JOSEPH EDMUND WRIGHT
OF
JOSEPH EDMUND WRIGHT
DOVER PUBLICATIONS, INC~ MINEOLA, NEW YORK
Bibliographical Note This Dover edition, first published in 2013, is an unabridged republication of the work originally published by Hafner Publishing Co., New York, in 1960. The 1960 edition was itself a republication of the work originally published as Volume 9 in the series, "Cambridge Tracts in Mathematics and Mathematical Physics" by Cambridge University Press, in 1908.
Library of Congress CataloginginPublication Data Wright, Joseph Edmund. Invariants of quadratic differential forms I Joseph Edmund Wright. p. em. "An unabridged republication of the work originally published by Hafner Publishing Co., New York, in 1960. The 1960 edition was itself a republication of the work originally published as Volume 9 in the series, 'Cambridge Tracts in Mathematics and Mathematical Physics' by Cambridge University Press, in 1908." Summary: "This classic monograph by a mathematician affiliated with Trinity College, Cambridge, offers a brief account of the invariant theory connected with a single quadratic differential form. A historical overview is followed by considerations of the methods of Christoffel and Lie as well as Maschke's symbolic method and explorations of geometrical and dynamical methods. 1960 edition"Provided by publisher. ISBN13: 9780486497686 (pbk.) ISBN10: 0486497682 (pbk.) 1. Differential forms. I. Title. QA381.W8 2013 515'.37dc23 2012046797 Manufactured in the United States by Courier Corporation 49768201 2013 www.doverpublications.com
PREFACE
THE aim of this tract is to give, as
far as is possible in so short a.
book, an account of the invariant theory connected with a single quadratic differential form. "It is intended to give a bird's eye view of the field to those as yet unacquainted with the subject, and consequently I have endeavoured to keep it free from all analysis not absolutely necessary. It will be found that the rest of the tract is independent of Chapters III and IV. These chapters are included so as to give an account, as far as possible complete, of the various methods that have been applied to the subject. The most successful method is that outlined in the remainder of the book. This method, begun by Christoffel, owes its modern development mainly to Ricci and LeviCivita, and it is hoped that this tract may induce some of its readers to turn to their papers. J. EDMUND WRIGHT. Fellow of Trinity College, Cambridge Associate Professor of Mathematics, Bryn Mawr College, U.S.A. TRINITY COLLEGE.
J'Uly 1908.
CONTENTS PREFACE
•
•
•
•
INTRODUCTION
CHAPTER
•
•
I.
Historical
II.
The Method of Christoffel .
•
•
III. The Method of Lie
•
v.
Applications : Geometrical . Dynamical •
•
•
•
•
•
•
1
•
•
•
•
5
•
•
•
•
9
•
•
•
•
29
•
•
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44
•
•
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•
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61
•
•
•
•
IV. Maschke's Symbolic Method
•
...
PAGE 111
80
INVARIANTS OF QUADRATIC DIFFERENTIAL FORMS INTRODUCTION
1. IN order to discuss in detail the geometry of a plane it is convenient to introduce coordinates. A point in the plane has two
degrees of freedom, and therefore to determine it two independent conditions must be satisfied. These conditions may be that two independent quantities• (e.g. the distances from two fixed points) take the values u, v at the point, and we then say that the coordinates of that point are u, v. If we suppose one coordinate given, the locus of the point will be a certain curvec, e.g. u = const., and, generally, a curve in the plane is given by a functional relation
f/l (u, v) = 0. For the metrical geometry of the plane we need an expression for the distance between any two points in terms of the coordinates of the points. Theoretically this can be calculated if the distance between an arbitrary point (u, v) and any neighbouring point (u+du, v+dv) is known. Suppose that (tD, g) and ($ + d$, y + dy) are rectangular cartesian coordinates of the two points, then :D, y are functions of u, v ; also if ds denote the distance between the two points, ds' =d:rr + dy2, or tJsl =Edu2 + 2Fdudv + Gdv 2, where
~~ ~~ E au + au '
F=~~ ay ay G=~~ ~~ a~u av +au av' av + &UI •
We have in fact a quadratic form in the variables du, d'D, for IY, and the coefficients of this form are functions of u, v. If E, F, and G are * These quantities have not necessarily any obvious geometrical significance. 1
2
INTRODUCTION
given it is possible to determine the equations of the straight lines of the plane in terms of u, v, to find the angle between any two of its curves, and, generally, to develope its metrical geometry. Now if the system of coordinates is given, E, F, G can be determined, and the, converse question at once arises, namely, if three functions E, F, G are given as three arbitrary functions of u, v, is it possible to take u, v as coordinates of the points in the plane so that the element of length shall be given by dr = Edu2 + 2Fdudv + Gdv' ? It appears that this is not possible unless a certain relation 1
K=. 2../EGP
{ a[
.F
aE
1
ao]
i; E../EGF2 a;· ../EGP" &;,
2 oF_ _ 1 BE_ F BE]} =o av ,JEGF2 ~u JEGF 2 &v EJEGF1 ti;J is satisfied for all values of u, v. This condition is also sufficient. If, instead of limiting ourselves to a plane, we consider any surface in three dimensional space, we have again two coordinates for any point ; the element of length is given as before by tlr =Edu2 + 2Fdudv + Gd'IJ, and there is a surface corresponding to any arbitrary functions E, F, G of a, v. (The particular case EG =F 9 is excluded.) It appears however that for a given surface the expression K has the same value at any given point on it, whatever coordinates u, v are chosen. Let u, v and u', 'II denote any two sets of point coordinates on the surface, then u'=f(u, v), v'=~(u, v), where /and~ are arbitrary functions ofu, v, and we have the theorem : If by ang tranllformatim~ u' =I (u, v), v' = q, (u, v), Edu9 + 2Fdudv + Gdvs becomes E'du'1 + 2F'du' dv' + G'dv'1, tlum K =K', wluwe K' is Kin tke accented 'IXJiriables. + !_ [
!a. Deflnition of a differential invariant. Any function of E, F, G and their derivatives satisfying this condition is called a di.ffwential in'INWiant of the form Edu9 + 2Fdudv + Gd'V'. The idea of a differential invariant may be extended by taking account of any families of curves on the surface, say ~ (u, v) = oonst.,
1, 2]
DEFINITION OF A DIFFERENTIAL INVARIANT
3
t/1 (u, v) =const., etc. When we transform to new variables u', v' we have Edu'+ 2Fdudv+ Gdv=E'du'1 + 2F'du'd'D' + G'dv'2, • (u, tJ) =~, (u', v'), Y, (u, v) =t/1' (u', v'), etc., and a differential invariant is defined as a function of u, v, E, F, G, 4>, y,, and their derivatives (u, v being regarded as independent variables) that has the same value whether written in the original or in the transforined variables. Invariants which involve only u, v, E, F, G and their derivatives are called Gaussian invariants, while those which involve also deri.. vatives of q,, t/1, etc. are called differential parameters. Thus for example K is a Gaussian invariant and _
1
{
aij;i ~
of/l a.p
a;;; '}
,1q, = EGFJ Bp;u 2Fa; iii+ Gav
is a differential parameter of the quadratic differential form Edu1 + 2Fdudv + Gdv. If the quadratic form is interpreted as the square of the element of length of a surface in space, K and A~ have also geometrical interpretations. K is the Gaussian or total curvature of the surface, and if 4~ = 1, the curves const. are the orthogonal trajectories of a fam~ly of geodesics on the surface. The extension of these ideas from two to m variables is immediate, and the quadratic differe11tial form in m variables may be regarded as the square of the element of length in the most general m dimensional manifold. The main point is that invariants are independent of the particular choice of coordinates, in other words they are intrinsically connected with the manifold itself. The course of ideas is as follows. We start with a given manifold, which possesHes certain properties. Some of these may be independent of each other, some may be consequences of certain others, and there are relations connecting these. We may develope the discussion on the lines of pure geometry, but we are compelled, sooner or later, to appeal to algebraic methods. These · methods involve the introduction of coordinates, and properties of the manifold are then expressed by means of algebraic equations. An algebraic expression has some interpretation in the manifold taken . together with the coordinate frame used, and a complication has been introduced, for the discussion will now involve those additional properties which are not intrinsic to the manifold, but arise out of the
"'=
4
INTRODUCTION
particular coordinate frame chosen. If however we work only with invariants, we avoid this latter class of properties and are able at the same time to use the powerful methods of analysis. The geometry of the manifold thus breaks up into two parts : (i) The determination of all invariants and all relations connecting them. (ii) The geometrical interpretation of all these invariants in the manifold. S. So far there have been considered only invariants arising through the quadratic form that is equal to M. These are all the invariants when we consider the manifold in itself, but if we suppose it existing in, say, Euclidean space of higher dimensions we introduce other invariants connected with the relation of that space to the manifold. For example, in the case of a surface in space, the totality of invariants is only given when two quadratic forms are taken account of, the additional one being that which determines the normal curvature at any point of the surface. The surface, in fact, is not intriusically determinate by means of the single form, but may be bent provided there is no tearing or stretching, or as we say, it may be deformed, without alteration to diP. (For instance any developable may be defo1·med into a plane.) The discussion when there are two or more forms is similar to that when there is only one. The invariants arising from the single form are called deformation invariants. Thus far it is suggested that the invariants are essentially connected with differential geometry. This is by no means the case. They are connected with a certain form, and any interpretation of this form leads to a corresponding interpretation for the invariants. Consider in fact any dynamical configuration with Lagrange coordinates ~, ~, ... , u,.,. The kinetic energy of this system is
l" f",l=l
ar,uru,, where ara is a function of the variables u,
and dots denote
derivatives with regard to the time. By a new choice of coordinates we effect a transformation of exactly the same type as that already considered, and again we have a series of invariants of a quadratic form, and these are those quantities which are dependent on the configuration itself as distinct from the particular system of coordinates.
CHAPTER I HISTORICAL
4. Group. Invariance necessarily carries with it the idea of a transformation. Suppose we have a set of transformations in any variables whatever, and suppose that each of the set leaves a 'certain function of these variables invariant, then any transformation compounded of two or more of the set will also lee,ve that function invariant. If any such transformation as this is. not one of the original set we add it to that set, and we may thus continue adding new transformations until we reach a closed set, that is one such that if you apply in turn any two of its transformations the result is another of its transformations. Such a set is called a GROUP, and it is clear that any invariant whatever is invariant under a group of transformations.
6. In the case considered in the preceding pages there are a certain number of quadratic differential forms l"' a,.,d:crd:c,, '1',
•=1
together with a certain number of functions ~(a?., ... , :c.), and the group of transformations :c~ =:c~, ('!Jl, ••• , y.), (i =1, ... , n), and we suppose that under a member of this group l"' ar,dterdm, becomes
"· •=1
ft.
~
..,.=1
a'r,d!lrdg,, and that
~
becomes ~'.
Then there are deducible
rela.tions for a'r, ~', and their various derivatives with respect to the y's, and for dm1 , ••• , d:c,. in terms of the original magnitudes ar,, ~, etc. In other words there exists a set of transformations for all the variables mentioned. It may be proved that this set is a gToup, and this group is said to be emteruled from the original group. o~r problem is the determination of all the invariants of this extended
grot:p.
6
HISTORICAL
6.
(CH. I
Christoffel.
There have been three main methods of attack. The first, his· torically, is by comparison of the original and transformed forms, and in this way invariants are obtained by direct processes. The fundamental work in this direction is due to Christoffel* (1869), though the first example of an invariant, the quantity K, was given by Gausst in 1827. Invariants which involve the derivatives of the functions are called differential parameters. Lame t, using the linear element in space given by d~ =dar+ dy2 + dz2, gave this name to the two invariants
{Al~)s =(~Y + (~Y + ~)' 
a2~
at~
As~ =a:r + iii
a2q, + a.z2 ,
and Beltrami§ adopted it for the invariants that he discovered, those involving first and second derivatives of a function ., taken with a form in two variables. lr1 the course of Christoffel's work there arise certain functions (i/crs); these were originally found by Riemann in 1861 in his investigations on the curvature of hypersurfaces. For a surface in space they reduce to the one quantity K. 7. Ricci and LeviOivita. To Christoffel is due a method whereby from invariants involving derivatives of the fundamental form and of the functions ~ may be derived invariants involving higher derivatives. This process has been called by Ricci and LeviCivita C()'IXJlriant derivation, and they have made it the base of their researches in this subject. These researches have been collected and given by them in complete form in the Matlt6matiscluJ Annalen II, and on their work they have based & calculus which they call Absolute differential calculus. They give & complete solution of the problem, and show that in order to determine all differential invariants of order p, it is sufficient to determine the algebraic invariants of the system : (1) The fundamental differential quantic, * Crtlle, Vol. 70 (1869) p. 48. t Duqui1itione1 gerurale• cif"ca 1uperjlcie• cu"'a1. ~ Ll~m
I of
1ur le1 coordonf&le1 cunJiligr&el (1859). Darboux, TMorie glnerale de1 I'Urjace•, Vol. m. pp. 198 Iff· gives an
Beltrami~&
work together with a bibliography. II Math. Ann. Vol, 54 (1901) pp. 12G •qq.
account
69]
HISTORICAL
7
(2) The covariant derivatives of the arbitrary functions ~ up to the order p., (8) A certain quadrilinear form G4 and its covariant derivatives up to the order p,  2 •.
8. Lie. 'rhe second method is founded on the theory of groups of Lie, and is a direct application of the theory given in his paper Ueber Di.fferentialinvarianten t. This theory involves the use of infinitesimal transformations, and the invariants are obtained as solutions of a complete system of linear partial differential equations. Our problem is discussed shortly by Lie t himself, for the case n = 2. ~orawski § considered this case in detail, and gave the invariants of orders one and two. He also treated the question of the number of functionally independent invariants of any order. C. N. Haskins II has determined the number of functionally independent invariants of any order. Forsyth~ has obtained the invariants of orders one two and three for a quadratic form in three variables, and of genus zero, that is to say, for ordinary Euclidean space. He has also obtained the invariants of the first three orders for any surface in space ••, that is, for two quadratic forms in two variables, one perfectly general, and the other connected with it by certain differential relations. The problem for the differential parameters has been solved by this method by J. E. Wright tt.
9. :Maschke. The third method is due to Maschket:, who has introduced a symbolism similar to that for algebraic invariants. He developes processes similar to that of transvection, whereby an endless series of invariants may be constructed. * Zoe. cit. p. 162. t Math. Ann. Vol. 24: (1884:) pp. 1;87 sqq. :1: Zoe. cit. § Ueber Biegungsinvarianten in Acta Math. Vol. :&VI (18921898) pp. 164:. 11 Tram. Amer. J'lath. Soc. Vol. 111 (1902) pp. 71, 91; also ib. Vol. v. (1904) pp. 167' 192. 1T Phil. Trans. Series A, Vol. 202 (1908) pp. 277883. ** Phil. Trans. Series A, Vol. 201 (1908) pp. 829402. tt Amer. J'oum. of Math. Vol. uv11 (1905) pp. 82884:2. ~ Trans • ..4.mer. Math. Soc. Vol. I (1900) pp. 197, 204:; and Vol. IV (1908) pp. 4:45469.
8
(cH. I
HISTORICAL
The geometrical interpretation of the invariants has been discussed at length by Forsyth*, and a considerable part of the work of Ricci and LeviCivita deals with geometrical applications. A general account of the whole subject was given by Maschke t at the StLouis Exposition, 1904. An account will now be given of these three methods. * See his two papers already quoted, and Rendiconti del Oircolo Matematico di Palermo, Vol. 21 (1906) pp. 115125.
t
Bt Louis Congress of Art• and Sciences, Vol.
1.
pp. 519, 580.
CHAPTER II THE METHOD OF CHRISTOFFEL
10. The quadratic form in two variables. Let there be two quadratic forms F 5 adal' + 2bd/l)dy + cdy?. and F'sAdXS+2BdXdY+OdY2, and suppose that :c, '!I may be ex .. pressed as functions of X, Y so that when these values are substituted in Fit becomes F'. We have then ada:' + 2bdf.Cd'!J + cdy2 = AdX2 + 2BdXd Y + Od Y 1•
In this we write d:r =
[i.ax + :;. d' Y, with a similar expression for d'y,
and the equation takes the form Pd.XS + 2QdXdY + Rd¥2= 0 where P, Q, R are certain functions of :c. y, X, Y and the derivatives of te, '!/ with regard to X and Y. Now X, Y are independent variables and therefore there exists no relation among the differentials dX, d Y, and hence P, Q, R are all zero. Thus the necessary and sufficient conditions in order that F shall be transformable into .F' are P =0, Q = 0, R = 0, or written at length.
a(:~y +2b :~a~+c (a~Y =A, aa a:c
( o:c ay
0.11
oy )
a11 a9
aa:;ray +b axay+ayax: +c a;tay=B, ote \2 o:c ey ( ay )51 a ( BYJ + 2b ay ay+ c ay = o. These three are differential eq nations of the first order for m, y as functions of X, Y. If they can be solved their solution gives the transformation whereby F is changed into F'. Now there are three equations, and they involve only two dependent variables tc, g; hence they cannot in general coexist unless there be relations between
10
THE METHOD OF CHRISTOFFEL
(CH. II
a, b, c, A, B, 0 and their derivatives. Our first problem is to find the conditions in order that they may coexist. By differentiation we obtain six equations in the six second derivatives of cc, g, and these may be solved for the second derivatives in question. If the original three are differentiated twice there are obtained nine equations involving third derivatives, and by means of the equations for second derivatives these may be reduced to a form in which they involve first and third derivatives only. There are only eight third derivatives of two functions z, y, each of two variables X, Y, and therefore by eliminating them from this last set of equations we get a new equation, which, since it involves first derivatives only, must be added to the original three equations. It happens that from these four equations the first derivatives can be eliminated and tl1us there is given a relation between a, b, c, A, B, 0 and their first and second derivatives. This relation is precisely K = K'. We can now proceed step by step to find the equations involving higher derivatives of :e, y, and then by elimination to find other relations among the coefficients a, b, c, A, B, 0 and their derivatives. In the case considered, that of two independent variables, these relations all follow from the equivalence of K and K'.
11. The quadratic form ln n variables. The general quadratic in n variables may be treated in exactly the same manner ; the statement of the work is much simplified by the use of cert&in abbreviations which we proceed to define. The form F itself is written I o,.,dm,.d~E, and the form F' is ~ a',.,dy,.dy.,
"· ,
'"·,
the summation being always from 1 to n for each of the
letters under the sign of summation. The y's are taken as the independent variables, and the :e's are assumed to be functions of these. The determinant of n rows and columns, whose elements are the quantities ar,, is called a. The cofactor of the tth row and sth column in a is written 4,.,. The quantity
! [aa a,t+a.!. a~a~ agAJ is written [gk, k], '/IJA 'IIJg 'IIJt and l [il, k] ~rt/a is wri~n {il, r}. k
AlKlfa is a'") (seep. 20).
[gk, k] and {gk, k} are called Christoffel's threeindex symbols of the first and second kinds respectively. The expression [gk, k] a_!_ [gi, k] + l a_!_ ~, ~1& JJ
({gi,p} [kk, .\] {gk,p} [ik,p])
1013]
11
THE CHRISTOFFEL SYMBOLS
is written (glclti) and is called sometimes Christoffel's fourindex symbol and sometimes a Riemann symboL In the case of a quadratic form in two variables, a2K is the only Riemann symbol. These symbols were discovered by Riemann, independently of Christoffel, in his researches on the generalisation of curvature for manifolds of '' dimensions. It is clear from the definition that [gh, k] =[ltg, k], and therefore that also {gk, k} = {kg, k}. Also, the threeindex symbols of the first kind are linear functions of the first derivatives of' the coefficients of F: conversely these first derivatives may be expressed linearly in terms of the symbols. We have in fact aaaik = ~~
[il, k] + [kl, i].
It is further to be noticed that just as the symbols of the second kind are expressed as linear functions of those of the first kind, so also those of the first kind may be expressed linearly in tkrms of those of the seQond kind. The typical equation is [ik, l] =~ az., {ik, v}. I'
19. The Riemann symbols are not all linearly independent.
We
deduce from the definition that there are atnong them the relations (gkik) = (gkki), (kgki) = (gkki), (iltlcg) = (glcki), (kigk) =(glcki), (gkki) + (gkik) + (gikk) = 0, and it readily follows that there are only /yn2 (n2l) independent
Riemann symbols. 18. We now pass on to consider the case of two general quadratic forms F and F'. Symbols derived from the form F' are distinguished by means of accents. If F can be transformed into F' we have the relation F = F'. This is equivalent to jn (n + 1) differential equations of the first order for the :c's as f11nctions of the g's, of which a typical • one 1s "~ a... a.v.. a~. , () ~~ a = a a.fJ • • • • • • • • • • • • • • . • • • • • • • • • 1 . r, B v9a. '!J• From these by differentiation are obtained ln51 (n + 1) equations for
second derivatives. Suppose these to be solved for the second derivatives : we get a set of equations of the type tP 3:r " { •L } am, om, ~ { R \.}' Oler ~~ ~~ + ol/lif tAl, ,. ;  :i:" = a~'~, A a ............(2). .i(f
"I! a. v9
i, k
cJ!}a. u8fJ
A
'!/A
12
THE METHOD OF CHRISTOFFEL
(CH. II
As in the particular case first considered, the number of equations of this type is exactly equal to the nun1ber of second derivatives. When, however, the set (1) is differentiated twice we obtain in1 (n + 1)1 equations involving third derivatives. There are only tn2 (n + 1) (n + 2) third derivatives of n functions of n variables, and thus by elimination are obtained !n2 (n + 1)2 ln2 (n + 1) (n + 2)=T\n2 (n2 l) new equations not involving third derivatives. These may be reduced, by means of the equations (2), to a set involving first derivatives only. Similarly from the equations for fourth derivatives may be deduced a new set of equations in first derivatives only, and so on for all higher derivatives. We call the set obtained from the equations involving 1·th derivatives the (r l)th set. The exception to this is the set (1), which is the first set. This is correct because there are no equations of the first order arising from the equations involving second derivatives. The number of equations in the (t· 1) th set is
(n+r1)! 1 !n (r ) (n 2) ! (r + 1) ! · The second set may be got directly from the equations of type (2). If we call the one given (o.{j), then
a
ag; (a,B) =0,
a Uufl (ay) =0,
a a and hence a,i,. (a,B) ay;(ay) = 0.
This last equation does not involve third derivatives and may be reduced by means of the equations (2) so as to be of the first order. The totality of new first order equations thus obtained are, after some algebraic modifications, reduced to the forn1
(ao~/3y )' =
~ ~
g,h, i,k
7~ 1 .) ozg (u~lt't ;kl "':fa.
oQJA
0.1:t oa:k ( ) a ~~ . . . . . . . . . . . . 3' "'9~ '!Jy "'liB ~
where a, {J, y, 8 take all values from 1 to n. These constitute the second set. As the number of linearly independent Riemann symbols is f?J:n2 (n2  1), the number of equations in this set is also /,;n2 (n' 1), as it should be. A simplification may now be introduced into the calculation of the remaining sets, for it may be seen without much difficulty that the third set may be obtained by differentiating the second set and eliminating the second derivatives involved by means of the equations of type (2), and in general the rth set may be obtained from the (rl)th in exactly the same manner.
1315]
THE QUADRILINEAR FORM
13
G4
14. The qua.drllinear form G4• We notice that the equations (1) and (8) are similar in form, and just as the former are the conditions for the equivalence of two quadratic differential forms, so the latter may be regarded as the conditions for the equivalence of two differential forms of the fourth order. Let there be four sets of differentials dy, dy, dg, d$, dx, d'8>:c, dx. Then if G4 l (glclti) d(1)tcgd(2):ekd(s):ehd(4)m,,
=
g,k, h, i
the relations (3) are equivalent to the single equation G4' = G4 • In this case we are compelled to take four different sets of differentials, for all the equations of (8) could not be obtained from a form in which any pair of the four were made equal to each other. In particular, for example, the form l (g/clti) d:c0 dtckdmhd:xi vanishes identically. Thus the second set of equations are the conditions for the equivalence of two quadrilinear forms. It will appear that this result is general, in other words that the rth set arises from the equivalence of two (r + 2)ply linear differential forms. 15. In fact let Gp. be any J.tply linear differential form, and Gp.' its transfonned. If the general coefficient of GP. is (i1 i2 .... ilL) then the equivalence of Gp. and Gp.' leads to a set of relations of type
(a.1 a.s • • • a.p. ), =
~ ~
ib ..., ip.
(. .
• ) ax, ami oya.l oytJ.a
a:x,IJ. ( ) ••• ••• ••• 4 oyD.p.
Zt Z2 • • • ~~~ L  . ! • • • • · • 
which are of the same form as (3). Differentiate (4) with respect to '!/«and substitute from (2) for the second derivatives of the :c's. Mter a little reduction the equation becomes of exactly the same form as (4) except that 1' is changed into 1' + 1. The new equations obtained may thus be regarded as conditions for the equivalence of two (p. + l)ply linear forms G~J+1 and G'~J.+l The relations which connect the coefficients of Gp.+l with those of Gp. are of type (i1 is ... i,.,. i)
=~ (i1is ... i,.) ;[ti~, A}(~ ... i,.) + {iig, A}(i1Ai8... i,.)+ ..
J. (5)
where there are 1' terms in the square bracket on the right, A replacing each of the letters ih is, ... , ip. in turn. A fonn such as G4 is called a covariant form of the original form
14
THE METHOD OF CHRISTOFFEL
(CH. II
F, and we now see that from any covariant form of order p. there may be derived another covariant form of order p. + 1. Further, the equivalence of the two quadratic forms F and F' leads to the equivalence of a series of covariant forms G., G,, ... derived from F, with the corresponding sequence derived from F'.
16. Sufficiency of the conditions obtained for the equi· valence of two forms. Now suppose that for given initial values of the variables y we can find initial values of the variables :c and of the first derivatives :
which make these two sequences equivalent and also make F=F'. Then from the equations (2) we can find uniquely the initial values of the second derivatives, and similarly we can find the values of the third and higher derivatives. There are no contradictions, since in addition to the relations given by the equivalence of the two sequences F, G4 , G5 , ••• and F', G4', G5', ••• there are only enough equations exactly to determine all the higher derivatives. Hence we can find for each of the x's its initial value and the initial values of all its derivatives for a given set of initial values of the y's, and it follows that the differential equations (1) can be formally satisfied, and there
fore that the transformation of F into F' is possible. We thus have the important result: The 'necessary and sufficient conditions i1~ O'rder tkat it sltall be possible to traruiform a quadratic jor1n F i'flto arwtlter quadratic form F are tkat tke equations in tke variables :c, g,
~,
derived from tM
equivakl~ce
of t/i,e two sequences F, G., G,, ... and F', G.', G,', ·~· sltall be algebraically compatible. We shall now prove this result in another way, and incidentally
show how the finite equations of the transformation may be obtained. Let the quantities :;: be denoted by u..', and consider a linear trans
formation between two sets of variables X and Y given by the scheme
"
~=~uCLiY". •=1
(i=l, ... ,n) •................. (6).
Now suppose that different sets of variables Y(l), Y..{3) = ~ .A ~.a. (>..a), ayfJ oya. Aa. where the A's are certain functions of the Zs andy's. By means of (11) these may be turned into equations of the same type, involving only however, the quantities (Ao.) where A takes values from 1 to k. The equations (9) may be reduced to two sets by means of (10) ; one is a set
:  x~.a. (Z
1 , ••• ,
z,., y) =(Aa), (~ = 1, 2, ... , It) ......(12),
and the other may be seen to vanish identically in virtue of (10) and (11). Now the conditions of coexistence of (12) may easily be seen to be the same as those for the set (9), and these al'e seen to be satisfied if all the quantities (Aa) are zero, (A= 1, ... , k). It follows from (11) that in this case all the remaining quantities (Aa.) vanish, and the Zs
18
THE METHOD OF CHRISTOFFEL
(CH. II
left arbitrary by the algebraic conditions must be determined to satisfy the equations obtained from (12) by making all the righthand members zero. These equations possess solutions involving It arbitrary constants, and we see that in this case the transformation ofF into F' is possible in a.: " different ways. 18. Connection of differential with algebraic invariants. In consequence of the theorems just proved, the problem of the equivalence of two quadratic differential forms is reduced to that of the equivalence of two sets of algebraic forms, where one set is obtained from the other by a linear transformation. The necessary and sufficient condition that it may be possible thus to transform one set of forms into another is that the algebraic invariants of the one set shall be equal to those of the other. It is convenient to extend our definition so as to include relative invariants ; a relative invariant is an expression which, under a transformation, repeats itself multiplied by some factor which depends only ou the transformation. Let I, 11, etc. denote a complete system of relative algebraic invariants for the first set, and 1', /1', etc. the corresponding complete system for the second set; we have /' = kl, /1' = k1lt, etc. and it is a known theorem that the quantities k are all powers of the determinant of the linear transformation. But this determinant is the Jacobian of the transformation performed on the variables :v; it therefore follows that tlte invariants /, l1, ... of the algebraic furms F, G4, ••• are a complete system of relative differential invariants fur tke quadratic differential form F, and if 'U/Tuler any trarulformation suck an invariant I becomes lei, then k is BO'me power of tlte Jacobian of the transformation. If we t.ake account of differential invariants which involve the magnitudes dm themselves, covariauts we may call them, it is clear that they correspond exactly to the covariants of the algebraic forms
F, G., .... 19. In the case where the equations do not determine the transformation of F into F' uniquely, it is easy to see that F must be transformable into itself, for since two different transformations give F' from F, the first of these followed by the inverse of the second gives a transformation of F into itself. Such a case arises, for example, when F is dt for a surface of revolution in space of three dimensions. It is clear that the conditions for this to be possible are expressed by the identical vanishing of certain of the invariants already obtained.
1720]
CONNECTION WITH ALGEBRAIC INVARIANTS
19
20. Differential parameters. Now suppose in the general case that we· wish to obtain all the invariants when account is taken of systems of functions f (xb ... , tttn) associated with the quadratic form. We have
~=:Sj[_~;
O'!Jr
t O:Vt Oyr
hence any invariant which involves only first derivates of I is taken account of by adding to the set of algebraic forms F, G, the linear form
l/i~, i
of'
where A=~ . tiJr,
For second derivatives the case is not so simple. We have, in fact, ilf = ~ ilf Omp CliVq + ~ Clj if'IVL •
oy,.ay,
:p, fl
OXp03Jq O'!Jr
ay,
p
ay,.oyB
03Jp
and the second term on the right shows that the second derivatives cannot by themselves be taken account of by means of a linear transformation. If the form F is used, however, there are the relations (2) for second derivatives ·of the x's with respect to the y's, and by means of these equations we have immediately f,'' _ 1 J; am,~ ,.,  .r>, q pq ay,. ay, ,
where
.J'

Jpq
02 /
03JpOaJq
lA {pn2' A} O:CA a.f
I
Now fpq differs from the corresponding second derivative of /by terms involving only first derivatives of/, and it follows that any function, and in particular any invariant, which involves only first and second derivatives of /, may be expressed as a function of the quantities f.,/IJfl. But it is clear from the equations of transformation of the quantities/_pq, that any invariant involving them may be taken account of by adding the quadratic l /pq~Xq' to our set of algebraic forms. JJ, fl
The extension is immediate, and just as the coefficients of G"+1 were obtained from those of Gp. (see equation (5)), so may the coefficients f 11qr of a cubic form be derived from those of the quadratic; any invariant involving only first, second, and third derivatives of I may be expressed in terms of f., /pq, f.qr, and is then seen to be an algebraic invariant of the forms F, G, and the three forms ul =.s 17P, u~ = ~ /pqXpXq', u. = l 1pqrx_pX'q' x,.". ~
p,q
p,q,r
20
[ca.
THE ABSOLUTE DIFFERENTIAL CALCULUS
11
Generally, any invaria.nt which involves derivatives up to the rth off is an algebraic invariant of the forms It~ G,, •.. , U't, U,, ... , Ur, where the coefficients of the successive forms U are calculated in exactly the same way as are the coefficients of the successive forms G. 21. The Absolute Differential Calculus. These ideas are at the base of the "Absolute Differential Calculus" of Ricci and LeviCivita. .A. system of functions Xr r, ... r.. (r1 , ••• , rm = 1, ... , n) is said to be covariant if the transformed system Yis given by y; l r ftls•••llll ax,l a.r,,, a:v,,. rlrll •••r,, = ;t,l . 0 ••• t a ,si, ...,Bm ":Jr1 y,., '!Jr,. 1
a
wtt
1
and the notation Xr r. is always used to denote an element of a covariant system. A contravariant system, an element of which is denoted by x(r•... r"'), is defined as one which has the transformation scheme 1
y
•••
l
(r1 ••• rm) =
, , ••• , '"' 1
X(•l ··· '•)
01!
O'l!
.i!.!.l. ...i!.!!
OIJI
~
o:c.,_ aa:,. ··· a:c,.,. ·
If X is any function of the variables m, and Y the same function expressed in the variables y, the equation Y =X shows that X may be regarded as belonging either to a covariant or to a contravariant system of order zero. Since the differentials dy satisfy the equations dy,. = ~ dx, ~ , the '
cJ:C,
differentials d.c form a contravariant system of unit order. The coefficients a,., of the fundamental quadratic form Fare an example of a covariant system of order two, and if the magnitudes a<JHJ.) are the coefficients of the form reciprocal to F (seep. 10), they are a contravariant system of order two. The laws of composition of systems are very simple : (1) Addition. If %,. Sr.... ,..,., are two covariant systems of the same order m, the system X,.•... r.,. + Srp.. r,. is covariant of the same order. This result holds also for contravariants. (2) Multiplicatioo. If x ...... r,., s......,, are two covariant systems of orders m and p, the system x,.1 ... r"' 'E,l ...•, is covariant of order m + p. This theorem also is true of contravariants. (3) Composition. If X ...... r ..p .. a, is a covariant system of order ?11 + p, and s) + l ~ {tq,1·z} XtrJ ...,.,_tqr,+t'".""'>} a.x, l=l q
...... (5"), and we have the equations of what may be called the contravariant
derivatives of a given system. It is to be noticed that the derivative of a particular element is indicated by writing an additional suffix or index at the right end of those denoting the original element ; for example the covariant derivative of Xpq with respect to ::c,. is Xpqr, and this is in general not the same as XFq· The laws for differentiation of sums and products of systems of the same kind are exactly the same as those for ordinary differentiation. For example, from the eqnations
Zpq= Xpq + Ypq, Zpqrst = Xpq
Yrsh
2224]
SECOND DERIVED SYSTEMS
23
it is easy to deduce the relations Zpqk =
X pqk + Y pqk,
Zpqrstk
= X pqk Y,.,t + X pq Yrstt,
and similarly for any covaria.nt or contravariant systems. If Z r1···rm "~ y(s, ...,,) ..t1.r "'V' rm8, ... s,, 1 •••
s., ..., Sp
the elements of the first derived system given by the equation Z,.l···rmk =
l
z,...
Y('a···•p)Xrp ..r.S:···Bpk + l
Bb ••• , Bp
s 1, ••• , Bp,
rmk
t
are readily seen to be
y<s•... Bpt) atk Xr(•••rm8l···'P'
and there is a corresponding formula for a composite contravariant system.
24. Now let X,.1...r. be any covariant system whatever, and form its second derived system. We have the identity
where ahkr,p is written for the Riemann symbol (kkrlp). ('rhis notation is justified since the Riemann symbols have been shown to be elemeuts of a covariant system of tl•e fourth order.) It thus appears that the element Xr,···rmhk is not in general equal to the element Xr1...rmkh· In fact, if all covariant differential operators are interchangeable, all the
Riemann symbcls must vanish identically. If the fundamental form is ~d~t2, all the Riemann symbols do vanish, and it readily appears from
'
the result of Christoffel already given that if these symbols vanish for a quadratic form, that form must be reducible to the sum of the squares of n perfect differentials. (There are some very particular cases of exception to this, of no importance in our theory.) In this case covariant differentiation reduces to ordinary differentiation. The number of linearly independent Riemann symbols has been proved to be ft n2 ( n2  1). In particular for n = 2 there is only one such symbol, G say, where G = (1212), and for n =3 there are six
symbols. The six equations obtained by Lame (see his Lefons sut• les coordonnees curvilignes) in connection with triply orthogonal systems of surfaces, are got by equating to zero the particular values the Riemann symbols take for the form Pd.'f!J+Qdy2+Rdz2. These six equations for a general quadratic form in th1·ee variables are given by Cayley ( Coll. Math. Papers, vol. XII. p. 13).
24
THE ABSOLUTE DIFFERENTIAl, CALCULUS
(CH. II
A certain amount of symmetry may be introduced into the case of n =8 if it is postulated that we may replace one index by another when their difference is 8. The linearly independent symbols may then all be expressed in the form ar+l, r+2, s+l, •+2t and if a,(rs) be written for this symbol, the system alied by the determinant of the transformation. If the t.ransformation is orthogonal, it and its contragredient are the same; also its determinant is unity, hence J is an absolute invariant. We therefore im1nediately obtain the result : The functionally independent set of invariants of orders up to and n
including the pth of the quadratic for.m l dul aud n associated i=l
functions ~(t), ••• , ..P'"'>, are J, and the orthogonal algebraic invariants of the system of 11ary forn1s A A(k), ( k = 1, 2, ... , n ; A= 1, 2, ... , p) 7l
(r = 1, 2, ... ,p)
l d'"ui U;. i=l '1f,
The linear forms l
tl'"u~ U1
may be omitted if account is taken of
'i=l
n
the invariants d'"c/l'",, and if the quadratic l
Ul~
is included we may
i=l
state the result in terms of a general linear transformation, since an orthogonal linear transformation is a linear transformation which leaves n
the quadratic 2 U 2 invariant. The final result is : i=l
The most general invariant is a function of the quantities arq,' D )2 (/"4>" U )2 • If n =2 the U's disappear and this expression becomes 2 1 ( 2 oF 02E o2 K = 2 (EG lf""J.) OuOv avt  Ou2 + ... , K
G)
where This is the wellknown Gaussian expression for K. Next let n =3 and let the space be Euclidean. Then we may take dr =dar + d'!f + dz2, and there is one expression U. Take the equation of the surface to be then U is F. \Ye get
F(x, y, z) = 0, If we substitute in the above general expression for K 1
K=
o2F
(¥1)' + (~i/ + (Wz'Y ~~,
a:cay' atp
CPF
a;;'
o:coz'
oy2 '
ayaz' 'iPF
o'JF
~:coz'
ayaz'
aF
oF
~:c '
aF
o2F
ay ,
rf~F
o.e1
o:c • oF ay oF
'
oF oz '
0
The two expressions forK given above have no apparent connection. They are, however, two interpretations of one and the same general formula.
* Maschke, Trans.
A mer. lllath. Soc. (1906), Vol.
VI.
p. 98.
CHAPTER V APPLIOATIONS
49. We now consider some applications of the theory of differential invariants. Suppose first that the quadratic form is interpreted as the square of the element of length in a certain ndimensional manifold. 'l'he invariants, though expressed analytically, are intrinsically connected with the manifold, apart from any frame of reference. They are differential, and involve only one set of the independent variables. They thus give the intrinsic character of the manifold at, and infinitely near, a particular point of it. If we take account of differential parameters, they express quantities intrinsically connected with the section of the fundamentaltnanifold by another manifold. But we have a set of quantities, defined geometrically, of just this type, that is to say they depend only on the manifold at or near some particular point and are intrinsically connected with it. Let us consider more in detail the particular case of a surface in Euclidean space of three dimensions. This is not intrinsically determinate if only one quadratic form, that for dB", is given. ~,or example, all developable surfaces have the. same quadratic form for ds2 as that of a plane. The catenoid (the surface of revolution obtained by revolving a catenary about its directrix) and the regular helicoid (the surface swept out by a line which is parallel to a fixed plane, intersects a fixed line perpendicular to the plane, and rotates uniformly about the fixed line as its point of intersection moves uniformly along that line) are two surfaces with the same quadratic form for dsl. Two surfaces which have the same quadratic form for dr are said to be applicable to each other. If we regard a surface as made of some perfectly flexible inextensible material, then it is clear that the surfaces into which it may be deformed are the surfaces applicable to it. There is a second quadratic form which we may associate with a
52
APPLICATIONS
[CH. V
given surface, namely that for ds2jp, where 1/p is the curvature of the normal section by a plane which meets the surface in the direction dufdv. The coefficients of the second form cannot be chosen arbitrarily if the first is given, though they involve a certain amount of arbitrariness. If both t.hese forms are given, the surface is intrinsically determinate. It follows that the geometrical magnitudes at a particular point on the surface are the differential invariants of two quadratic forms. Similarly, the geometrical quantities connected \vith any curve U = 0 on the surface may be expressed in tenns of differential paratneters that involve two quadratic forms. 'fhus the principal radii of curvature of the surface, the normal, geodesic, aud principal curvatures and the torsion of a given curve on the surface are differential parameters of this type. Among all these geometrical quantities, there are some that are the same for all surfaces applicable to each other. Such are, for example, the total curvature I/R1R2 of the surface, the angle of intersection of two curves on the surface, the geodesic curvature of any curve on the surface, and many others. These, it is clear, do not depend on the second quadratic form, and they are therefore invariants and parameters of a single quadratic form, that for dil'. It follows, conversely, that any invariant or parameter of the form for ds2 represents some geometrical magnitude associated 'vith the surface, and the magnitude is the same for all surfaces applicable to that given. On this account, such iuvariants are called deformation in variants.
50. Geometrical interpretation of invariants. In order to apply the invariant theory, we have now to interpret our invariants in terms of geometrical n1agnitudes, and also to interpret geometrical magnitudes in terms of invariants. It is perhaps worth while to point out the advantages gained by the use of invariant theory. In the first place we are able to apply in the simplest possible way the methods of analysis to geometrical problems, for we express all our data analytically, and yet avoid extraneous properties which arise through the relations of our surface to a particular coordinate frame chosen. Again, when we express a given quantity as far as possible by means of invariants, it may happen that its expression only involves invariants, and thus its invariance becomes intuitive. Also, if we know that a given quantity is invariant, we can often determine its invariantive form for some simple choice of coordinates, and then we are able
49, 50]
GEOMETRICAL INTERPRETATION OF INVARIANTS
53
to write down its expression for any coordinates whatever. An example of this has already been given in the case of Laplace's equation 02Jr c2V a2V a(Cs + (Jy2 + ?z2 = 0. In this case we are dealing with Euclidean space of three dimensions for which ds2 = dx2 + dy + dz2, and the equation, in the notation of the Absolute Differential Calculus, is l a('·s> Vr8 = 0, for this particular T, 8
quadratic form.
It follows at ouce that 02V
02JT
a:r
02V
+ cy2 + oz2 • is au invariant of a general quadratic for1n in three variables, and its general expression is lar.Ys, A2c/J = la (rs) c/Jrs,
A~= ~a(rs) ~r~8,
A ( ~'
then these three quantities are obviously invariants, and 1 2 A1'= 02 , A(v,Av)=ofJav'
ao
A2v = _!__
~
(A) =  _!_0 aoov ..!. Op.
AO ov 0
Hence
1
P
A2'V
JF:V
+
3
1 A (v, Av) 2 (Av)i .
It follows that 1/p is an invariant, since the A's are all invariants, and further, the geodesic curvature of any curve q, (u, 1') = const., on a surface for which the quadratic form ds' is perfect]y general, is
_ A2c/> +!A (cf>, Acf>) J Acp 2 (~cf>)i . This example is a good illustration of the advantages derived from the invariant theory. We start with 1
aA
 AO &v'
54
APPLICATIONS
(OH. V
and express it as far as possible in terms of invariants. It happens that it is entirely expressed by means of invariants, and hence it is itself an invariant. Further, when it is thus expressed in terms of invariants only, its general value may be at once written down. We also note that the differential equation for all families of geodesics on the given surface is 2~ 2 cp. A~=~ (9>, A~), and this may be turned into a differential equation for the geodesics themselves by writing dv a,p;a~ du =au ov'
with a corresponding expression for the second derivative
·
d2v du9
•
61. Another application of the differential invariant theory can be made by means of the theory of algebraic invariants. All our differential invariants and parameters are invariants of algebraic forms. By means of the algebraic theory we can determine syzygies or algebraic relations connecting these invariants, and such syzygies, when expressed in terms of the geometrical magnitudes of the surface, lead to algebraic relations among these apparently independent quantities. Also the coefficients of the various forms are algebraically independen~ and thus all such relations are given by syzygies. A surface is intrinsically determinate if two quadratic forms are given, and hence all its geometrical magnitudes are invariants of two quadratic forms. We are thus able, by means of the invariant theory, to determine which among these magnitudes are independent, and, by means of syzygies, to determine all the relations connecting those that are not independent. For illustrations of this part of the subject the reader should consult the latter part of Forsyth's memoir t.
52. The case of a surface in ordinary spacethe quadratic form in two variables. Confining ourselves to a quadratic form in two variables, we suppose that there are associated with the form two functions ~ (u, v), t/1 (u, v). In this case there is only one Riemann symbol, the quantity G. We * For other examples the reader should consult Wright, Vol.
(1906), p. 879. t Phil. Trans. (1908), Ser. A, vol. 201, pp. 369 •qq. XII.
B1dl • .Amer.
Math. Soc.
5052]
55
THE SURFACE IN ORDINARY SPACE
may therefore take instead of the series of forms G4 , Gr,, ••• ,the quantity K, which is an absolute invariant, and the covariant derived forms of K. The set of algebraic forms is now (i) the fundamental quadratic, (ii) cp, .p, K, and their successive covariant derived forms. All the deformation invariants and parameters of a surface are therefore given by the algebraic invariants of these forms. If the quadratic form is Edu2 + '2Fdudv + Gdv2, we have the invariants
~...1..~
=~a(rs)...t.~r~s =_!_{Ea§. H ov 2F~~+G~ Olt ov au I } ' ~ (q, ) =la cp ,,, =_!__ { E aq:, a.p  F (~cp a.p + 05!_ B4') + G ~ a_r_} H ov av au av av ote au o·u ' (...1.. ·'') = ~ = _!__ a.p  ~ 01} H {ocp OU O·V OU OV ' ...1..
and .P cut at an angle a, it may be easily proved that cos a= 11 ( q,, .p) sin a= ® ( cp, t/1) Jb.cp.~~' J~cp.~~' and the intet'Pretation is therefore cos2 a + sin~a = 1. 'fhe invariants are all interpreted geometrically when the geometrical interpretation of lirp is obtained. Let dn denote the perpendicular distance between the curves cp and c/J + dcp at the point (u, v), then dcp a~ ilu acp dv dn = au dn + av dn . Also, if ds denote an infinitesimal arc of the curve c/l = const.,
01 ~ + ~ t!!=o ouds a~ds ' E
+ 2F
'
::.!. ~ {! (G ocfl  F~)} + _!_ ~ {_! (E~ F~)}. Hou H olt ov Hov H av au
Beltrami's method of obtaining it is by an application of Green's 'fheorem. In fact it is easy to prove that if cfl and .p are two functions regular inside a closed contour on the surface, and if du_ denotes an element of area of the contour, ds an arc of the boundary,
if 1Jr·rA ( cfJ, 1/J) du =  j t/t ocfl fin ds  }J 1/1 ~"' du, and it follows, since all the other ter1ns of the equation are invariants, ~nd since the contour is arbitrary, that 4 2 cfl is an invariant.
. 54. All the parameters may be expressed in terms of three of them, and their derivatives. The invariants ll., ®, l12 once obtained, we can calculate from them many others, for example Jl./1 ~' il. ( rp, b.~), 42 (b.~), t1 ( b.cfl, 4ep), etc. We shall prove that all invariants can be obtained by algebraic combinations of these, a result due to Beltrami. Suppose tbat the quadratic form has been transformed so that cfl, tf! are the parametric curves, then the coefficients may be calculated without trouble and we have iJs2
[email protected] ( ~' 1/1) {Al/;d"'~  2L\ ( "'' 1/1) d#l/1 + At/Kit/}}. Now let I be any invariant involving the two functions .., "')/0 ("'· "').
Hence all the derivatives may be expressed by means of the ®'s,
5255]
57
THE COMPLETE SYSTEM OF PARAMETERS
and the result follo,vs. If the invariant contains only one function rp, we may take c/J, and either b.c/J or b.2cp, as the two independent variables of the form. It follows, in this case, that all the invariants may be obtained by repeated application of the operations d, b.2 to the single function q,. Although we thus have a method whereby all invariants can be obtained, the result is not complete, for we have no clue as to the inde· pendence of the invariants, and frequently they are not thus expressed in their simplest for1ns. We know from the algebraic theory that there are three independent invariants of the second order involving one function cfl. These must be b.b.cf>,
a (cf>, b.cfl),
b.2 cf>.
There are four of the third order, and these are included in b.b.b.cfl,
b. (.P, b.b.tfl), b. (b.cfl,
b.~cf>),
b. (b.2cp),
~ (c/l,
b.2c/J), etc.,
but we have no im1nediate method of showing what are the relations that connect the last set. 55. Another method for deriving invariants of order 1· + 1 fro1n those of order r is the following : Let I be any invariant, and let ds1 denote an arc of the curve cf> = coust. We have
di = oi du + ~I dv dsl
ou dsl av dsl
~ d·u + OJ!. dv _ 0 J
au dsl ov dsl 
'
E(du)2 + 2F(~) (dv_) + G(t!!)~ = 1. ds1 ds1 ds1 ds1 Hence di/dsh which is obviously an invariant, and is of order one higher than I, is ®(I, cfl)/ J~. Simila:rly dljds2 =®(I, t/1)/ JiS:¥, where ds2 is an arc of the curve .y = const., and any function of these two quantities is an invariant of order r + 1, where I is of order r. In particular, suppose that , tf! are such that a (cp, tf!) = 0, then the curves cp =const., .p =const. cut at right angles. An invariant of order r + 1 is + ds1 ds2
(d[)2 (d[)2,
and this is readily seen to be AI; it thus does not involve rp, .p explicitly. For example, the Gaussian invariant of the third order may be written either b.K or (dK/ds1) 2 + (dK/ds2)1, where dsh ds2 are any two infinitesimal arcs through u, v cutting at right angles.
58
[cH. v
APPLICATIONS
56. Geometrical properties expressed by the vanishing of invariants. We next consider the geometrical properties involved iu the vanishing of certain invariants. If ® ( cfJ, .p) =0, from what we have already seen, 4> and .; cut at an angle zero; therefore they are coincident, that is to say, ~ is a function of ~. If A ( c/l, .p) = 0 the curves cfl and .p cut at right angles. Now At/> is A ( c/J, f/J) and hence, if 4cJ> = 0, the curve cfl =const. is at right angles to itself. This curve (of course imaginary) must therefore be such that the tangent at any point meets the circle at infinity. Such curves are analogues of the straight lines '!I + i:c =lc in a plane. Along such a curve ar~>
a~ OV
:. du, + .: dv = 0 Ott,
'
and it therefore follows that Ed~t2 + 2Fdztdv + Gdv2 = 0, that is to say ds' = 0. The curves are hence such that the distance between any two points on one of them, measured along that curve, is zero. Consider the more general case of Acfl = f (cfJ). Let the curves .P be chosen at right angles to the curves cp, then A ( cp, 1/1) =0, and therefore ..7 2 df/>2 dt/12 = t:t. c/l + ~' ((18
or if du=dcpjJJ·(cf>), ds2 = dlt2 + 0 2dl/J, where 0 is a function of u and .y. The curves u =const. and the curves cfl = const. are obviously the same, since ~"is a fttnction of c/J. Now if 1/p is the geodesic curvature of the curves t/J = const., and
Hence, in our case, 1Ip is zero. 'rhe 1/J curves are therefore geodesics, and the u's cut them orthogonally. Hence, if b.cJ> =f (cfl), the curves ~ = const. are the orthogonal trajectories of a family of geodesics.
57. In order to develope the properties of 11 2~, we consider the following problem. It is proposed to express the fundamental quadratic in the form P (.x) dp2 + 2Q (.x) dpdg + R (OJ) dq2, · where P, Q, R are given functions of the single argument :c, and :c, p, q are functions of u, 'l' to be determined.
56, 57]
THE VANISHING 01•' PARTICULAR INVARIANTS
59
We have 11p = R!(PR Q2) == S, say, then Sis a known function of :c. Also Ra:c_Q~
11 (p, t::. p ) = S and
'
papR  Q2oq '
~
®
, ( ) a2 , , p, 11p = 8 'J· PRQ 2
{!_ R _!_ Q } P  J p R _ Q2 op J p R _ Q2 aq J p R _ Q2 •
112 _
1
From these equations, by elimination of :c, o:cfop, orcfoq, there is obtained a relation between the invariants of the first and second orders of p. This relation, being invarianti ve, may be written in terms of the original form, and we thus have an equation of the second order for JJ as a function of u, f'· Taking any solution of this equation, we obtain without difficulty an equation of the first order for q, and then te 1nay be obtained by elimination. Consider, for example, the case in which Q = 0, R =P =:c. We now have for the quadratic form, m (dy + dq2), and 1 Ap = ~·
1 ax A (p, Ap) = {ij Op'
8 (p, J.\p) =
1 a:c
/? aq•
A1p= 0.
The second order differential equation is thus 112 p = 0. · Conversely, let p be auy solution of ll. 2 p = 0, and let a function q be chosen so that 4 (p, q) = 0.
We have
!{! ~ {! (EOJ!.F~)}=o ouH (G~FBJ!.)}+ au av ovH av au ' E?J!. av ~F(?E ov au ~ ov +~~)+ av au G?£ oze ~ au =0. From the first of these equations it follows that
(o
(E
_!_ OJ!.F~) =Of"_ _!_ ap F~) =Jt. H au av ev' H ov au au' where f is a certain function of u and f', and the second. equation becomes
~~~~=0 au av av au •
Hence q is a function of/, and without loss of generality we may take it to be exactly j. We can now calculate Ag in terms of the derivatives of p, and we have in fact dq = Ap. Thus the quadratic form
becomes
i.p (dp' + dq'), and hence a: is 1/J.\p,
60
[ca. v
APPLICATIONS
Nolv if the quadratic form can be \vritten a· (dJJ2 + dq2), the surface is said to be divided iso'lnet'rically, or ?·sothermally, by the systems of curves p and q, and "·e have the important result that the solutions of ~2P = 0 give the fan1ilies of isothern1al curves on the surface. We notice that, if one conjugate pair of solutions JJ, q of i12c/J =0 has been deter1nined, then taking these as independent variables, we have _ 1 (o2) =~ + ::::; , .. :r tJP.. oq.. and hence the 1nost general solution of i1 2 cp = 0 is given by cJ> =It (p + iq) +A (p iq), ~.,cp
whereft and}; are arbitrary functions of their arguments. Other examples of this theory (given by Darboux) are dSJ = cos2 :rdp2 + sin2 xdq2, in which case ~ (JJ, ~p) = 2~2P (11JJ 1),
ds2 =:cdp2 +
and
~
for which Exantples.
(p,
~p) =
! dq_B,
i1p i12]J.
Find the relation among the in\'"ariants of p if ds2=dp2fsin 2 x+dq2fcos2 .7). 9 (ii) ProYe that if the equations A2'/J==O, t. ( ~~cJ>~cJ>), cJ>) = 0, have a (i)
common solution, the quadratic can be reduced to LiouT"ille's form, ds 2=( U + l') (du2+dv2), where U is a function of u only, T7 a function of v only, and show how to find the parametric curves u, v.
As another illustration, consider the curves cp for which 112c1>/b.cfl =I Cc/J). If we take these for the parametric curves u, and their orthogonal trajectories for the curves v, \ve have dil =A 2dtt2 + C 2d#, and
~.u = }oo~, (~)'
Au=
1~·
Hence our eq nation becomes
~(log~) =/(u), and therefore 0 = AeU+ v where U is a function of u only, V a function of v only. Hence dt =A 2e2U [e 2u du 2 + e2Y dv2] =A (du' 2 + dv' 2 ),
57' 58]
61
.APPLICABILITY OF TWO SURFACES
where tl is a function of tt, v' a function of v. The curves tit' are the same as the curves¢, and it follows that if l12¢/~¢ =/(¢),the curves cp form an isothermal system.
58. Applicability of two surfaces. We no'v consider the general problem of the applicability of two surfaces. Let the quadratics for the two surfaces be Edzt2 + 2.Fdltdv + Gd'lf,
E 1dlt12 + 2F;du1dv1 + G1dVt2•
We first calculate the invariant K for the t\VO forms, and we must have K(lt, v) = K1 (ut, vt)· This gives one equation for tt1, v1 in terms of u, v unless K is a constant. As \Ve can have no relation between tt, v· alone, X. 1nust in this case be also the same constant. Let K =:: K 1 = a, a constant. Choose on the first surface any point P, and take as para1netric curves v the geodesics through this point, the curves 'lt being their orthogonal trajectories. We have then for this surface, ds2 = du2 + 0 2d#', and 0 vanishes with tt, whatever be the value of v. Similarly, the second quadratic may be reduced to 2 1 o2C o C 2 2 2 dzt1 + 0 1 dvt • No'v K = ~ 2 , ai1d therefore ~ 2 + aO= 0. Hence 0 lllt tJlt 0 = V sin Jau. Similarly 01 = V1 sin J azt1. If, then, we take u =u1 , Vdv= V1dvt, the quadratics are transformed into each other, and we see that in this case the surfaces can be applied so that a given point on the first corresponds to any given point on the second, and so that a given geodesic through the point on the first corresponds to any given geodesic through the corresponding point on the second. Now suppose that K and Kt are not constants. We must have AK = ~'IG, where the index denotes that the invariant is formed with reference to the second of the two forms. There are thus obtained t\vo equations for 'lt1 , e1 as functions of u, v. If these equations are inconsistent, the surfaces are not applicable. 'fhere are two other possibilities ; either they are independent of one another, or one is a consequence of the other. In the second case it is easy to see that AK must be a function of K, and A'IG 1nust be the same function of K 1 • Suppose that ~K =.,f(K), and form the invariant A 2K. Again there are the three possibilities before mentioned. If
A2K = ¢ (K), A2' (Kt) = cJ> (K1), take the parametric curves u so that du = dK/f(K).
Then A (u) = 1,
62 ~ 2 (u)
APPLICATIONS
= F(u),
(cu. v
say, and ds' = du2 + (J2d'f1, where the v's are the
orthogonal trajectories of the u's. Also .:11u= ~~; hence 0=
VeJF(u)du,
where Vis a function ofv. Now take Vdv=dv', and it follows that ds'::: du'J + 6 JF(u) du dv'B. Similarly, for the second surface ds' = dull + e JF(ua)dul dvl'2, and the surfaces are deformable into each other by means of u = u1 , v =v1 + const. It thus follows that if K=K1, b.K=li'Kt, 112K=~2'K1, lead to only one equation between u, v, u1, v11 the surfaces are applicable to each other in a single infinity of ways. Now l~t the three equations just mentioned lead to two equations between u, v, u1, v1. Suppose first that K and 4K are independent of ea.ch other, and take them for the variables in the first form. Similarly, let ~ and ~, K1 be taken for the variables in the second form. Then, if the surfaces are applicable to each other, these two forms must nO\V be the same. It follows at once from the general expression for the form given on p. 56 that we must have ~11K = 11'~'K,., 11 (K, ~K) = 11' (K,_, fl..,~). If K and AK are dependent on one another, we take K and A 2K as the variables of the form, and the necessary and sufficient conditions become in this case
tl.tl.sK =11'~;lit, 11 (K, 11aK) = fl.' (I1t, ~~,K1) in addition to K = K1, tl.K =fl.' K1, A2K = Aa'K1. 59. The quadratic form in three variables. For this case we only note the significance of a few of the more important invariants of lowest order. The geometrical interpretation of those of orders one, two and three for a form of rank zero, the case of ordinary Euclidean space, has been considered in detail by Forsyth in his memoir, already quoted, on the differential invariants of space. We note that there are six Riemann symbols. The explicit expressions of these have been given by Cayley, and they have been discussed in detail by Lame for the case in which the parametric
58, 59]
'l'HE QUADRATIC FORM IN THREE VABIABLES
63
surfaces cut orthogonally, iu his book on Curvilinear Coordinates. If they all vanish, the space is Euclidean. 'fhe invariant Ac/> la('·')~rq,, is (df/>/dn) 2, where dn denotes an element of arc perpendicular to the surface cp = coust. The invariant ~ (f/>, t/1) E la(ra) c/Jr'/1s
=
(~) (~) cos a,
•
IS
where
a
is the angle between the surfaces ,P, ap, and d1l1 is normal to ®(c/J, af!, x) '1.c(rllt)q,,..;,xe
is equal to
=
deb) (d"') .0 (dii il1i"t (d)() dn~ Sin
.p.
'
where sin 0 is the sine of the solid angle at which the three surfaces v,.w,, 2Bv,.w, linear an~ homogeneous in the six quantities '· V1W1 1 VtWI, 11aWs, 'V~s + 'VaW1, VsW1 + 'lhWs, V1W1 + VaW1, and, in addition, A ( u, v) laurv, s lu(•)v, =0, l1 (u, w) lu(•)w, = 0. Now from u,(l)vt + u.."1,.. Letf,. =,.,_>..,1,., then sincef,., =f.,. and
we deduce that
P.Ant,. + plyniJ~Jr'>.;l• =P.~/1 + ~~}niJ'Att,~Jr • (i . ij If we multiply this equation by >..,., w, p.. These equations are linear in p. and the ~'s, when these quantities are eliminated we have an equation ~ ((a)) = 0 of degree '" 1 for w. Suppose that the roots of ~are all simple. Then to any root, say w," there corresponds a set of values for the 'A's. We thus have n 1 congruences which are obviously orthogonal to [n]; they are orthogonal to each other. For, multiplying the second of the above equations by 'At(q) and summing for all values of q, we have l (Xqr + w~~,aqr) A~~,..,
,.
r
is the cosine of the angle between f/l and.;,
ucos (n.P) u~ cos (vt/1) = p cos (p,.p).
This shows that any line perpendicular to the line n and to the line p. is also perpendicular to the line v ; in other words : TluJ tangent to tlte line of geodesic curvature of any t'rajectory lies in tke plane determined by t'M tan.gent to tkat trajectory and tke tangent to til£ line offorce throogk tke poi'llt. If, in particular, ·the forces X are all zero, A,"' is proportional to v,. or else u and v,. vanish identically. Now [n] and (v] are at right
7779]
FIRST INTEGRALS 01~ THE SYSTEM
(ii)
angles, and hence the first case gives n1inimallines. Excluding these, we have the result: If the external forces are all zero, the trajectories are the geodesics of the manifold, and they are described with constant velocity. When the field of force is not zero, we deduce, if in (iv) 've take l/J to be (k], any one of the congruences of the ennuple, u=p cos (p.n), 1 U ')'nhn = p cos (p.k), (k = 1, 2, ... , n 1) which are the equations in invariautive form.
79. First Integrals of the system (ii). The n equations (ii) are linear in the second de1·ivutives of the ala with respect to t. If they are solved for these derivatives they take the form (v) .x, =X(')  l {1·s, i} x,.tt,. (i = 1, 2, ... , n) '1',
s
Let J' =co'lst. be a first integral of these equations, then f is a function of the quantities m and :t, and dj ~OJ .. ~OJ. t = ~a::c, + ...,i fi(Ci ~ x, d , 'llJf
(vi)
=l
~ ..l"' (i) + li
i t 1Xi
{:£ :hi  ~ l flfCi
fiXt .,, 1
{rs, i} Xr:t,} .
'!'his last expression must he zero identically. Suppose in particular that f is a polynomial in the first derivatives :t, and let the terms of highest degree in these derivatives be u. Then, since (vi) vanishes identically, the coefficients of the various quantities Xr in it must vanish se:parately, and hence (vii)
~
~
i
{au . aw,
flJ#,
au ~~ {rs, "'} fJJr. x,. } = _ 0. a;w, .,.,,
Therefore u =const. must be an integral of the equations (ii) when all the X's are made zero. That is to say, to any polynomial first integral of the general set (ii) there corresponds a homogeneous polynomial first integral of the differential equations for geodesics, in the manifold for which dSJ = l a,.,d:c.,.d(l),. '1',8 Assume that u = ~ c.,.J·· .r. tbr, •••:hr,., f't' ••• , t'm
then the system c is covariant of the rth order, and (vii) becomes l ,.h••
Cr 1••• rmrm+l Xr1• • • d:r•+l
'"'"'+1
EE 0.
84
APPLICATIONS
(CH. V
80. Homogeneous linear first integrals of the equations
for geodesics. For example, suppose that m is unity, th()n == lcr.Tr:
ft
r
and (vii) becomes r,s
'fherefore Cra + Csr =0. (r, s = 1, 2, ... , n) These conditions show that the quadratic form must admit the infinitesimal transformation lc(r)
aj,
r
O:Cr
and hence the necessary anrl sufficient condition that the equations for geodesics of a manifold have a linear first integral is that the quadratic form of the manifold admit an infinitesimal transformation. If the geueral system (ii), where the X's are not zero, also has the linear solution u = const., the retnaining terms from (vi) give ~crX(r)
=0,
r
and hence the additional condition is that the path curves for the infinitesimal transformation must be orthogonal to the lines of force.
81. Quadratic first integrals. Systems (ii) that possess a quadratic first integral are of particular interest, for a reason that will appear later. Suppose the first integral to be l c,.,thrtbs = const. r, s
Then c is a covariant system of the second order with reference to l ar8 dxrdx, for which Cra = Csr, aud our conditions are t",B l CrseX rXsXe := 0, ~ Crs X (r) Xa := 0. ~~t
~8
These give the equations (viii) (ix)
Crat
+ Cstr + Ctrs =0, ~cr,X(r)
= 0.
(r,s,t=l,2, ... ,n) (s= 1, 2, .•• , n)
T
Hence the necessary and sufficient conditions that the equations for geodesics possess a homogeneous quadratic integral are the equations (viii). If, in addition, a general dynamical system for which the X's are not zero possess that integral, the equations (ix) must also be
8082]
85
QUADRATIC FIRST INTEGRALS
satisfied for nonzero values of the X's, and hence the discriminant of the quadratic l Cr8 XrX8 must vanish. If this discriminant vanish, the 'I",B
ratios of the X's can be determined, in general uniquely, from the equations (ix). If the quadratic integral is not homogeneous, let it be 2 CrsXrOJ8 + '2brtbr + U =const. r,s
r
In this case the system must possess the linear integral 2,. brtbr =const., and hence also the quadratic integral 2crsXrXs + u =const. The con.. ~· ditions for this last are (viii) and lcrsX(r) + te, =0. (8 = 1, 2, •.. , n) (x) ,. One solution of (viii) is obvious, for we know that the first derived system of the coefficients ars of the quadratic form is identically zero. Hence one quadratic first integral of geodesics is ~ a,.stb,.dJ, and since larsX(r) ,.
=Xr,
'r, B
the equations (x) give X,.+ Ur = 0, that is to say, the
forces X must be actual first derivatives of a function u. The system of forces is therefore conservative, and the first integral is the energy integral. . In addition to the solution Crs =Or, the general form does not possess a quadratic first integral for geodesics, and all the forms which do possess such integrals have not yet been determined, though many classes of such forms have been obtained. The case of n =2 has been completely solved by Liouville.
82. Systems which have the same tr~eotories. Suppose that we have any dynamical system for which the equations are the set (ii). Any solution whatever is given by definite expressions for the n variables a:1 , ••• , rc. as functions of the single parameter t. Thus to each solution there corresponds a curve in any n dimensional space, the trajectory of the solution. Now suppose that we have any other dynamical system d
(xi)
(au) au
dt1 oa:,.'  aa:,. = Y,.,
(r =1, 2, ... , n)
where 2 U =l c,.,a:,.' a:,', and accents denote derivatives with regard to t1 • ,., 8
We enquire what are the conditions that the trajectories of this system are the same as those of the former system. If the trajectories are the same in the two systems, let a:, =if (t), (i = 1, 2, ... , n), be a solution of
86
[ca v
APPLICATIONS
the former; then if Xt = F, (tt) is the corresponding solution of the latter, that is to say the solution that gives the same trajectory, these two expressions for :e, must become the same when we choose for t an appropriate function of t 1 • We write t = 6 (tt) and take l ar8dterdXs r, 8 as the fundamental form, and we also make the assumption that the. discriminants of the two quadratic forms do not vanish. Confining ourselves to a particular trajectory, 've have ''6'2, :Xr = X,,fl , Xr = Xr 6" + Xr I
'LJI
II
•
Xr + l {pq, t'} :hpdJq ::: X