.183)

3 #I

is the harmonic pole of the point A0 with respect to the foci X, = A; zjAo, j # i. B u t by (6.155), the points F; coincide with t h e points Zi = Ai - liAo determining the location of th e second normal I,. Note t h a t if th e conjugate net C o n a submanifold Vm is the conjugate system, i.e. if I;, = 0, i , j # k, j # 6 , then equations (6.179) take the form:

+

(I;,

+ z ) w j = 0.

It follows t h a t t h e foci X, of t h e straight line AoAi are determined by t h e formulas: Xi = Ai - ':,A0 a n d correspond to the focal displacement along t h e curve Cj. We can now justify o u r assumption 7;. # 0 used in formulas (6.162). If lj = 0, then formulas (6.160) and (6.158) imply t h a t

-

-lij = 0 a n d

= li.

B u t this implies that all foci of th e straight line AoAj coincide, an d if m > 2, t h en the latter fact means that a conjugate system o n Vm is conic (cf. Section 3.6). We excluded this case from consideration in this section.

NOTES 6.1. In the recently published book [Sto gab], Stolyarov considers the topics closed to that of this chapter. 6.2. Schouten and Haantjes in [SH 361 considered the geometry of the general normalized submanifolds V" in spaces with a projective connection, and as an application of their studies, they investigated a hypersurface in P". The general normalized submanifolds V" in P" were considered by Grove in [Gro 391, Norden (see his papers [N 351, [N 371, [N 45a], [N 471, [N 481 and the book [N 761) and Atanasyan in [At 58bl. The central normalization for submanifolds in the affine space was considered by Atanasyan in [At 521 who called this normalization axial. Ivlev in [Iv 901 showed that a torsion-free projective connection is invariantly associated with a codimension two normalized submanifold V" C P" and a tangentially degenerate normalized submanifold V" C P" and found the fields of geometric images defined by the curvature tensor of this connection. 6.3. The normal connection in a Euclidean space was actually considered by E. Cartan in his lectures [Ca 601. The curvature tensor of this connection was called by Cartan the geodesic torsion of a submanifold. For a normalized submanifold in a space of constant curvature, the normal connection was investigated by Chen in [Ch 731, Chakmazyan in [Cha 77a], [Cha 77b] and [Cha 78b] and Lumiste and Chakmazyan in [LC 811. For a normalized submanifold in an affine space, the normal connection was investigated by Akivis and Chakmazyan in [AC 751 and by Chakmazyan in [Cha 77~1, [Cha 84b], [Cha 871 and [Cha 891 (see also the book [Cha 901). Submanifolds V" admitting a parallel p-dimensional subbundle in the normal bundle were considered

206

6.NORMALIZED SUBMANIFOLDS IN A PROJECTIVE SPACE

by Chakmazyan in [Cha 761, [Cha 77a], [Cha 77b], [Cha 77d], [Cha 77e], [Cha 78a], [Cha 78b], [Cha 801, [Cha 84b] and [Cha 891 and Lumiste and Chakmazyan in [LC 741 (see also the book [Cha 901). Normalized submanifolds in a projective space P" were considered by Norden in the book [N 761. Apparently, the normal connection on such submanifolds was first considered in this book. 6.4. Submanifolds with flat normal connection were considered in [Ch 731, [Cha 77~1,[Cha 77d], [Cha 78b], [Cha 831, [Cha 871 [Cha 891 and Lumiste and Chakmazyan in [LC 811 (see also the book [Cha 901). Theorems 6.1 and 6.2 are proved in this book for the first time. Theorem 6.3 can be found in [Cha 77d] and [Cha 78b] (see Theorems 3 and 4 in [Cha 901, pp. 26-27). Note that the relation between the flatness of a connection and the one-sided stratifiability of some pair of subspaces was noticed by Lumiste in [Lu 651 and [Lu 671. For the conjugate-harmonic normalization of rn-conjugate system this fact was proved in [G 661 by Goldberg. The conjugate, harmonic and conjugate-harmonic normalizations of rn-conjugate systems were first studied by Goldberg in [G 661. He also studied pairs of m-conjugate systems with a common conjugate-harmonic normalizations (see [G 691). The notion of a pseudofocus was introduced by Bazylev in [Ba 65b] for a planar multidimensional nets. Later in [Ba 661 he considered the pseudofoci on the tangents to conjugate lines of V" c P". However, in the latter case the location of pseudofoci depends on the choice of the normal subspace n,. Theorem 6.4 was proved by Chakmazyan in [Cha 77d] and [Cha 78b] (see Theorem 1, p. 37 and Lemma, p. 31 in [Cha 901). Theorem 6.5 is proved in this book for the first time. 6.5. The problem of construction of an invariant normalization of an m-dimensional submanifold V" in an f i n e space A" and a projective space P" for the general case and different special cases was considered by many authors. The affine normal for a surface V 2in an affine space A3 was known for a long time (see, for example, the book [B123]). Weise in [We 381 suggested a scheme for a construction of an invariant normalization of V" in A", where n 5 !jrn(rn 3). His scheme is based on the existence of a nonvanishing differential invariant. Atanasyan in [At 58.1 used Weise's scheme to construct theinner geometry of V" in A". In 1951, in [Kli 521, Klingenberg gave a scheme which is valid for any rn and n. Another scheme which is valid in a centroaffine, affine and projective space was suggested by Liber in 1952-1956, first in the case n 5 ! j r n ( r n + 3 ) and later for any rn and n (see [Lib 521, [Lib 531, [Lib 551, [Lib 561, and [Lib 661). Using the normal objects introduced by Hlavatf in [H 491 (who also proved their existence), Shveykin in [Shv 551, [Shv 561 and [Shv 581, constructed these objects and applied them to find an invariant normalization of a submanifold V" in an affine space A". In [Lo 5Oa] and [Lo 50b], Lopshits studied the problem of an invariant normalization of a hypersurface and a submanifold of codimension two in an equiaffine n-space. Pyasetsky found an invariant normalization of a hypersurface and a submanifold of codimension two in a complex affine n-space (see [Pya 53a], [Pya 601 and [Pya 611) and of a hypersurface in a centroaffine n-space (see [Pya 53bl). Izmailov considered the problem of an invariant normalization for a two-dimensional submanifold in an affine n-space (see [Iz 541 and [Iz 57bl). The survey paper [Shv 661 contains the most important results on the problem of normalization of a submanifold V" in an affine space A". In [Ud 631, Udalov constructed an invariant normalization of a submanifold V" in a projective n-space for the case when n = ("';*) - 1. In another paper [Ud 681, Udalov considered the same problem for general rn and n provided that the

+

Notes

207

submanifold V" possesses two nonvanishing relative invariants. In the case rn + 1 < n < frn(rn 3), an invariant normalization of V" in P" was constructed by Berezina (see [Ber 64.1 and [Ber 64b]), and in the cases 1 < n - m. < irn(rn + 1) and i m ( m 1) < n - rn < irn(m I) bm(m l)(m 2) by Ostianu (see [Os 641 and [Os 661). Berezina and Ostianu followed Weise's scheme in their constructions. Ermakov constructed an invariant normalization of V" in P" with some restriction on the dimensions of osculating subspaces of higher orders of V" (see [E 651 and [E 661). Laptev in [Lap 591, constructed an invariant normalization of a submanifold V" in a space with an affine connection where n 5 +m(m+ 3). If frn(rn 3) < n 5 $rn(rn l)(rn 2) and some additional conditions are satisfied, an invariant normalization of a submanifold V" in a space with an affine connection was found by Laptev and Zaits in [LZ 631 (see also [Lap 641). Izmailov constructed an invariant normalization of a two-dimensional submanifold in a space with an affine connection (see [Iz 611). Invariant normalizations of a family of multidimensional planes in P", of a distribution of rn-dimensional linear elements in a space with a projective connection, of a hyperplanar distribution in P",of manifolds imbedded in a space with a projective structure was considered in the series of papers authored or co-authored by Ostianu (see [Os 691, [LO 711, [Os 711, [Os 731, and [OB 781). For a more detailed description of the developments in the construction of an invariant normalization of submanifolds in different spaces see the survey papers [Lap 651 and [Lu 751 by Laptev and Lnmiste. The described construction of an invariant normalization of V" c P", n < irn(rn 3), which is intrinsically connected with its geometry is due to Ostianu (see [Os 641 and [Os 661). 6.6. Invariant normalizations of submanifolds V" of some special types in the space P" were constructed by Atanasyan and Vorontsova for cones V" with a pdimensional vertex and (p 1)-dimensional generators (see [AV 631, [AV 651 and [Vo 64]), by Goldberg for the Cartan submanifold (see [G 70]), by Ostianu for a submanifold V" with a given net on it (see [Os 70]), by Akivis for a submanifold V", rn > 2, n < 2m, carrying a conjugate net (see [A 70b]), and by Ivlev and Lnchinin for a submanifold V" of codimension two (see [IL 67])-in the latter case a conjugate net on V, always exists. In [Lib 641, Liber indicated a new scheme for construction of an affine connection and subsequently of an invariant normalization of a two-dimensional submanifold V 2 in P". His scheme is based on the construction of a differential p-form and the corresponding p-web defined in a differential neighborhood of order r , where ( r l ) ( r 2) < 2n 5 ( r 2 ) ( r 3). For a more detailed description of the developments in the construction of an invariant normalization of submanifolds of special kinds in different spaces see the survey papers [Lap 651 and [Lu 751 by Laptev and Lumiste. In this section, following the paper [A 70b], we construct a normalization of submanifolds V" c P" carrying a conjugate net of lines.

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209

Chapter 7

Projective Differential Geometry of Hypersurfaces The theory of hypersurfaces Vn in a projective space Pm+' occupies a special place in the projective differential geometry of submanifolds. First of all, this is explained by the fact that this theory is the closest generalization of the theory of surfaces in a three-dimensional projective space. The projective differential geometry of two-dimensional surfaces in a three-dimensional projective space was studied in many books and papers in the 1920's - 1930's. The monographs of Fubini and Cech [FC 261 and [FC 31]), Finikov ([Fi 37]), Bol ([Bo 5Oal) and Lane ([La 321 and [La 421) were devoted to this theory. In these monographs, the investigations in this field conducted in the years preceding monographs' publication were summarized. As to the proper theory of hypersurfaces (i.e. the case when m > 2), this theory was also studied in great detail. The deepest results in this theory are due to Laptev (see his papers [Lap 531 and [Lap 651). In this present chapter we will present the basic results of this theory and introduce some new concepts.

7.1

Basic Equations of the Theory of Hypersurfaces

1. As in the case of an arbitrary submanifold V" in a projective space P", we associate a family of moving frames with any point t of a hypersurface V" in a projective space Pm+' in such a way that the point A0 of these frames coincides with the point L , the points A ; , i = 1,. . . , m,belong to the tangent subspace Til)(Vm) which is a hyperplane of Pm+lif V" is a hypersurface.

210

7.

PROJECTIVE

DIFFERENTIAL GEOMETRY OF HYPERSURFACES

In such a frame, the initial equation of the hypersurface wg=O, n = m + l , and

V" has the form: (7.1)

dAo = w,OAo + wbA,

The forms w6 are basis forms of the hypersurface Vm. As we did in the previous chapters, we set w; = wi. Exterior differentiation of equation (7.1) leads to the single exterior quadratic equation: w ' A w Y = 0.

(7.3)

Applying Cartan's lemma to this equation, we find that

~1 = b i j ~ 3 ,b..83 - b3.%.

(7.4)

(cf. (2.21), (2.94) or (6.2)). The forms wT define the Gauss mapping y(V") of the hypersurface V"' into the Grassmannian G(m,m 1) which is the dual projective space P"':

+

y : V"

+ G(m, m

+ 1) = P"'.

(7.5)

The hypersurface Vm is tangentially nondegenerate if its Gauss mapping y is nondegenerate, i.e. if

b = det(b;j)

# 0.

(74

The determinant b is called the discriminant of second order of the hypersurface V"'. The Gauss mapping y(Vm) defines a hypersurface Vm' in the space P"'. In what follows in this chapter, we will assume that the hypersurface Vm is tangentially nondegenerate, i.e. condition (7.6) holds, and we will study the hypersurface Vm along with its dual image V"' . By relation (7.6), the 1-forms w? are linearly independent, and they are basis forms on the hypersurface Vm'. The generating element of the hypersurface Vm' is a hyperplane a" = A0 A A1 A . . . A A,. The differential of the latter hyperplane has the form: da" = - ~ , " a "- W ~ Q '

(7.7)

(cf. (1.72)). The single second fundamental form @(z) of the hypersurface V" is written in the form:

This form defines the asymptotic cone of second order

C of the Theory of Hypersurfaces 7.1 B Q S ~Equations

211

in the tangent hyperplane Til)(Vm). Since

d2Ao

@(2)A, (mod Ti')),

equations (7.9) define directions along which the tangent hyperplane Til)(Vm) has a tangency of second order with the hypersurface Vm. The second fundamental form of the hypersurface Vm*, which is defined by the formula:

@iz)

= -w;wiao

@a"

differs from the form follows:

@(2)

(mod a", ai),

only in sign. By (7.4), this form can be written as

where biJ is the inverse tensor of the tensor bij . Since the asymptotic lines on the hypersurface Vm are determined by the equation @(2) = 0, and on hypersurface Vm* they are determined by the equation = 0, the Gauss mapping y preserves asymptotic lines. The same is true for conjugate directions. To study further the geometry of the hypersurface Vm, we take the exterior differentials of equations (7.4):

[dbij - bikwjk - b t j w f

+ b;j(w: + w,")] A

W'

= 0.

(7.10)

As earlier, in order to make our notation shorter, we will use the operator V:

V b ; j = dbij - bike; - b k j e f ,

(7.11)

where @ = w i - bjw:. Application of Cartan's lemma to equation (7.10) gives:

where the quantities b;jk are symmetric in all lower indices. These quantities are connected with the third order neighborhood of a point z of the hypersurface Vm. If the point A0 is fixed, i.e. if u i= 0, equations (7.12) give

V a b . . - b13. . ("00 - 7Fn). n 13 -

(7.13)

Since bjkbki = 6!, we find from relations (7.12) that

Vb" where

+ b"(u:

- w,") = -bki j w k ,

(7.14)

212

7.

PROJECTIVE

DIFFERENTIAL GEOMETRY OF HYPERSURFACES

and

i'

b v - b iP b j q b p q k .

(7.16)

If w i = 0, relation (7.14) gives

VJbi' = -biJ(r:

- T,").

(7.17)

Formulas (7.13) and (7.17) confirm that the quantities b i j and b i j are relative tensors. In the same manner as in Section 2.1, we can prove that 6w' = - ( T j - 6jr,O)wj,

(7.18)

or V6W'

= 0.

(7.19)

Using formulas (7.13) and (7.19), we obtain:

6@(,) = (To" - 4 ) @ ( 2 ) .

(7.20)

In the same manner, we find that

bat,) = (r: - 7T,")@i2,.

(7.21)

Relations (7.20) and (7.21) prove that the forms @(2) and are relatively invariant, and this must be expected. Note that on a tangentially degenerate hypersurface, the relatively invariant second fundamental form @(2) defines a conformal structure or a pseudoconforma1 structure. Such a structure on a hypersurface of the space Pn was studied by Sasaki in [Sas 881 and by Akivis and Konnov in [AK 931. 2 . Exterior differentiation of equations (7.12) and application of Cartan's lemma to the exterior quadratic equations obtained as a result of this differentiation leads to the following equations:

Vbijk

+

bijk(W,"

- Uo")

+ 3b(ijW:) - 3b(ijbk)lwL = bijklW',

(7.22)

where the parentheses mean the cycling, for example,

and the quantities b;,kl are symmetric in all lower indices. These quantities are connected with the fourth order neighborhood of a point t of the hypersurface

Vm. Let us find how the quantities b i , k are changed under admissible transformations of moving frames associated with a point z E Vm. To do this, we set wi = 0 in equations (7.22). This gives:

V a b j j k = bijk(rg

-

x,") - 3b(ij(a:, - b k y a ; ) .

(7.23)

7.1

Basic Equations of the Theory of Hypersurfaces

213

I t follows from these equations that the quantities { b i j , bijk} form a differentialgeometric object of third order which is associated with the hypersurface Vm. In addition, equations ( 7 . 2 3 ) show that the quantities bjjk do not form even a relative tensor since expressions ( 7 . 2 3 ) contain the forms T: and TI, which determine the displacements of the first and second normals associated with the point Ao. However, the quantities bijk allow us to construct a relative tensor associated with the third order neighborhood of a point G of the hypersurface Vm. To do this, we contract the quantities bijk with the relative tensor bik and define 1 b . - -b3kbb

tjk.

(7.24)

"m+2

Using relations ( 7 . 1 4 ) and ( 7 . 2 2 ) , we find that (7.25)

where the quantities l j j are expressed in terms of the object bijkl (associated with the fourth order neighborhood) as follows: lij

= -biklbSI

+ bk'bklij.

(7.26)

If wi = 0, equation ( 7 . 2 5 ) implies v6bk

= - T k 0 + b k l T ,I .

Note that formulas (7.23) and ( 7 . 2 7 ) contain the same forms order to eliminate these forms, we introduce the object:

(7.27) T:

- akin!,. In

Applying the operator V6 to this object, we find that

V s B i j ;= ~ Bijk(B:

- T,").

(7.29)

It follows from these equations that the quantities B i j k form a tensor. This tensor is called the Darboux tensor of the hypersurface V m . It follows from relations ( 7 . 2 8 ) that the tensor Bijk is connected with the tensor bij by the relations: Bijkbi3 = 0.

(7.30)

which are called the conditions of apolarity of the tensor Bijk to the tensor b;,. The Darboux tensor determines the cubic form: 'P(3)

=B ~ ~ ~ w ~ J w ~ .

(7.31)

This form is defined in the third order neighborhood. By relations ( 7 . 2 9 ) and (7.19), we have

214

7. PROJECTIVE DIFFERENTIAL GEOMETRY OF HYPERSURFACES

q 3 )

= (4 - 4 * ( 3 ) .

(7.32)

This proves that the form Y(3) is relatively invariant. Comparing formulas (7.21) and (7.32), we see that they contain the same factor 70"- T,". Thus, the ratio of these forms: (7.33)

is an absolute differential invariant. This invariant is called the Fubini linear element since in the case of a surface V 2 in P 3 , this invariant is a projective differential invariant introduced by Fubini (see [FC 311, p. 6 6 ) . The cubic form (7.31) defines the cone (7.34)

of third order in the tangent hyperplane T'')(Vm). Darbouz cone of the hypersurface Vm.

This cone is called the

3. In this section we encountered the condition (7.30) of apolarity of the tensors b,, and B,,k. We will also encounter in this chapter the apolarity condition of two (0, 2)-tensors. Since in the literature we could not find a description of a geometric meaning of this type apolarity conditions for n > 2 (for n = 2 such a description can be found in the book [SS 59]), we will fill this gap in this subsection. Let g,j and h,, be two symmetric (0, 2)-tensors and let g,, be nondegenerate. The tensor h,, is said to be apolar to the tensor g,J if

q"h,, = 0. (*I Here i,3 = 1,. . . , n, and 9;' is the inverse tensor of the tensor g,,. Next, in a vector space L", we consider two cones: g,,z'zJ = 0

and h , , Z t Z J = 0. (h) Condition (*) means that there ezists a frame { e l , . . .,en) which is conjugate with respect to the cone ( 9 ) and inscribed into the cone (h). We will prove this by induction with respect to n. If n = 2, then the cone (h) has two different zero directions. Taking the vectors el and e2 of a frame along these directions, we reduce the equation of the cone (h) to the form

2h1zz'z2 = 0,

where hlz

# 0. Because of this, the apolarity condition (*) takes the following form: g12h12 = 0.

This implies g'* = 0, i.e. the vectors e l and ez are conjugate with respect to the cone

(PI.

7.1

Basic Equaiions of the Theory of Hypersurfaces

215

Suppose now that om statement is true for any k less than n. We will prove that it is valid for k = n . To prove this, we take the vector en of a frame in such a way that en belongs to the cone (h) but not the cone (g). Then h,, = 0 and gnn # 0. Consider a subspace L"-' C L" which is conjugate to the vector e, with respect to the cone (g), and take the remaining vectors e l , . . . ,e,-1 of the frame in L"-'. In this frame, the matrix g of the quadratic form on the left-hand side of (g) takes the form

and thus the inverse matrix 9-l takea the form:

This implies that the apolarity condition (*) reduces to the relations:

h,,gUU = 0. The latter condition is the restriction of the condition (*) to the subspace L"-'. Under the induction assumption, there exists a frame { e l , . . ,en-l] in L"-' which is conjugate with respect to the cone guvzyz' = 0 and inscribed into the cone hy,zYzU = 0. The vectors { e l , . . . , en-l] together with the vector en form a basis in the space L" satisfying the required condition. Note that if n > 2, the apolarity condition for tensors g;, and hi, is not symmetric, i.e. in general, the condition (*) does not imply the condition

.

g;,h'' = 0. Consider now the apolanty condition for (0, 3)-tensor h,,k and (0, 2)-tensor g i j . Suppose that these tensors are both symmetric and, as earlier, the tensor g;, is nondegenerate. Let y = y'e, be an arbitrary vector in L". Then this vector defines (0, 2)-tensor h;,(y) = h i , k y k , and the latter tensor defines the second order cone I j htjkz Z

y

k

=o,

(**I

called the second polar of the vector y with respect to the cubic hypercone h,,kZiz'Zk = 0. Since the apolarity condition g'jhijk

=0

implies that gi'hi,(y) = 0

for any vector y E L", the geometric meaning of apolarity of two (0, l)-tensors proves that it is possible to inscribe a frame into any cone (**) in such a way that this frame is conjugate with respect to the cone (g). For the case n = 2, in the book [SS 591 (see 523), the following geometric meaning of the a p o l h t y condition gi'h,,k = 0, i , j , k = 0,1,2, is given: if the equation hi,kzizjzk = 0 has three mutually distinct solutions e l , e ~ , e s and , each of the uectors uq, a, b, c = 1,2,3, which is harmonic conjugate of the vector e. with respect to

216

7 . PROJECTIVE DIFFERENTIAL GEOMETRY OF HYPERSURFACES

the uectors eb and e, ( t h e indices a, b and c are mutually distinct), i s also conjugate to the vectors ea with respect to directions defined b y the equation g i , x ' d = 0, i.e. g ( e a , aa) = 0.

7.2

Osculating Hyperquadrics of a Hypersurface

1. T h e tangent hyperplane Til)(Vm) has a first order tangency with the hypersurface V" at the point z. The order of their tangency along the asymptotic lines defined by equations (7.9) is equal to two. I t is natural to try to find a hypersurface Q of second order that has a second order tangency with the hypersurface V" in all directions emanating from the point Ao. We will write t h e equation of a hyperquadric Q relative to a first order frame associated with the point A0 E V" in the form:

Q(z,z) = A,,t"z" = 0,

U,U

= O , l , . . . , m +1 .

(7.35)

Since the hyperquadric Q must pass through the point Ao, we have (7.36)

Q(Ao,Ao) = Aoo = 0.

Since the hyperquadric Q is tangent to V" at A D ,the following condition must be satisfied:

Q(Ao 1.e.

Q(Ao

+ dAo, Ao + ~ A o=) 0,

+ w,OAo + w i A i , A0 + w,OAo + J A j ) = 0.

Taking into account relations (7.36) and neglecting infinitesimals of order higher than one in the latter equation, we obtain

Aoj = 0.

(7.37)

As a result, the equation of the hyperquadric Q takes the form: 2Aonz0zn

+ Aijziz' + 2Ajnzizn+ A,,(E")'

=0.

(7.38)

Moreover, we have Ao, # 0 since otherwise the hyperquadric Q degenerates into a hypercone. We normalize equation (7.38) by imposing the condition:

Aon = -1.

(7.39)

Finally, requesting that the hyperquadric Q has a second order tangency with the hypersurface V " , we find that

7.2

Osculating Hyperquadrics of a Hypersurface

&(A0

217

+ dAo + j1d Z A o , + dAo + -21d 2 A o ) = 0. A0

Taking into account relations (7.36), (7.37) and (7.39) and neglecting infinitesimals of order higher than two, we obtain

A.. 13 - 6'.3..

(7.40)

After these simplifications, the equation of the family of osculating hyperquadrics Q takes the form:

= 6ijxizj

ZZ'Z"

+ 2 A j , ~ ~ r "+ A , , ( Z " ) ~ .

(7.41)

We see from this equation that the family of osculating hyperquadrics depends on m 1 parameters. Let us find directions along which a hyperquadric (7.41) has a third order tangency with the hypersurface V"'. Such directions are called characteristic directions. To find them, we should substitute coordinates of the point

+

c

= A0

1 1 + dAo + -d2Ao + -d3Ao 2 6

into equation (7.41) of a hyperquadric Q. We write the differentials d2Ao and d3A0 in the form:

d2Ao = @A0 d3Ao = @A*

+ RdAi + Cl;j.A,, + Q t A j + @:An.

(7.42)

If we substitute coordinates of the point c into equation (7.41), then the terms of orders 0, 1 and 2 cancel out since the hyperquadric has a second order tangency with the hypersurface V"', and the third order terms give 1 -a;

3

By (7.42), the forms

+ Qtw," = 6ijwiCl{ + Ai,fl;w*.

nt,0:

(7.43)

and @;j. have the form:

+ +

+ +

0; = dwi w:ui d w ; , q = wiwr = b.!3 .wiwj, a; = an; R3,ujn n;w;.

(7.44)

Hence, equation (7.43) takes the form: 1 -(do;

+ Cliwjn + OGw,") + Q ~ L J ,=" bijCl$J + Ai,R;w'.

3 To complete our calculations, using (7.44) and (7.12), we find dR;:

d.Qt = dbijw'wj + 2b,jwidwj . . = ( 6 i k W f bkjWb - bij(w: W , " ) bijkWk)W*wJ +26ijwi((n{ - w t w i - w k w i ) , = ( - b ; j ( 3 ~ , 0 u:) bijkwk)w'wj 26ijwi0{.

+

+

+

+

+

(7.45)

218

7. PROJECTIVE DIFFERENTIAL GEOMETRYOF HYPERSURFACES

Substituting this expression into equation (7.45), after uncomplicated calculations we obtain (7.46)

Thus, the characteristic directions, along which the hyperquadric Q has a third order tangency with the hypersurface V"' at the point Ao, are defined by equation (7.46). These directions form a cubic cone for each osculating hyperquadric (7.37). A hyperquadric (7.41) is called the Darboux hyperquadric if the cone (7.46) coincides with the Darboux cone (7.34). Since the Darboux tensor B i j k is defined by formula (7.28), the cone (7.46) coincides with the Darboux cone if and only if A,i = bi.

(7.47)

Because of this, the equation of the family of Darboux osculating hyperquadrics has the form:

+ 2biziz" + Ann(Z")'. (7.48) We remind that in formulas (7.47) and (7.48) n = m + 1. Since this family 2c0cn = bijz'd

depends on one parameter, it is called the pencil of Darboux osculating hyperquadrics. We will prove the following property of hyperquadrics of this pencil.

Theorem 7.1 If one of osculating hyperquadrics has a third order tangency with a hypersurface Vm at a point I E Vm in any direction emanating f r o m the point 2 , then this hyperquadric i s a Darbouz hyperquadric, and the Darbouz tensor of the hypersurface V"' vanishes at this point. Proof. In fact, if a hyperquadric of the family (7.41) has a third order tangency with a hypersurface V" at a point c E V", then equation (7.46) has to become an identity at this point, that is we must have

bijk - 3An(ibjk) = 0. Contracting this relation with the tensor b'j, we find that

Ani = b;.

(7.49)

(7.50)

Substituting these values of A,; into equation (7.41), we see that this equation coincides with equation (7.48) of the family of Darboux hyperquadrics. On the other hand, if we substitute these values of A,,, into equation (7.49), we find that Bijk = 0. 2. It is easy to see that the second fundamental tensor bij of a hypersurface Vm vanishes at all points of V"' if and only if the hypersurface V'" is a part of a hyperplane. The Darboux tensor possesses a similar property. We can now prove the following result.

7.2

Osculating Hyperquadrics of a Hypersurface

219

Theorem 7.2 A hypersurface Vm of a projective space Pm+lis a hyperquadric or a p a d of a hyperquadric if and only if the Darboux tensor of this hypersurface is identically equal to 0 . Proof. Necessity. Consider a hyperquadric Q which is defined in a frame {A,} by the equation:

Q ( t , X ) = Au,,tUtV= 0, u, v = O , l , .

. .,n.

(7.51)

Since the frame {A,} is a moving frame, the coefficients A,, of equation (7.51) depend on the location of this frame. Let us find the differential equations which the coefficients A,, of equation (7.51) must satisfy in order that the hyperquadric Q be fixed in the space P". To do this, first of all, we find the conditions for a point

X = zUA,

(7.52)

to be invariant. Since the equations of infinitesimal displacement of a moving frame have the form dA, = w:A,,

(7.53)

we have:

+

d X = (dzU tVw,U)A,.

(7.54)

Since the coordinates of the point X admit multiplication by an arbitrary factor, the condition for this point to be invariant has the form dX = uX. By equation (7.54), this implies dxU

+ xvw,U = uxU.

(7.55)

Similarly, the condition for the hyperquadric Q to be invariant has the form dQ = PQ.

(7.56)

Differentiating equation (7.51) and using formulas (7.55), we find that dA,, - Auww,U,- A,,W:

= BA,,,

(7.57)

where 0 = p + 2u. We now assume that a hypersurface Vm is a hyperquadric Q. We associate a frame of first order with Q placing the point A0 into a point 3: E Q and placing the points At into the tangent hyperplane to Q at the point x = Ao. As a result, we obtain the following relations: A00 = Aoi = 0.

(7.58)

In addition, we will normalize the equation of the hyperquadric Q by the condition

220

7. PROJECTIVE DIFFERENTIAL GEOMETRY OF HYPERSURFACES

Aon = -1.

(7.59)

Let us write equation (7.57) for different pairs of values of the indices u and v. If u = v = 0, we obtain an identity by means of relations (7.58) and (7.1). If u = 0 and v = i, by means of relations (7.58), (7.59) and (7.1), we find easily that

= Aijw'.

w;

(7.60)

Comparing these equations with equations (7.4), we find that

Aa,j . - b8.j.

(7.61)

Next, if u = 0 and v = n, we obtain

0 = Aniwi

- w:

- u,".

(7.62)

If u = i and v = j, using all relations obtained earlier, we find that

+

Vbij = bij(w: - u,") 3 A n [ k b i j p k .

(7.63)

Comparing this relation with equation (7.12), we arrive at the following equation: bijk

= 3An(kbij).

(7.64)

Substituting these expressions into formula (7.24), we obtain: bk

= f i n k , where n = m + 1.

(7.65)

If we use this equation, we find from formula (7.28) that Bijk

= 0,

(7.66)

i.e. the Darboux tensor of the hyperquadric Q is identically equal to 0. Suficzency. For u = i and v = n, from relations (7.57) we find that

VAi, =

-UP + bijw; + (Annbij + A,;A,j)w;,

(7.67)

and if u = v = n, we have

Assume that condition (7.66) holds at any point of a hypersurface V". Then, a t any point of V" its Darboux hyperquadrics has a third order tangency with I/". We prove that there is one fixed hyperquadric among the Darboux hyperquadrics. This means that for some value of Ann$all conditions of invariance, considered above, will be satisfied provided that relation (7.68)

‘7.2

Osculating Hyperquadrics of a Hypersurface

221

holds. In fact, equations (7.57) of invariance of hyperquadric (7.48) are identically satisfied for the following pairs of u and w: u = 0,w = 0; u = 0 , u = i and u = 0, v = n. If u = i, v = j , condition (7.57) is satisfied by means of equation (7.63). Suppose that u = i and v = n. Substituting the values A,, from equations (7.65) into equations (7.67), we obtain:

Vbi =

-UP +

bijw:

+ (A,,bij + bjbj)w’.

(7.69)

Comparing this equation with relation (7.25), we find that ljj

= A,,bij

+ bjbj.

(7.70)

Contracting this relation with b’j, we find that

A,, = - 1b S J ( l , j - b i b j ) . ”

m

(7.71)

One can easily check that the quantities A,, satisfy relations (7.68). Hence, the hyperquadric Q is fixed, and the hypersurface Vm is a part of this hyperquadric. 3. Theorem 7.2 is closely connected with the problem of existence of a tangentially nondegenerate hypersurface with a parallel second fundamental form. In an Euclidean and affine geometries this problem was considered by many authors (for the Euclidean space, see, for example, ‘$3 of the book [Ch 811 by Chen and the joint papers [CV 801 and [CV 811 by Chen and Vanhecke). However, there is a projective interpretation of this problem. To study the problem of a parallelism of the second fundamental form, we must naturally define an affine connection on the hypersurface in question. As we saw in Chapter 6, such an affine connection is defined if the hypersurface is normalized by means of first and second normals. Suppose that such a normalization of a hypersurface Vm c Pmtlis given. Suppose also that the vertex Amtl of a frame associated with Vm lies on the first normal n,, and the points A l , . . . , A m belong to the second normal 1,. Then in formula (7.12), Vbij is the covariant differential of the second fundamental tensor b i j , and the quantities b i j k form a tensor since the fact that first and second normals are fixed implies the vanishing the forms 7ri and xf, in equations (7.23) and the following form of these equations:

In this formula the tensor b i j k is the covariant derivative of the second fundamental tensor b j j . The condition for the second fundamental form to be parallel in the affine connection indicated above can be written as Vbjj = 0

7. PROJECTIVE DIFFERENTIALGEOMETRY OF HYPERSURFACES

222

(see [Vil 721). By (7.12), this implies

- w,") = b i j k w k .

-bij(w:

It follows W:

- W," = - a k W k

and bijk

Let us find the Darboux tensor (7.24) we find that

= 3b(ij'Yk).

Bijk

- - Q k1( b . . a $j '-m+2 and thus, by (7.28), we have B i j k = 0. This proves the following result.

of the hypersurface Vm. From equation

b.

k

-k

bjkai

+bkiaj),

Theorem 7.3 The second fundamental form of a tangentially nondegenerate hypersurface Vm c Pm+l is parallel with respect to an a@ne connection induced b y a normalization of Vm if and only if the hypersurface Vm is a hyperquadric or a part of a hyperquadric. We have proved the necessity of the condition of this theorem. Its sufficiency follows from the fact the Darboux tensor B i , k of a hyperquadric is identically equal to 0.

7.3

Invariant Normalizations of a Hypersurface

1. In this section we will construct an invariant normalization of a hypersurface Vm in a projective space Vm+', i.e. we will find a congruence of onedimensional first normals n, and a pseudocongruence of (m - 1)-dimensional second normals 1, which are intrinsically connected with a hypersurface Vm. The first normal n, at a point A0 = x is spanned by the point A0 and a point

+

(7.72)

+~ i A o

(7.73)

Yn = An y i A i , and the second normal I, at a point A0 = x is spanned by points Zi

Ai

(cf. formulas (6.91) and (6.92)). The normalizing objects of the first and second kinds, yi and z i , must satisfy the equations: Vayi

= y'(.," - .,") - ?Ti

(7.74)

7.3

Invariant Normalizations of a Hypersurface

223

and V62, = -Ti",

(7.75)

which are obtained from equations (6.95) and (6.96) by setting a = n. We can establish a connection between the first and second normals of a hypersurface V"' at a point A0 by requesting that these normals are polarconjugate to one another with respect to the pencil of Darboux osculating hyperquadrics. To do this, we consider the quantities: yi

= b;jyt.

It follows from equations (7.74) and (7.13) that these quantities equations:

(7.76) yi

satisfy the

Vay; = - b i j n f , ,

(7.77)

We now set uj

= z; - y i .

(7.78)

-(UP

(7.79)

Using (7.75) and (7.77), we obtain V ~ U=;

- b;jIrjn).

Comparing this equation with equation (7.27), we see that by setting ~i= b i ,

(7.80)

we have equation (7.79) be satisfied. By this reason, we set t; = y; -I-bi.

(7.81)

We will now prove that the first and second normals, which are defined by the objects yj and z; connected by relation (7.81) are polar-conjugate t o one another with respect to the pencil (7.48) of Darboux osculating hyperquadrics. To do this, we write the bilinear equation which is polar to the quadratic equation (7.48):

z0tn+ Z " ~ O = b i j z ' t j

+ bi(zi