Selected Papers of Kentaro Yano

SELECTED PAPERS OF KENTARO YANO

NORTH-HOLIAND MATHEMATICS STUDIES

Selected Papers of KENTARO YANO

Edited by

MORIO OBATA

1982

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM 9 NEW YORKO OXFORD

70

"North- Holland Publishing Company. I982

All rights reserved. No part ofthis publication may be reproduced. stored in a retrievalsystem, or transmitted, in any form or by any means, rlecrronic, mechanical, photocopying, recording or otherwise. wirhour rhepriorpermission of rhe copyright owner.

ISBN: 0 4 4 4 8 6 4 9 5 4

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD

Sole disrriburors f o r the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

PRINTED IN T H E NETHERLANDS

Table of Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII Kentaro Yano-My Old Friend, by Shiing-shen Chern . . . 1x Notes on My Mathematical Works, by Kentaro Yano . . . . . . . . X I Bibliography of the Publications of Kentaro Yano . . . . . . . . . . X X X V Les espaces i connexion projective et la gioniitrie projective des “paths” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Sur la theorie des espaces A connexion contorme . . . . . . . 71 On harmonic and Killing vector fields . . . . . . . . . . . . . . . . . . . 130 On n-dimensional Riemannian spaces admitting a group of motions of order i n ( n - 1) 1 . . . . . . . . . . . . . . . . . . . . . . . . I38 On geometric objects and Lie groups of transformations (with N.H.Kuiper) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 On invariant subspaces in an almost complex X,,, (with J. A. Schouten) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 On real representations of Kaehlerian manifolds (with 1. Mogi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 A class of affinely connected spaces (with H. C. Wang) . . Einstein spaces admitting a one-parameter group of conforma . . . . . . . . . . 219 transformations (with T. Nagano) . . . . . . . . . . Harmonic and Killing vector fields in co ct oricntable Rie. . . . . . . . . . . . . . . . 230 niannian spaces with boundary . . . . . . Projectively flat spaces with recurrent curvature (with Y . C. 241 Wong) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On a structure defined by a tensor field f of type ( 1 , l ) satisfying f ” + f =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Prolongations of tensor fields and connections to tangent bundles, I. General theory (with S. Kobayashi) . . . . . . . . . . . . 262 Some results related to the equivalence problem in Riemannian 279 geometry (with K. Nomizu) . . . . . . . . . . . . . . . . . . . . . . . . . . . Vcrtical and complete lifts from a manifold to its cotangent bundle (with E. M. Patterson) . . . . . . . . . . . . . . . . . . . . . . . . . 289 Almost complex structures on tensor bundles (with A. J. Ledger) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Differential geometric structures on principal toroidal bundles (with D. E. Blair and G . D. Ludden) . . . . . . . . . . . . . . . . . . 327 Kaehlerian manifolds with constant scalar curvature whose

+

*

Numbers in brackets refer to the Bibliography.

V

Bochner curvature tensor vanishes (with S . Ishihara) . . . . . . . 337 [303] Notes on infinitesimal variations of submanifolds . . . . . . . . . . . . 345 [309] CR submanifolds of a complex space form (with A . Bejancu 355 and M . Kon) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VI

Foreword Professor Kentaro Yano has made great contributions to Differential Geometry for nearly fifty years since 1934 when he wrote his first paper. In recognition of this, the publication of this Selecta volume was planned by his former students and his colleagues on the occasion of celebrating his 70th birthday, following a Japanese custom. This volume consists of his own selection of papers which rcminiscc over the times, places, and/or persons in his long career in niatheniatical research. I t should be mentioned that, besides these particular mathematical research papers and books, he has an enormous number of books and cssays written in Japanese. Among them are some enlightening introductions to modern mathematics, some cultural essays regarding mathematics or relativities, and some textbooks of mathematics at various Icvels. Perhaps he is one of the most well-known mathematicians among the Japanese public because of these social and cultural activities. Both Professor Yano and I owe debts of gratitude, cspccially to Professor S. S. Chern for contributing so gracious an introductory essay, to the editors of all the journals involved in this volume for generously permitting reproduction from the originals, and finally to Mr. A. Sevcnster of North-Holland Publishing Company for his care in the production of this volunic from its conception. May 1982 Morio Obata

VII

This Page Intentionally Left Blank

Kentaro Yano-

Old Friend

By Shiing-shen CHERN

Yano and I differed by about four months in age, he being the younger one. It must have been an accident that both of us got interested in the same area of mathematics, as a result of which our paths have crosscd repeatedly. We first met in Paris in the fall of 1936 when we both did the natural thing that a differential geometer would do, e.g., to be close to the great master Elie Cartan. When Cartan met his students Thursday afternoons, we were in the hallway outside his office in “Institut Henri PoincarC”. One could not fail to notice Yano’s capacity for hard work. Thc library of the “Institut” was at that time a big room walled by bookshelves with tables in the middle. Yano’s presence almost cqualcd that of the librarian. We were then both working on “projective connections”. He was writing his thesis and I wrote two little papers. Yano is a differential geometer in the best tradition of Ricci, Levi-Civita, and Schouten. H e is a great expert on tensor analysis. Here the fundamental notion is that of a vector bundle. For its analytical treatment one needs a field of bases (or frames). Tensor analysis is based on the principle of choosing the natural bases of a local coordinate system. The idea is both natural and simple. Throughout the years a standardized notation has been developed, which is understood by evcryonc in the field and which detects easily errors of computation. In general I believe the bases should not be tied up with local coordinates in order to allow the freedom, generality, and simplicity, as amply demonstrated by the method of moving frames. Unfortunately this has led to a proliferation of notations and one has to rely on tensor analysis for communication. The great service of tensor analysis deserves appreciation. I believe these “Selected Papers” will tell more about Yano’s mathematical works than anything I can say. These papers show ;I breadth and depth which can only be the result of a long span of activity. Perhaps his books: 1 ) (with S. Bochner) Curvature and Betti numbers; 2 ) The theory of Lie derivatives and its applications; 3 ) Integral formulas in Riemannian geometry; 4) (with M. Kon) Anti-invariant submanifolds; give a fairly good picture of the scope of his works. The first book is hi\ work best known outside of differential geometry. T h e second is a treatmcnt of transformation groups in generalized spaces whose study, in both the local

IX

and the global aspects, should have a promising futurc. Yano knows “cverything” in the litcrature of differcntial gcomctry. His knowledge of niathcmatics in general is extcnsivc, as denionstratcd by his many inathematical books and publications in Japancsc. He is clearly the model of a master who commands our great admiration.

X

Notes on My Mathernatical Works By Kentaro YANO

I was born on March 1st. 1912 in Tokyo. In 1922, when I was ten years old and in the fifth year class of primary school, the famous German thcoretical physicist Dr. Albert Einstein visited Japan and gave lectures o n the theory of relativity at Tbhoku University and the University of Tokyo. He also gave conferences for laymen at Sendai, Tokyo, Nagoya, Kyoto, Osaka, Hiroshima and Moji. The Japanese people were very curious about Einstein himself, who created a theory called the theory of relativity and used it to predict the so-called Einstein effect: “the path of the light from a star at a very far distance is bent to the side of the sun when it goes through the strong gravitational field made by the sun”. This prediction can be tested only when the solar eclipse occurs. They were also curious about the theory of relativity created by Einstein. The people believed the rumor that the theory of relativity was so difficult to understand that there were only twelve people in the world who knew the real meaning of the theory of relativity. But my father, who was a sculptor, encouraged me by saying “I do not know how difficult the theory of relativity is to understand, but it was not created by God, but was created by a human being named Albert Einstein. So, Kcntaro! I am sure that if you study hard you may understand some day what the theory of relativity is.” When I was a student in senior high school, I found, in the appendix of the textbook of physics, an introduction to the theory of relativity. I tried to read it very carefully and I thought that I could understand what the theory of relativity was. But my teacher of physics, Prof. T. Yamanouchi taught me that there were two theories of relativity: the theory of special relativity and the theory of general relativity, and the part that I could understand was just the first part of the theory of special relativity. He also taught me that to understand the theory of general relativity we must study the differential geometry, especially the Riemannian geometry and its generalizations. So, I immediately decided to go to the Department of Mathematics of the University of Tokyo to study differential geometry. In 1931, I was able to pass the entrance examination of the University of Tokyo. Early in the 1930’s, the most active school of differential geometry in Japan was that of Tbhoku University in Sendai. The Tbhoku school was studying the theory of ovals and ovaloids and also the theory of curves and surfaces in Euclidean, affine, projective and conformal spaces.

XI

Among my classmates, there were three, Mr. Hiroshi Kojinia, Mr. Toshio Seiiniya and myself, who wanted to study differential gcoinctry. Following the suggcstion of Mr. Kojinia, wc dccided to read together the famous book J. A. Schouten, Der Ricci-Kalkul, Springer Verlag, Berlin, 1924. I also began to read by myself other famous text books: H. Wcyl, Raum, Zeit, Materie, Springer, Berlin, 1921. L. P. Eisenhart, Riemannian geometry, Princcton Univcrsity Press, 1926. L. P. Eisenhart, Non-Riemannian geometry, Amer. Math. SOC.Colt. Publ., VIII (1927). T. Levi-Civita, The absolute differential calculus, Blackic and Son, London and Glasgow (1927). E. Cartan, Leqons sur la gkomktrie des espaces de Riemann, GauthierVillars, Paris, (1928). I graduated from the University of Tokyo in 1934. When I was a graduate student of thc Univcrsity of Tokyo, I read original papers of H. Weyl, those of L. P. Eisenhart, J. M. Thomas, T. Y. Thomas, 0. Vcblen and J. H. C . Whitehead of the Princeton school, those of E. Cartan and those of D. van Dantzig, J. Haantjcs and J . A. Schouten of the Dutch school. I liked most the ideas of E. Cartan and bcgan to look for a chance to go to Paris and to study this “ncw” diffcrcntial geometry undcr thc dircction of Elie Cartan. I found that the French Govcrniiient cvery year invited six Japanese students called “boursiers”, thrce in the field of literature and three in the field of science, and let them study in France for two ycars if they passed an examination in the French language. So I started to refresh my French, and two years later I could pass this examination. At that time, it took more than 30 days to go to Paris from Tokyo using boat and train transportation.

Projective connection I mct Professor EIic Cartan at “lnstitut Henri Poincari” in Paris in thc fall of 1936. Herc I met Professor S. S. Chern, who kindly wrote a very nice article for this “Selected Papers”. Profcssor Chern and I wcrc both interested in projective conncctions. I tried to find the geometrical nicanings of the old results in the projective theory of affine connections. For example, from the stand point of Elie Cartan, the so-called projective change of affine connections of Weyl t=;,

=

r:,+ a;p, + a:p,

can be intcrpreted as the change of the plane at infinity of the projective frame [A,, A , , - . . , A , , ] attached to each point A , , of the inanifold [12]. J. H. C. Whitehead was trying to define a projective parametcr on the path

XI1

~

d2x dsz

+ rh. d x j .~

ds

~

dx2 ds

=o,

s being the so-called affine parameter. But, attaching to each point A,, of the manifold a projective frame and defining a path as a curve satisfying

p being a function of the parameter t, we can see that t is actually a projective parameter and is defined by

{ t , s} =

~

1 n-1

R.ji

dxj

dxt

7-ds 3

where { t , s} is the Schwarzian derivative of t with respect to s and R , , the Ricci tensor of [18]. I was also interested in the ( n 1)-dimensional affinely connectcd manifold used to represent an n-dimensional projectively connected manifold. The Princeton school used an ( n I)-dimensional manifold described by ( x o , x ' , . , x " ) and with an affine connection rF2( K , 2, p, Y, . . . = 0, 1, 2, . . , n) satisfying

+

+

P

'I 1

= Cl,,

r;,= S; , r;<, : functions of

XI,

x?, . . . , x n only

,

( x ' , x 2 , . . . , x") being the coordinates of a point of the n-dimensional projcc-

tively connected manifold. The Dutch school used an ( n 1)-dimensional manifold dcscribcd by the so-called curvilinear homogeneous coordinates (x", s',x:, . . . , with an affine connection satisfying

+

XI<)

r;,

r f i= r;p, r ; , x 2 = 0 , r;,,: homogeneous functions of x2 of degree

I, (xo, x ' , . . . ,x " ) and (px", px', . . . , px"), p f 0 , representing the same point of the n-dimensional projectively connectcd manifold.

+

-

I found that these two schools used the same kind of ( 1 1 1)-dimensional affinely connected manifold, that is, both of them assumed that thc ( n 1)dimensional manifold admits a vector ficld tc and a symmctric afinc connection such that

+

where F1 denotes the operator of covariant differenation with respect to T;g2 and R,,,' the cuavature tensor of The set of equations above is equivalent to the set

r;<,.

XI11

VIEc = S; and V,VIEc + Rvp;EY= 0 , and the first equation means that the vector field CK is concurrent and the second means that the vector field f " defines an infinitesimal affine collineation [18]. I was also very much interested in the theory of the non holonomic spaces of G. Vranceanu and its application to the so-callcd unified field theory [8], [91, 1101.

Conformal geometry

I came back to Tokyo in 1938 and then tried to develop the theory of manifolds with conformal connection again using Cartan's method [26]. The theory of manifolds with normal conformal connection is cquivalent to the conformal geometry of Riemannian spaces. A circle in a Riemannian manifold is defined as a curve whose first curvature is constant and whose other curvatures arc all zero. The circle thus defined is called a geodesic circle of the Riemannian space. Under a conformal change of the Riemannian metric the circle thus defined does not change into a circle of the same kind. If we use Cartan's method, a circle in a space with a normal conformal connection is defined as a curve satisfying

p being a function of the parameter t and [A,, A,, . . . ,A , , , A,] being a conformal frame attached to each point A,, of the space. Under a conformal

change of the Riemannian metric the circle thus defined changes into a circle of the same kind. So this is called a conformal circle [20]. We can easily see that the parameter t is of projective character. A conformal transformation g J ,+ p2gjt of the Riemannian metric changes geodesic circles into geodesic circles if and only if the function p satisfies Vpp2 - p,.p2 = q%,2

7

p2 = a log p

P

9

where V , denotes the operator of covariant differentiation with respect to the Riemannian connection and 9 is a certain function. A conformal transformation g,, + p'g,, is called a concircular transformation if p satisfies the equation above, and the study of the properties of a Riemannian manifold which are invariant under concircular transformations is called concircular geometry [33], [351, [361, [371, [461. From the beginning, I had a very competent collaborator, Professor Yosio Muto. Using the theory of manifolds with normal conformal connection,

XIV

I could develop, with Professor Y. Muto, the theory of curves and submanifolds in the conformal geometry of Riemannian manifolds [34], [39], [40], [41l, [42l, [441, [451.

Lie derivatives Now during the second world war 1939-1945, I studied thc operator found by W. Slebodzinski in 1932 and named Lie derivation by D. van Dantzig and its applications to the study of groups of transformations in Riemannian manifolds and manifolds with affine, projective or conformal connection. We consider a geometric object field Q ( x ) and an infinitcsiinal point transformation

(“1

X h

= Xh

+ Xh(X)& ,

where X ” ( x ) is a vector field and E is an infinitesimal. The object has the value Q ( X ) at the deformed point (X”). We drag back this object from the deformed point (X”) to the original point ( x ” ) by a transformation inverse to (*I, obtaining ’Q(x) at the original point ( x h ) . Then

L,Q

=

lim [’Q(x) - Q(x)]/E 6-0

is called the Lie derivative of Q(x) with respect to X h . The second world war ended in 1945, and then I startcd to writc down in English the results on Lie derivatives and the groups of transformations which I got during the war and I could publish the book [I] in 1949.

Holonomy groups My colleague Prof. Shigeo Sasaki emphasized the importance of the study of holonomy groups of manifolds with various structures. I am vcry proud of having some joint papers with him on holonomy groups [58], [80], [86], r911.

Curvature and Betti numbers I knew that during the war Professors Oswald Veblen and John von Neumann of the Institute for Advanced Study were developing the theory of spinors in projective geometry and Professor Salomon Bochner of Princeton University had started the study of “Curvature and Betti numbers” using a theorem of Green, the lemma of E. Hopf and results of W. V. D. Hodge. So I wanted to go to Princeton and study the new method of Professor

xv

Bochner in order to develope the global differential geometry instead of the local one which I had been studying. Fortunately, Professor Veblen was kind enough to invite me to the Institute for Advanced Study as his assistant. Thanks to him, I could attend the International Congress of Mathematicians held in 1950 at Harvard University. He encouraged me to master the method of Professor Bochner. I worked very hard and read all the papers on this topic. The following theorem is well known: Theorem. In order for a vector field X ” to be harmonic in a compact orientable Riemannian manifold, that is, for the I-form cz associated with X” to be a harmonic form, it is necessary and sufficient that

gjiV,ViXh

-

K,”X‘ = 0 ,

where g” are the contravariant components of the metric tensor, V I the operator of covariant differentiation with respect to the Riemannian conncction and K L h the mixed components of the Ricci tensor. Corresponding to this I could prove the following: Theorem. In order for a vector field in a compact orientable Ricmannian manifold to be a Killing vector ficld, it is necessary and sufficient that X’l

gjiVjPiXh + K i k X i = 0 ,

ViXi = 0

.

Using this theorem I could prove [97] Theorem. In a compact orientable Riemannian manifold, an infinitesimal affine collineation is an isometry. This was the first result I could get at Princeton. After that I was able to writc some papers on “curvature and Betti numbers” [99], [loo]. I also had the help of Professor Deane Montgomery of the Institute for Advanced Study and could prove the following: Theorem. A necessary and sufficient condition that an n-dimensional Ricmannian space for n>4 and n#8 admits a group of motions of order t n ( n - I ) 1 is that the space is the product spacc of a straight line and an ( n - 1)-dimensional Riemannian spacc of constant curvature (this is equivalent to the fact that the space is conformally flat and admits a parallel vector field) or that the space is of negative constant curvature [ I O l ] . Later, in collaboration with Professor H . C. Wang I could generalize some of the results of this paper to results in an affinely connected manifield [ 1211. Professor Bochner suggested that I write a book on “Curvature and Betti numbers” gathering all the results on this topic known up to that time. I did my best to write it. H e was kind enough to write some very important and interesting supplements (we can call them papers) for this book [II]. This book is a very good souvenir of my stay in Princeton.

+

XVI

In order to write this book I studied complex tensor analysis and Kaehlcr geometry. I also enjoyed many nice conversations with Professor Albert Einstein who was developing the unified field theory based on a non-symmetric tensor field. Visiting Professor at Rome Now 1 met Professor Enrico Bonipiani who was at Pittsburgh University as a visiting professor. He invited me to attend the International Congress on Differential Geometry which would be held in Italy in 1953 and thcn to teach at the University of Ronic and at Istituto Nazionalc di Alta Mateniatica. I met at this Congress many distinguishcd European differential geometers: A. D. Alexandroff from Russia, W. Suss, E. Kaehler, W. Blaschke, H. Rund, R. Sauer, W. Klingenberg from Germany, P. Finsler, H. Hopf, B. Eckniann, H. Gugenheimer from Switzerland, W. V. D. Hodgc, E. T. Davics, A. G. Walker, T. J. Willniore from England, W. Fcnchcl from Denmark, J. A. Schouten, J. Haantjes, N. H. Kuiper from the Nctherlands, L. Godcaux from Belgium, P. Vinccnsini, C. Ehresmann, P. Liberniann. A. Lichncrowicz, G . Leeb from France and many ltalian differential geometers. After this Congress, I visited Paris, Durham, Leeds, Southampton, Amstcrdam, Leiden and Marseillcs and stayed at the University of Marseilles for a month as a visiting professor giving lectures on almost coniplcx structures. These lectures were published in [I 131. I came back to Rome in the fall of 1953 and gave lecturcs on groups of transformations in generalized spaces at Istituto Nazionalc di Alta Matematica. I gave niy lectures in French, but the note was translated into Italian and was published as [III]. During my stay in Rome, I collaborated with Y. Tashiro in Tokyo studying geometric objects [ 1041, with M. Ohgane in Tokyo studying six-dimensional unified field theory [105], with I. Mogi in Tokyo studying Kachlerian manifolds with constant holomorphic curvature [103], with E. T. Davies in Southampton studying connections in a Finsler space [ 1071 and thc so-called contact tensor calculus [ 1 101 and with S. Sasaki at Princeton studying pscudo-analytic vectors on pseudo-Kaehlerian manifolds [ 1191. Main theorems in [lo31 are the following: Theorem. If a real representation of a Kaehlerian manifold satisfies the axiom of holomorphic planes, then the manifold is of constant holomorphic curvature. Theorem. A necessary and sufficient condition that a real rcprcsentation of a Kaehlerian manifold admits a holomorphic free mobility is that the manifold is of constant holomorphic curvature. Theorem. In a Kaehlerian manifold of positive constant holomorphic curvature k>O, the distance between two consecutive conjugate points is constant

XVII

and is given by

2n/dk Visiting Professor at Amsterdam

I receivcd an invitation to study in Anistcrdani for 6 months after thc 1954 International Congress of Mathematicians which was held at Amsterdam i n September of 1954, as a visiting professor of the Mathematical Centre at Amsterdam and the University of Amsterdam. Thanks to this invitation I attended the Congress and gave a lccture [106]. Just before the Congress, thc second edition of Ricci-Calculus of Professor J. A. Schoutcn had been published and hc was showing a copy of this book to people. My first textbook on tensor analysis and its geometrical applications was the first cdition of this book published in 1924. I am very proud of the fact that many of papers 1 wrote since 1934 are quoted in the second edition of this book. This second edition contains sections on Hcrniitian manifolds and Kaehlerian manifolds, but docs not contain any section on almost complex manifold. Professor Schouten, thc director of the Mathematical Centre, showed great interest in a manifold with almost complex structure F," and in the tensor

N , , " = F,'d,F,"

-

F,'a,F," - (d,F,'

-

d,F,')F,'' ,

found by his pupil Albcrt Nijenhuis and which is now callcd the Nijenhuk tensor. If the almost complex manifold is of class 12"we could prove that in order for an almost complex manifold to be a complcx manifold, it is necessary and sufficient that thc Nijcnhuis tensor vanishes. For thc case i n which the manifold is of class C ' this fact was later provcd i n 1957 by A. Newlandcr and L. Nirenberg. I am very proud of having 4 joint papers with professor J. A. Shouten on almost complex manifolds and the Nijenhuis tensor [ 1 IS], [I 161. [ 1 1 71. [ I 181. I also collaborated with Professor N. H. Kuipcr who came to Amsterdam to attend the seminar held at the Mathematical Centre. Following his suggestion we discussed mathematics on the sight-seeing boat on the canal and could write two joint papcrs [109], 11221.

Summer Research Institute at Seattle

In 1956, the American Mathematical Society held thc Summer Research Institute on Differential Geometery at the Univcrsity of Washington. The organizers were Professors C. B. Allcndoerfer, H. Buscniann, S. S. Chern and H. Samelson. I could participate in this Summer Institute as the only

XVIII

one differential geometer from outside of the USA. I met here Professors W. Ambrosc, L. Auslander, E. Calabi, H. Flandcrs, T. T. Frankel, L. W. Green, R. Hermann, E. T. Kobayashi, S. Kobayashi, B. Kostant, H. E. Rauch, I . M. Singer, H. C. Wang and other American geoiiieters. In this Summer Rescarch Institutc 1 gave a series of lectures on integral formulas and thcir applications. The lectures werc published as [127]. Using a result in this paper, I gave a rather simple proof of the following theorem of Matsushima [ 1231. Theorem. In a compact Kachler-Einstein manifold whose curvature is positive, a contravariant analytic vector ficld 11 can be written in the forni I I = v F w in one and only one way where v and w are both Killing vector fields and F the complex structure tensor of the manifold.

+

Visiting Professor at Southampton The University of Southampton invited me to be a visiting profcssor for three months in 1958. Thanks to this invitation, 1 attended the 1958 International Congress of Mathematicians held at Edinburgh. 1 also collaborated with Professor E. T. Davics in the study of the so-callcd fibred spaces [ 1341. At Southampton I also studied harmonic and Killing vcctor fields in Ricniannian spaces with boundary [126], [ 1281, [ I 311. I n collaboration with T. Takahashi, I gcneralized these results t o harmonic and Killing tensor fields [126], [129]. In 1959, in collaboration with Professor T. Nagano, I proved the rather important following theorem [ 1 301 : Theorem. Let M be a connected completc Einstein space of dimensions n > 2 and of class C ’ and suppose t h a t a vcctor field on M generates globally a onc-parameter group of non-honiothctic conformal transformations. Then M is isometric to a simply connected space of positive constant curvature. In particular M is homomorphic to the sphere S”.

Visiting Professor at Hong Kong In 1960, Professor Y . C. Wong invitcd mc to be a visiting profcssor to the University of Hong Kong for 6 months. For the students I gave lectures on tensor calculus and its applications to Rkmannian geometry. For the staff I gave a lecture on the differential geometry of complex and almost complex spaces. This lecture was later published by Pergamon Press [ V ] in 1965. In collaboration with Professor Y. C. Wong, I studied projectively flat spaces for which the covariant derivative of the curvature tensor is proportional to the curvature tensor itself [142]. I also collaborated by correspondence with Professor M. S. Kncbelnian of Washington State University in order to generalize sonic of results on homo-

XIX

thetic mappings of Riemannian spaces obtained by Professor Knebelman [ 1411. By correspondence I also collaborated with Professor Richard Blum of the University of Saskatchewan in the study of the imbedding of a Riemannian space in a conformally Euclidean space [144] and with Miss M. Ako of the Tokyo Institute of Technology in the study of almost analytic vectors in almost complex spaces [145].

Symposium at Zurich Now, in June of 1960, an International Symposium on Differential Geometry and Topology was held in Zurich. I attended this symposium directly from Japan and gave two lectures. One was on an almost Hermitian manifold which satifies

V,F,'"

+ V z F j h= 0 ,

where V j is the operator of covariant differentiation with respect to the Hermitian metric and F t h is the almost complex structure tensor. I called such a manifold an almost Tachibana manifold, but now some people call this kind of manifold a nearly Kaehler manifold. The other lecture was on what T. Nagano and I call a geodesic vector field [143].

Visiting Professor at Seattle In 1961, Professor C. B. Allendoerfer of the University of Washington, Seattle, was kind enough to invite me to be a visiting professor for three months, April, May and June. Professor A. Nijenhuis was there also and I had no duty to teach, so I could really enjoy the stay at Seattle collaborating with Professors Allendoerfer and Nijenhuis. Let F be an almost complex structure tensor. Then it satisfies

I being the unit tensor, and thus it satisfies F ~ + F = o . On the other hand, let (p, f , 7) be a set of a tensor field p of type (1, l), a vector field f and a 1-form v defining an almost contact stracture. Then we have p*x=

-x + ? ( X ) f ,

pt = 0 ,

v(pX) = 0

, v(f)

= 1

for any vector field X , Thus we can see that the tensor field p satisfies

xx

'p3+y=O.

Thus to study the almost complcx structure and the almost contact structure simultaneously we have only to study a structure defined by a non-zero tensor field f of type ( I ,1) satisfying

f"+f=O. Such a structure is now called an f-structure. I started the study of an fstructure at Seattle [139], [148].

Visiting Professor at Southampton and Liverpool In 1962, Professor E. T. Davies of the University of Southampton invited me to be a visiting professor for a month at Southampton and Professor A. G . Walker of the University of Liverpool invited me to stay for two months at Liverpool. Thanks to these invitations, 1 could attend with my wife the 1962 International Congress of Mathematicians held at Stockholm. Since I was very much interested in the tangent bundle at that time, 1 studied, with Professor Davies, tangent bundles of Finder and Riemannian manifolds [I471 and. with Dr. A. J. Ledger of the University of Liverpool. linear connections on tangent bundles [ 1491. At the Tokyo Institute of Technology, 1 had a very competent colleague Professor Shigeru Ishihara. During my stay at Liverpool, I collaborated with Professor Ishihara by correspondence and studied the integrability conditions of the f-structure [150].

Visiting Professor of Brown University In 1963, Professor Katsumi Nomizu of Brown University invited me to be a visiting professor for six months. I really enjoyed the stay at Brown University and collaboration with Professor Nomizu [ 15 11, [ 1531, [ 1571, [170]. An infinitesimal transformation X on a Riemannian manifold M is said to be strongly curvature-preserving if L,y(L"lLK) =0 ,

172

=

0, 1,2, . . . ,

where L , denotes Lie derivation with respect to X and V°K = K , K being the curvature tensor of M . Professor Nomizu and I proved [I511 the following theorem : Theorem. Let M be an irreducible analytic Riemannian manifold of dimension IZ 2 2. Then a strongly curvature-preserving infinitesimal transformation is necessarily homothetic. If furthermore, M is complete, then X is Killing.

XXI

Visiting Professor at Berkeley

In 1965, Professor S. S. Chern of the University of California, Berkeley invited me to be a visiting professor at his university for three months. 1 had no duty, so I could spend all of my tinic to the collaboration with m y former student Professor S. Kobayashi. I bclieve that my joint paper [167] with Kobayashi laid down thc foundation of the thcory of prolongations of tensor fields and connections to tangent bundles. In Bcrkelcy I also studicd thc conjecturc : A compact Riemannian manifold with constant scalar curvature admitting a one-parameter group of conformal transforinations which is not that of iconietrics is isometric to a sphcrc. I could prove [I631 the following: Theorem. If M is compact, oricntablc. of dimension I I > 2, K = const., and which is not an isomctry : admits an infinitesimal confornial transformation 9’’ L,,g,, = 2pg,,, pfconst., such that

Jar

G,,p’p‘dV is nonnegative, dV being

the volume element of M , then M is isonlctric to a sphere, where

K , , and K being respectively the Ricci tensor and scalar curvature of M and pz the gradient of p.

Professor Chern was kind cnough to present this paper to the National Academy of Science. Since then 1 tried vcry hard, in collaboration with S. Sawaki [184], 11961, [207], S. I. Goldbcrg [204], M. Obata 12061 and H. Hiraniatsu 12571, [284], [286] to prove this conjecturc, but I always needed sonic additional conditions. Quite recently N. Ejiri showed that thc conjecture itself is not truc ( N . Ejiri, A negative answer to a conjecturc of conformal transformations of Riemannian manifolds, J . Math. SOC.Japan, 33 (1981), 261-266).

Fibred spaces

In studying the five-dimensional Riemannian manifold used by Kaluza and Klein to establish a unified theory of gravitation and elcctrornagnctisin and also the ( n 1)-dimensional affinely connected manifold used to represent an n-dimensional projectively connected manifold, I got thc idea of fibred space. A fibred space is dcfincd in thc following way. Let M and M be differentiable manifolds of dinicnsion n 1 and 17 rcspectively. We assume that and M are both of class C and we suppose that there exists a differentiable mapping r r : M , which is onto, of maximum rank everywhere, and of class C ” . Then, for each point P of M ,

+

+

XXII

the inverse image n - ' ( f ) is a one-dimensional submanifold of M . Denoting n - ' ( f >by F,, we call F,, the fibre over a point P of M . We assume that the fibre F , = r - ' ( P ) is connected for any point 1' of M . Now we consider in M a vector field C, which is nonzero everywhere and tangent to the fibres, and a 1-form 7 satisfying

?(C) = 1 and Lg

=

0 (niodg) ,

where L denotes the operator of Lie derivation with respect to C, The set ( M , M , r , 7 ) satisfying the conditions above is called a fibred space. We call M and M the total space and the base space respectively. C and 7 are called the structure vector field and the structure I-form rcspectively. The mapping is called the projection mapping of the fibred space. I n collaboration with Professor Ishihara. or, I would say, by the help of Professor Ishihara, 1 could dcvelope the theory of fibrcd spaces "621, [172], [173], [179]. In 1965, The United States-Japan Seminar in Differential Geonietry was held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, during the week of June 14-19. I could meet not only my old American friends, Professors K. Bott, E. Calabi, S. S. Chern, J. Eells, S. Helgason, B. Kostant, H. E. Rauch, I . M. Singer, H. C. Wang, J. H. Sanipson, but also m y old Japanese friends staying in United States, S Kobayashi and K . Noniizu. My coauthor Professor Noniizu gave the lecture [ 1.571 and Professor Ishihara the lecture [158].

c,

Cotangent bundles In 1966, Professor E. M. Patterson of the University of Aberdeen, Scotland invited me to his univcrsity as a visiting professor for three months, May, June and July. Thanks to this invitation, I could attend the International Congress of Mathematicians held at Moscow in August on the way back to Japan. Since I was, at that time, very much interested i n tangent bundles, I wrotc a paper on tensor fields and connections on cross-sections i n the tangent bundles of a differentiable manifold [ 1771. Professor Patterson was kind enough to present this paper to the Royal Society of Edinburgh. I already had some results on tangent bundles. But since I did not havc any results on cotangent bundles, 1 collaborated with Professor E. M. Patterson to study vertical, complete and horizontal lifts from a manifold to its cotangent bundle [1741, [1751. I also wrote a paper on tensor fields and connections on cross-sections in the cotangent bundle [1781.

XXlll

G . A. Miller Visiting Professor In 1968, Profcssor S. I. Goldberg of the University of Illinois invited me to his university as G. A. Miller Visiting Professor. Since Professor Coldberg was interested in the so-called f-manifolds and in Riemannian manifolds admitting an infinitesimal conformal transformation, we collaborated in the study of thcsc problems and wrote the joint papers [1971, [1981, [1991, [203], 12041. Professor Morio Obata who was then at Lehigh University visited me at the University of Illinois and stayed a few days in Urbana. We wrote the joint paper [206]. When I was at the Institute for Advanced Study, I used Green’s theorem to study “Curvature and Betti numbers”. Since then I was very much interested in the integral formulas applied to various problems [ 1231, [ 1261, [1271, [1311, [1431, 11631, [1841, [1871, [1901, [1921, [1961, [2021, “2051, [207], [208]. During my stay at the University of Illinois, 1 could complete the manuscript of my book “Integral formulas in Riemannian geometry”, [VI] which was published by Marcel Dekker Inc., New York, in 1970.

Canadian Summer Research Institute

In 1969, Professor Tanjiro Okubo of McGill University invited me to bc a visiting professor of the Canadian Suiiimcr Research Institute, at Queen’s University in Kingston, With Professor Okubo I studied tangent bundles of generalized spaces of paths [2251. Professors D. E. Blair and G. D. Ludden of Michigan Statc University at East Lansing came to Kingston to participate in this Canadian Summer Research Institute. Professors Blair, Ludden and I studied induced structurcs on submanifolds of codimension 2 of almost complex nianifolds [209]. We found that the induced structure consists of a tensor field f of type ( I , 1 ), two vector fields U , V , two 1-forms u , v and a scalar I satisfying f’=-I+ uof

u@U+v@V,

=Iv,

v o f

ju= - I V ,

= -lu,

fV=IU

u(U) = 1 - I’ , u(V) = v(U) = 0 , v ( V ) = 1 - I’ .

We call such a structure (f, U , V , u,v,2)-structure. If the ambient space is an almost Herniitian manifold, we have moreover

XXIV

We call such a structure an (f, g, I I , v,])-structure. Later I studied, in collaboration with Professor M. Okumura, ( j , g , 21, v,])-structures in detail, [200], [21 I]. If we consider structures induced on a hypersurface of an odddimensional sphere, we get similar structures [217].

Mathematische Forschungsinstitut, Oberwolfach Now, in the summer of 1969, an International Symposium on Differcntial Geometry in the large was to be held at Mathematische Forschungsinstitut, Oberwolfach. We flew first from Canada to Paris and enjoyed two weeks of vacation in Paris. During this period we flew from Paris to Southampton and attended the party to commemorate the retirement of Professor E. T. Davies from the University of Southampton at the age of sixty five. Professor Davies told us that after his retirement from the University of Southampton he would go to Canada and would teach at a Canadian university and he would invite us to that Canadian university. From Paris, we went to Oberwolfach and could meet many European and American differential geonicters including Professors S. S. Chern, E. Calabi, S. Helgason, I. M. Singer, S. Kobayashi. When I had met Professor Chern in 1956 at the Summer Research Institute held at Seattle, Professor Chern said to me “Yano! We are the oldest here”, and this time he said to me “Yano! We are still the oldest here”. In Oberwolfach, 1 gave a lecture on conformal transformations in Riemannian manifolds [218].

Visiting Professor of Michigan State University

I n Scpteinber of 1970, the International Congress of Mathematicians was held at Nice. I got an invitation from Professor T. J. Willmore of the University of Durham to spend a fcw days at his university and an invitation from Professors D. E. Blair and G. D. Luddcn of Michigan State University to spend two months with thcm on the way to Tokyo from Nice. My colleague Professor S. Ishihara also got an invitation from Professor Willmore to stay at the University of Durham for two months. So we came back from Nice to Paris, and from Paris we went to Newcastle through

xxv

London. We met at the Newcastle Airport Professor M. Obata who also would spend two months at the University of Durham. After having enjoyed a few days stay at the University of Durham, we left Newcastle Airport on September 17, and passing through London and Chicago we arrived at Lansing Airport and met Professors Blair and Ludden and Dr. Bang-yen Chen. I collaborated with Professors Blair and Ludden to study “our” structure that we had found in Kingston and we wrote papers [219] on the intrinsic geometry of S” x S” and [247] on differential geometric structures on principal toroidal bundles. Of course I could collaborate also with Dr. Bang-yen Chen to study mainly conformal properties of submanifolds of Riemannian manifolds and wrote the joint papers [213], [220], 12211, [223], 12321, [2351, [2401, [2411, [2421, 12431, [244], [249]. Professor C. S . Houh of Wayne State University in Detroit was a very faithful mcniber of the seminar at Michigan State University, and so I could also collaborate with him and could write joint papers with Houh and Chen [236] on structures defined by a tensor field of type ( 1 , l ) satisfying ‘p4 k ‘p2 = 0 and [248] on quasi-umbilical submanifolds of codiniension 2 in a space form. As I was enjoying the collaborations with these geometers, I got a telegram from my university, Tokyo Institute of Technology, asking me to come back immediately since I was elected Dean of the Faculty of Science. So, I came back to Tokyo and accepted this position. I had already made a contract with the University of Hong Kong to go there and teach for three months. But I found that the Dean should not be absent for so long time, so I was forced to cancel this appointment with the University of Hong Kong. Thus I decided to go to Hong Kong by my own expense to explain the situation and beg their pardon for cancelling the appointment.

Retirement from Tokyo Institute of Technology

In 1972, I retired from Tokyo Institute of Technology at the age of 60. For this occasion, my colleague S. Ishihara and my former students, S. Kobayashi, I. Mogi, T. Nagano, M. Obata, T. Takahashi and Y. Tashiro edited “Differential geometry in honor of Kentaro Yano” Kinokuniya BookStore Co. LTD., Tokyo, Japan, 1972. I am very proud of the fact that many world famous differential geometers, R. L. Bishop, D. E. Blair, R. B. Brown, B. Y. Chen, S. S. Chern, E. T. Davies, P. M. D. Furness, S . I. Goldberg, P. J. Graham, A. Gray, C. C . Hsiung, A. J. Ledger, P. Libermann, A. Lichnerowicz, G. D. Ludden, S. S. Mittra, A. Nijenhuis, H. E. Rauch, T. J. Willmore, Y. C. Wong and my

XXVI

Japanese colleagues contributed their papers to this volume. In 1972, Colloquium Mathematicum wanted to celebrate the fortieth anniversary of the discovery of the Lie dcrivativcs by Professor W. Slebodzinski. So I dedicated the paper [239] on isometries to Professor W. Slebodzinski on this anniversary. In the same year, I collaborated with Professor S. Ishihara to study local fibering and transversal hypersurfaces of contact metric manifolds and dedicated the paper [238] to Professor A. Kawaguchi on his seventieth birthday. Early in the 1970’s, Professor S. Ishihara started the study of manifolds with almost quarternion structures. In 1972, Miss M. Ako and I studied the integrability conditions for almost quarternion structures and dedicated the paper [233] to Prof. Y. Katsurada on her sixtieth birthday.

Collaboration with Professors Blair, Chen and Ludden In 1972, Michigan State University invited me again to be a visiting professor for two months, May and June, so I could again collaborate with Professors Blair, Ludden and Chen. With Professors Blair and Ludden, I studied differential geometry of complex manifolds similar to the Calabi-Eckmann manifolds [252]. With Professor Chen, I studied mainly conformally flat submanifolds and wrote joint papers [240], [241], [242], [243], and [244]. We dedicated [244] on special conformally flat spaces and canal hypersurfaces to Professor S. Sasaki on his sixtieth birthday. I also dedicated the joint paper [245] with S. Ishihara and M. Konishi and the joint paper [246] with Y. Muto on almost cotangent structures to Professor S. Sasaki.

Visiting Professor at Waterloo In 1972, Professor E. T. Davies of the University of Waterloo invited me to his University for two months, October and November, Suppose that therc is given, in a 2n-dimensional differentiable manifold M , a tensor field Q of type (1,l) such that Q’ = 0 and that the rank of Q is n everywhere. Such a tensor field Q defines an almost tangent structure on M which is then called an almost tangent manifold. In Waterloo, Professor Davies and I collaborated to study differential geometry of almost tangent manifolds and finished the paper [260] in Waterloo. We dedicated the paper to Professor B. Segre on his seventieth birthday. Professor Davies also invited Dr. F. Brickell of the University of Southampton to Waterloo. So I could collaborate with Dr. Brickell. Dr. Brickell showed his interest in the idea of concurrent vectors (See [47]) and so we

XXVII

wrote a joint paper on concurrent vector fields and Minkowski structures [255].

Weyl and Bochner curvature tensors In 1973, the 20th Summer Research Institute of American Mathematical Society was held at Stanford University from July 30 to August 17. I went to Palo Alto directly from Tokyo and enjoyed a very nice three week period with my American friends. There I gave a lecture on manifolds and submanifolds with vanishing Weyl or Bochner curvature tensor [261]. Let M" be a Riemannian manifold with the metric tensor g j t and denote by K k j t h ,K j t and K the curvature tensor, the Ricci tensor and the scalar curvature respectively. Then the Weyl curvature tensor W k j t his defined to be Wkjth

= KkjCh

+

6:cjf

- s"jkf

+ c,hgjt

- Cjhgkl

Y

where n-2

1 2(n - l)(n - 2)

Kgjt

9

C,h

=

cktgfh

a

On the other hand, let M" be a real n-dimensional almost Hermitian manifold with almost complex structure tensor Fth and with almost Hermitian tensor g , t . Denote by K k j P , K f t and K the similar tensors as above. Then the Bochner curvature tensor is defined to be

where

Symposium in Korea In 1973, the first Symposium on differential geometry in Korea was held from September 14th to 18th in Seoul and Kyunpook Universities. I was invited to this symposium and gave two lectures, one on Bochner curvature tensor and the other on various structures introduced on a differentiable manifold.

XXVIII

100th birthday of Levi-Civita An international congress to celebrate the 100th anniversary of the birth of T. Levi-Civita (1873-1941) was held at the Accademia Nazionale dei Lincei in Rome in 1973 on December 17-19, and Professor E. T. Davies and I were invited to this congress. Professor Davies was asked to give a lecture entitled “The influence of Levi-Civita’s notion of parallelism on differential geometry” and I was asked just to attend the ceremony. So, when I was leaving Waterloo, 1 said to Professor Davies “See you in Rome”. But unfortunately, Professor Davies died suddenly by heart attack on October 8th, 1973. So Accademia Nazionale dei Lincei asked me to give a lecture having the same title as that of Professor Davies in his place. The University of Waterloo sent me a very rough draft of Professor Davies’ lecture, but it was so difficult to read his hand writing and I had less than two months to prepare it. I finally decided to write down my own manuscript of this lecture. I went everyday to the Department of Mathematics of the University of Tokyo and worked very hard to write down the manuscript of this Iecture. This was published as [265]. I hope very much that I said what Professor Davies wanted to say. In Rome, I met many European geometers, E. Bompiani, H. Freudenthal, L. Godeaux, E. Kahler, A. Lichnerowicz, B. Segre, G. Vranceanu, A. G. Walker and others. Professor Walker said to me “Yano, you are a child here”.

Circles and extrinsic spheres In 1973, I collaborated with Professor K. Nomizu of Brown University who was at Bonn and studied the circles and extrinsic sphers in Riemannian geometry [254]. We proved, for example, Theorem. Let M ” , n >= 2, be a connected submanifold of a Rimannian manifold M”‘. If, for some r > 0, every circle of radius r in M “ is a circle in M”’, then M ” is an extrinsic sphere in M ” ‘ . Conversely, if M ” is an extrinsic sphere in MI”, then every circle in M “ is a circle in M”‘.Here, an extrinsic sphere means a subminifold which is umbilical and has parallel mean curvature vector. In 1974, the International Congress of Mathematicians was hcld in the campus of British Columbia University at Vancouver from August 2 1st to 29th. I attended the Congress and met many of my friends from Europe, the United States and Canada.

XXIX

Professor Ludden in Tokyo In 1974, Professor G , D. Ludden of the Michigan State University came to Tokyo and spent April and May at Tokyo Institute of Technology to collaborate with me and with my staff. At that time we are very much interested in the so-called anti-invariant (totally real) submanifolds of a Kaehlerian or Sasakian manifolds, we concentrated in the study of this topic and could write the joint papers [256], [264], [280] on totally real submanifolds. I also collaborated with people at Michigan State University by corrcspondence and wrote the joint paper [268] on what we call nearly Sasakian structures and [271] on what we call semi-invariant immersion. Bochner curvature tensor In 1975, I collaborated with Professor S. Sawaki of Niigata University and could prove [262] the following: Theorem. Let M?’be a p-dimensional totally umbilical and totally real submanifold of an n-dimensional Kaehlerian manifold whose Bochner curvature tensor vanishes, (4 5 p < n , 8 5 n ) , then M p is conformally flat. This is a generalization of a theorem of Blair. I collaborated with Professor B. Y. Chen and could prove [266] thc following: Theorem. In order that a Riemannian manifold of dimension n > 3 is conformally flat, it is necessary and sufficient that there exists a (unique) quadratic form Q on the manifold such that the sectional curvature K(o) with respect to a section o is the trace of the restriction of Q to u , i.e. K(o) = trace Q/a, the metric being also restricted to o. Theorem. In order that the Bochner curvature tensor of a Kaehlerian manifold vanishes, it is necessary and sufficient that there exists a (unique) hybrid quadratic form Q such that the sectional curvature K ( a ) with respect to a holomorphic section u is the trace of the restriction of Q to o, i.e. K(u) = trace Q/o, the metric being also restricted to o.

Complex conformal connections In order to find a geometrical meaning of the Bochner curvature tensor, I considered what I call a complex comformal connection [258]. First of all, we prove Theorem. In a Kaehler manifold with Hermitian metric tensor g,, and complex structure tensor F,” , there exists an affine connection which satisfies DkeZPglc= 0

,

xxx

DkeZPFjt =0

where p is a scalar function and

where

We call such an affine connection a complex conformal connection. Then we can prove Theorem. If, in an n-dimensional Kaehlerian manifold (11 2 4), there exists a scalar function p such that the complex conformal connection i n the above theorem is of zero curvature, then the Bochner curvature tensor of the manifold vanishes. As to the Bochner curvature tensor, I have a joint paper [283] with Professor L. Vanhecke. I tried to obtain results in Sasakian manifolds similar to thosc mentioned above and wrote (2781, where the so-called contact Bochner curvature tensor appears. In May 1976, the Korean Mathematical Society celebrated its 30th anniversary and I received an invitation from Professor E. Pak, the prcsident of the Society, to come to Seoul, to attend the ceremony and to give lectures. To celebrate its 30th anniversary in 1976 means that the Korean people established the Korean Mathematical Society immediately after the end of the Second World War. I gave three lectures in Seoul, “Infinitesimal variations of submanifolds” for specialists, “Complex structure and its generalisations” and “Recent topics in differential geometry” for non-spccialists.

Anti-invariant submanifolds To study anti-invariant submanifolds of a Kaehlerian manifold and of a Sasakian manifold, I collaborated with Professors Blair, Ludden and Okumura, but also with Professor M. Kon, who is now associate professor of Hirosaki University, [267], [2721, [2731, [2741, [2751, [2771, 12791, [2801, [2901. Gathering all these results, Professor M. Kon and I published the lecture note [VIII].

XXXI

Infinitesimal variations In 1977, Professors U-Hang Ki and Jim Suk Pak visited Tokyo Institute of Technology, and since I was rather interested in the infinitesimal variations of submanifolds [293], [303], [306], I studied with them infinitesimal variations of submanifolds and the generalizations of what I call an (f, g, u, v, 2)structure [291], [292], [294], [295]. CR submanifolds The 1978 International Congress of Mathematicians was held at Helsinki from August 15 to 23. I attended this Congress flying to Helsinki directly from Tokyo, and could see again my European and American friends. I met there Professor Aurel Bejancu of Polytechnical Institute of Jassy, Roumania, who gave a lecture entitled “CR submanifolds of a Kaehler manifold”. Let be an almost Hermitian manifold with almost complex structure tensor J. A submanifold M of @ is called a CR submanifold of @ if there exists a distribution D : x + D, on M satisfying the following two conditions: ( 1 ) D is invariant, that is, JD, = D, for each x E M . ( 2 ) The complementary orthogonal distribution D,: : x -+ D+ c T , ( M ) is anti-invariant, that is, J D i c T , ( M ) l for each ’xE M, T , ( M ) and T , ( M ) I being tangent space and normal space of M at x respectively. A similar definition applies to submanifolds of a Sasakian manifold. Since this is a very nice generalization of anti-invariant submanifold and since the theory of the so-called f-structure [ 1391 plays rather important r d e in the theory of CR submanifolds, Professor Kon and 1 are now concentrating on the study of CR submanifolds in Kaehlerian manifolds and Sasakian manifolds [299], [304], [305], C3071, 13091, [3101, [3111, [3121.

Chern Symposium

In 1979, a symposium on differential geometry was held at the University of California, Berkeley to commemorate the retirement at 68 of Professor S. S. Chern from his university on June 25 through 30. 1 flew from Tokyo to San Francisco and attended this Chern Symposium. Professor Chern’s reputation attracted more than three hundred differential geometers from all over the world. A participant of this Symposium was saying that he had never seen such a good set of differential geometers. I met again my old friends, Professors M. F. Atiyah, F. Hirzebruch, N. H. Kuiper, E. Calabi, L. Nirenberg, R. Osserman, I. M. Singer, Chen-Ning Yang, L. Auslander, W. Boothby, J. Eells, T. Frankel, E. Pitcher, H. Samelson,

XXXII

A. H. Taub, Y. C. Wong, C. S. Houh, C. J. Hsu, A. Gray, W. F. Pohl, F. Warner, L. Vanhecke, S. T. Yau, M. P. do Carmo, R. Bott, S. Helgason, H. H. Wu, J. A. Wolf, etc. In June 1980, I was able to attend the meeting of Southcast Asian Mathcmatical Society and met again Professor S S. Chern who gave a nice lecture on web geometry. I also met Dr. K. P. Mok of Hong Kong Polytcchic who was intcrestcd in the theory of tangent and cotangent bundles. In August 1980 I visited again the University of California, Berkclcy, and attended International Congress on Mathematical Education and could see Professors H. Freudenthal and L. K. Hua.

Semi-symmetric metric connections

In [201], I studied the semi-symmetric metric connections in a Riemannian manifold. If an affine connection satisfies

r:, sJ; = r;,- rtJ= q p , - a:p,

,

pz being a covector, then the connection is said to be semi-symmetric, and

if the operator 8, of covariant differentiation with respect to

r:, satisfies

g,, being a Riemannian metric tensor, it is said to be metric.

A semi-symmetric metric connection

rl;iis given by

{:%} being the Christoffel symbols formed with gr, and p h = p , g z h . I could prove the following: Theorem. In order for an n-dimensional Riemannian manifold ( n 2 3) to be conformally flat, it is necessary and sufficient that the manifold admits a semi-symmetric metric connection whose curvature tensor vanishes identically. If the torsion tensor of the affine connection TFLsatisfies

for a certain tensor T," of type ( 1 , l ) and a covector p i , the connection is said to be quarter-symmetric. It seems to me that some people are now interested in the quarter-symmetric metric connection [3 131, [3 141.

XXXIII

Kaehlerian manifold admitting a holomorphically projective vector Once I studied very hard the Riemannian manifolds with constant scalar curvatures admitting a conformal Killing vector field. But now I am studying the Kaehlerian manifolds with constant scalar curvature that adinit a nonaffine holomorphically projective vector field. A vector field X ” is called a holomorphically projective vector field if it satisfies

+

L , { : t } = F,F,X” X’(K,,,’” == 6FpL 6:lp, - plF,‘Fth - p ( F , ‘ F j h ,

+

where { ! L } are Christoffel symbols of a Kaehlerian manifold, V j the operator of covariant differentiation with cspect to {!,}, Kl,,‘” the curvature tensor, p , a covcctor and F,” the complex structure tensor, L , denoting the Lie derivative with respect to X / L . Professor H. Hiramatu and 1 could prove recently [308]: Theorem. If a complex n( > 1 ) dirnensional compact connected and siniply connected Kaehlerian manifold M with constant scalar curvature K admits a non-affine holomorphically projective vector field X h , then M is isometric to a complex projective space CP“ with Fubini-Study metric and of constant holomorphic sectional curvature K / n ( n 1).

+

Infinitesimal variations of submanifold Once I studied Killing, affine and conformal vector fields [II]. But presently I am studying infinitesimal isometric, affine and conformal variations of submanifolds of a Riemannian manifold and getting results similar to those in [II] [303].

XXXIV

Bibliography of the Publications of Kentaro Yano Books and Monographs [I] Groups o f transformations in generalized spaces, Akademeia Press Company LTD., Tokyo, Japan (1949). [11] (with S. Bochner) Crrrvatur-e and Betti Numbers, Annals of Mathematics Studies, Number 32, Princeton University Press, (1953). [III] Gruppi di trasformazioni in spazi geometrici differenziali, Istituto Matematico, Roma, 1953-1954 (mimeographed), [IV] The theory o f Lie derivatives and its applications, North-Holland Publishing Co., Amsterdam ( 1957). [V] Differential geometry on complex and almost complex spaces, Pergamon Press, London (1965). [VI] Integral formulas in Riemannian geometry, Marcel Dekker Inc., New York (1970). [VII] (with S. Ishihara) Tangent and cotangent bundles, differential geometry, Marcel Dekker Inc., New York (1973). [VIII] (with M. Kon) Anti-invariand submanifolds, Marcel Dekker Inc., New York (1976). Papers 1934 [ 1 ]

On the linear displacements in the generalized manifold, Proc. Physico-Math. Soc. Japan, 16 ( 1 934), 3 18-326.

1935 [ 2 ] On the theory of linear connections in the manifold admitting homogeneous contact transformations, Proc. Physico-Math. Soc. Japan, 17 (1935), 39-47. [ 3 ] On the metric space K , , Proc. Physico-Math. Soc. Japan, 17 (19351, 163-1 69. [ 4 ] (with Y. Muto) On the connections in X , associated with the points of Y , , , Proc. Physico-Math. Soc. Japan, 17 (1935), 379-390.

[ 5 ]

1936 (with Y. Muto) On the connections in X , associated with the points

xxxv

of Y,,,, 11, Proc. Physico-Math. Soc. Japan, 18 (1936), 1-9. (with Y. Muto) Notes on the deviations of geodesics and the fundamental scalar in Riemannian space, Proc. Physico-Math. Soc. Japan, 18 (1936), 142-15’2. [ 7 1 Sous-espace d’un espace admettant le parallklisme absolu, T o k y o Butsuri-Gakko Zassi, 535 (1936), 1-10.

1937 L a thtorie unitaire des champs proposie par M. Vranceanu, C. R. Acad. Sci. Paris, 204 (1937), 332-334. Sur la thCorie unitaire non holonome des champs, I. Proc. PhysicoMath. SOC. Japan, 19 (1937), 867-896. Sur la thdorie unitaire non holonome des champs, 11. Proc. PhysicoMath. Soc. Japan, 19 (1937), 945-976. Sur les espaces non holononics totalement gkodbsiques, C. R . Acad. Sci. Paris, 205 (1937), 9-11. Sur le changement des codfficients d’une connexion projective, C. R . Acad. Sci. Paris, 205 (1937), 637-639. Sur les Cquations des gCodCsiques dans une variCtC d connexion projective, C . R. Acad. Sci. Paris, 205 (1937) 829-831. (with Y . Muto) Sur la thiorie des spineurs, Proc. Physico-Math. SOC. Japan, 19 (1937), 413-435. 1938 L a relativitk non holonome et la thdorie unitaire &Einstein et Maycr, Mcrthematica, 14 (1938), 124-132. Remarqucs relatives h la thkorie dcs espaces i connexion conforme, C . R. Acad. Sci. Paris, 206 (1938), 560-562. Sur l’espace projectif de M. D . van Dantzig, C. R . Acad. Sci. Paris, 206 (1938), 1610-1612. Les espaces :i connexion projective et la gLomdtrie projective des “paths”, Annales Scientifiques de 1’Universitt de J u s ~ y ,24 (1938), 395-464. (Thesis, University of Paris). Sur la nouvclle thiorie unitaire de MM. Einstein et Bergmann, Proc. It?z/). Acad. T o k y o , 14 (1938), 325-328. Sur les circonfdrences gbnkralisCes dans les cspaces i connexion conforme, Proc. I t n p . Acucl. Tokyo, 14 (1938), 329-332. (with Y. Muto) Sur la dktcrmination d’une connexion conforme, Proc. Physico-Math. Soc. Jupun, 20 ( 1 9 3 8 ) , 267-279. The non holonomic representation of projective spaces, Proc. PhysicoMath. Soc. Japan, 20 (1938), 442-450. 1939 Sur la connexion de Weyl-Hlavatp et la gComCtrie conforme, Proc.

XXXVI

I m p . Acad. Tokyo, 15 (1939), 116-120. Sur les Cquations d e Gauss dans la gComCtrie conforme des espaces d e Riemann, Proc. Imp. Acad. Tokyo, 15 (1939), 247-252. Sur les Cquations de Codazzi dans la giomktrie conforme des espaces d e Riemann, Proc. I m p . Acad. Tokyo, 15 (1939), 340-344. Sur la thCorie des espaces A connexion conforme, J . Fac. Sci., I m p . Univ. Tokyo, 4 (1939), 1-59. (Thesis, University of Tokyo). (with Y. Muto) Sur les transformations de contact et les espaces d e Finder, TShoku Math. J . , 45 (1939), 295-307. (with Y. Muto) A projective treatment of a conformally connected manifold, Proc. Physico-Math. SOC. Japan, 21 ( I939), 270-286.

1940 Sur la connexion d e Weyl-Hlavat? et ses applications h la gComktrie [ 29 1 conforme, Proc. Physico-Math. SOC.Japan, 22 (1940), 595-621. [ 3 0 1 (with St. Petrescu) Sur les espaces mitriques non-holononies complCmentaires, Disqiiisitiones Mathernaticae et Physicae, 1 ( 1940), 191-246. Conformally separable quadratic differential forms, Proc. I m p . Acacl. Tokyo, 16 (1940), 83-86. Sur quelques proprittks conformes de I/, dans V7,,dans I/,,, Proc. I m p . Acad. Tokyo, 16 (1940), 173-177. Concircular geometry I, Concircular transformations, P roc. I m p . Acad. Tokyo, 16 (1940), 195-200. (with Y. Muto) Sur la thkorie des hypersurfaces dans un cspace ii connexion conforme, Proc. I m p . Acad. Tokyo, 16 (1940), 266-273. Concircular geometry 11, Integrability conditions of ,o,~”= $g,,”, Proc. Imp. Acad. Tokyo, 16 (1940), 354-360. Concircular geometry 111, Theory of curves, Proc. I n l p . Acad. Tokyo, 16 (1940), 442-448. Concircular geometry IV, Theory of subspaces, Proc. I m p . Acatl. Tokyo, 16 (1940), 505-511. [ 38 1 [ 39

1

1941 (with Y. Muto) Sur la thiorie des hypersurfaces dans un espace ii connexion conforme, J a p . J . Math., 17 (1941 ), 229-288. (with Y. Muto) Sur la thkorie des espaces zi connexion conforme normale et la giomitrie conforme des espaces d e Riemann, Proc. I m p . Acad. Tokyo, 17 (1941), 87-94. (with Y. Muto) On conformal arc length, Proc. I m p . Acad. Tokyo, 17 (1941), 318-322. (with Y. Muto) On the generalized loxodroiues in the conformally connected manifold, Proc. I m p . Acacl. Tokyo, 17 ( 1941 ), 455-460. (with Y. Muto) Sur la thCorie des espaces connexion conforme

XXXVII

normale et la gComCtrie conforme des espaces de Riemann, J . Fac. Sci., Imp. Univ. Tokyo, 4 (1941), 117-169. 1942 [ 43 1 Les espaces d’ClCments linkaires h connexion projective normale et la gCometrie projective gCnCrale des paths, Proc. Physico-Math. SOC. Japan, 24 (1942), 9-25. [ 44 1 (with Y. Muto) Sur le thkorime fondamental dans la gkomitrie conforme des sous-espaces riemanniens, Proc. Physico-Math. Soc. Japan, 24 (1942), 437-449. [ 45 1 (with Y. Muto) On the curves developable on two-dimensional spheres in the conformally connected manifold, Proc. Imp. Acad. Tokyo, 18 (1942), 222-226. [ 46 1 Concircular geometry V. Einstein spaces, Proc. Imp. Acad. Tokyo, 18 (1942), 446-451. [ 47 1

r481

r 49 1 [ 50

1

1943 Sur le parallClisme et la concourance dans I’espace de Riemann, Proc. Imp. Acad. Tokyo, 19 (1943), 189-197. Sur les Cquations fondamentales dans la gkomCtrie conforme des sousespaces, Proc. Imp. Acad. Tokyo, 19 (1943), 326-334. Sur une application du tenseur conforme C,, et du scalaire conforme C., Proc. Imp. Acad. Tokyo, 19 (1943), 335-340. Conformal and concircular geometries in Einstein spaces, Proc. Imp. Acad. Tokyo, 17 (1943), 444-453.

1944 Projective parameters in projective and conformal geometries, Proc. Imp. Acad. Tokyo, 20 (1944), 45-53. (with T. Adati) Parallel tangent deformation, concircular transformation and concurrent vector field, Proc. Imp. Acad. Tokyo, 20 (1944), 123-127. l-531 Projective parameters on paths in D. van Dantzig’s projective space, Proc. Imp. Acad. Tokyo, 20 (1944), 210-215. r 54 1 Uber eine geometrische Deutung der projektiven Transformationen nicht-symmetrischer affiner Ubertragungen, Proc. Imp. Acad. Tokyo, 20 (1944), 284-287. 55 1 On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20 (1944), 340-345. [ 5 6 1 (with K. Takano) Sur les coniques dans les espaces Q connexion affine ou projective I, Proc. Imp. Acad. Tokyo, 20 (1944), 410-417. r 5 7 1 (with K. Takano) Sur les coniques dans les espaces B connexion affine ou projective ll., Proc. Imp. Acad. Tokyo, 20 (1944), 418424.

XXXVIII

(with S. Sasaki) Sur les espaces B connexion conforme normale dont les groupes d’holonomie fixent une sphiire B un nombre quelconque de dimensions, I., Proc. I m p . Acad. Tokyo, 20 (1944), 525-535. [ 59 1 Sur les espaces B connexion affine qui peuvent reprtsenter les espaces projectifs des paths, Proc. I m p . Acad. Tokyo, 20 (1944), 631-639. [ 60 1 Subprojective transformations, subprojective spaces and subprojective collineations, Proc. Imp. Acad. Tokyo, 20 (1944), 701 -705. 1945 Sur les espaces B connexion affine qui peuvent representer les espaces projectifs des paths 11, Proc. I m p . Acad. Tokyo, 2 1 (1945), 16-24. Sur les espaces B connexion affine qui peuvent reprtsenter les espaces projectifs des paths 111, Proc. Imp. Acad. Tokyo, 2 1 (1945), 97-103. Sur la thtorie des espaces 2 hyperconnexion euclidienne I, Proc. I m p . Acad. Tokyo, 21 (1945), 156-163. Sur la thtorie des espaces B hyperconnexion euclidienne 11, Proc. Imp. Acad. Tokyo, 2 1 (1945), 164-170. Bemerkungen uber infinitesimale Deformationen eines Raumes, Proc. Imp. Acad. Tokyo, 2 1 (1945), 171-178. (with K. Takano) Conics in D. van Dantzig’s projective space, Proc. Imp. Acad. Tokyo, 2 1 (1945), 179-187. Sur la dtformation infinitesimale des sous-espaces dans un espace affine, Proc. Imp. Acad. Tokyo, 2 1 (1945), 248-260. Sur la deformation infinittsimale tangentielle d’un sous-espace, Proc. Imp. Acad. Tokyo, 2 1 (1945), 261-268. Quelques remarques sur un article de M. N. Coburn intitule “A characterization of Schouten’s and Hayden’s deformation methods’,, Proc. Imp. Acad. Tokyo, 2 1 (1945), 330-336. Lie derivatives in general space of paths, Proc. Imp. Acad. Tokyo, 2 1 (1945), 363-371. On the fundamental differential equations of flat projective geometry, Proc. Imp. Acad. Tokyo, 2 1 (1945), 392-400. On the flat conformal differential geometry I, Proc. I m p . Acad. Tokyo, 2 1 (1945), 419-429. On the flat conformal differential geometry 11, Proc. Imp. Acad. Tokyo, 2 1 (1945), 454-465. 1946 On the flat conformal differential geometry 111, Proc. Japan Acad., [741 22 (1946), 9-19. I 7 5 1 On the flat conformal differential geometry IV, Proc. Japan Acad., 22 (1946), 20-31. [ 76 1 Quelques remarques sur les groupes de transformations dans les espaces A connexion linCaire I, Proc. Japan Acad., 22 (1946), 41-47.

XXXIX

(with K. Takano) Quelques remarques sur les groupes de transformations dans les espaces 2I connexion IinCaire 11, Proc. Japan Acad., 22 (1946), 69-74. Quelques remarques sur les groupes de transformations dans les espaces 21 connexion IinCaire 111, Proc. Japan Acad., 22 (1946), 167-172. (with Y. Tomonaga) Quelques remarques sur les groupes de transformations dans les espaces 2I connexion IinCaire IV, Proc. Japan Acad., 22 (1946), 173-183. (with S. Sasaki) Sur les espaces ii connexion conforme normale dont les groupes d’holonomie fixent une sphkre 2I un nombre quelconque de dimensions 11, Proc. Japan Acad., 22 (1946), 225-232. (with Y. Tomonaga) Quelques remarques sur les groupes de transformations dans les espaces A connexion IinCaire V , Proc. Japan Acad., 22 (1946), 275-283. (with K. Takano and Y. Tomonaga) O n the infinitesimal deformations of curves in the space with linear connection, Proc. Japan Acad., 22 (1946), 294-309. (with Y. Muto) Note sur le thCorhme fondamental dans la gComCtrie conforme des sous-espaces riemanniens, Proc. Japan A cad., 22 (1946), 338-342. [ 84 1 [ 85

1

1947 On the flat projective differential geometry, l a p . J . Math., 19 (1947), 385-440. Quelques remarques sur les groupes de transformations dans les espaces A connexion IinCaire VI, Proc. Japan Acad., 23 (1947), 143-146.

1948 (with S. Sasaki) Sur la structure des espaces de Riemann dont le r 861 groupe d’holonomie fixe un plan 2I un nombre quelconque de dimensions, Proc. Japan Acad., 24 (1948), 7-13. Union curves and subpaths, Math. Japonicae, 1 (1948), 1-9. (with K. Takano and Y. Tomonaga) On infinitesimal deformations of curves in the space with linear connection, Jap. J . Math., 19 (1948), 433-477. 1949 Sur la thCorie des dCformations infinitCsimales, J . Fac. Sci. Univ. Tokyo, 6 (1949), 1-75. (with T. Adati) On certain spaces admitting concircular transformations, Proc. Japan Acad., 25 (1949), 188-195. (with S. Sasaki) On the structure of spaces with normal projective

XL

connexions whose groups of holonomy fix a hyperquadric, or a quadric of (N-2)-dimensions, Tbhoku Math. J . , 1 (1949), 31-39. 1950 Note on the conformal theory of curves, Tensor, N . S., 1 (1950), 6-13. (with T. Iniai) On affine collinations in projectively related spaccs, J . Math. SOC.Japan, 1 (1950), 287-288. 1951 (with H. Hiramatu) Affine and projective gcomctries of systcm of hypersurfaces, J . Math. Soc. Japan, 3 ( 1951), 116-1 36. On groups of homothetic transformations in Riemannian spaces, J . Indian Math. Soc., 15 (1951), 105-117. 1952 (with H. Hiramatu) On the projective geometry of K-sprcads, Coinpositio Math., 10 (19S2), 286-296. On harmonic and Killing vector fields, Ann. of Math., 55 (1952), 3 8-45. (with M. Ohgane) On unified field theories, Ann. of Math., 55 (1952), 318-327. Some remarks on tensor fields and curvature, Ann. of Math., 55 (1 952) 328-347. (with S. Bochner) Tensor-fields in non-symmetric connections, Ann. of Math., 56 (1952), 504-519. 1953 On n-dimensional Riemannian spaces admitting a group of motions of order + n ( n - 1 ) 1, Trans. Amer. Math. Soc., 74 (1953), 260279. On Killing vector fields in a Kaehlerian space, J . Math. SOC. Japan, 5 (1953), 6-12. 1031 (with I. Mogi) Sur les variCtCs pseudokahlkriennes il courbure holomorphique constante, C . R . Acad. Sci. Paris, 237 (1953), 962-964.

+

1954 1041 (with Y. Tashiro) Some theorems on geometric objects and their applications, Nieuw Arch. Wisk., 2 (1954), 134-142. 1051 (with M. Ohgane) On six-dimensional unified field theories, Renrliconti di Matematica e delle sue applicazioni, 13 (1954), 99-132. 1061 On pseudo-Hermitian and pseudo-Kahlerian manifolds, Proc. International Congress of Mathematicians, 1954, Amsterdam, Volume I l l , 190-1 97.

XLI

(with E. T. Davies) O n the connection in Finder space as an induced connection, Rend. Circ. Mat. Palerrno, 3 (1954), 409-417. Sur la correspondence projective entre deux cspaces pseudo-hermitiens, C . R. Acad. Sci. Paris, 239 (1954), 1346-1348. (with N. H. Kuiper) On geonictric objects and Lie groups of transformations, Indag. Math., 17 (1954), 41 1-420. (with E. T. Davies) Contact tensor calculus, A n n . Mat. Piira A p p l . , 37 (1954), 1-36. Geometria conforme in varietsi quasi-hermitiane, Rendiconti dell’ Accademia Nazionale dei Lincei, 16 (1954), 449-454. (with H. Hiramatu) On groups of projective collineations in a space of K-spreads, J . Math. Soc. Japan, 6 (1954), 131-150. 1955 Quelques remarques sur les variCtCs ii structure presque complexe, Bull. Soc. Math. France, 83 (1955), 57-80. On three remarquable affine connexions in almost Hermitian spaces, Indag. Math., 17 (1955), 24-32. (with J. A. Schouten) On an intrinsic connexion in an X,, with an almost Hermitian structure, Indag. Math., 17 (1955), 1-9. (with J. A. Schouten) On the geometrical meaning of the vanishing of the Nijenhuis tcnsor in an X,,, with an almost complex structure, Indag. Math., 17 (1955), 133-138. (with J. A. Schouten) On invariant subspaccs in an almost complex X,,, Indag. Math. 17 (1955), 261-269. (with J. A. Schouten) On pseudo-Khhlerian spaces admitting a continuous group of motions, Indag. Marh., 17 (1955), 565-570. (with S. Sasaki) Pseudo-analytic vectors on pseudo-Kahlerian manifolds, Pacific J . Math., 61 (1955), 987-993. (with I. Mogi) On real representations of Kaehlerian manifolds, A n n . of Math., 61 (1955), 170-189. (with H. C. Wang) A class of affinely connected spaces, Trans. Amer. Math. Soc., 80 (1955), 72-92. 1956 (with N. H. Kuiper) Two algebraic theorems with applications, Indag. Math., 18 (1956), 319-328. 1957 Sur un thCorhme de M. Matsushima, Nagoya Math. J . , 12 (1957), 147-150. (with T. Nagano) Some theorems on projective and conformal transformations, Indag. Math., 19 (1957), 451-458.

XLII

1958

On Walker differentiation in almost product or almost complex spaces, Indag. Math., 20 (1958), 573-580. Harmonic and Killing tensor fields in Riemannian spaces with boundary, J . Math. SOC. Japan, 10 (1958), 430-437. Some integral formulas and their applications, Michigan Muth. J . , 5 ( 1 9 9 9 , 63-73. Sur les vecteurs harmoniques et vecteurs de Killing dans un espace de Riemann B frontikre, C . R. Acad. Sci. Paris, 247 (1958), 10851087. 1959 (with T. Takahashi) Some remarks on harmonic and Killing tensor fields in a Riemannian space with boundary, The Golden Jubilce Commemoration Volume ( 1958-1959), Calcutta Math. Soc., 439446. (with T. Nagano) Einstein spaces admitting a one-paranicter group of conformal transformations, Ann. of Math., 69 ( 1959). 45 1-46 I . Harmonic and Killing vector fields in compact orientable Riemannian spaces with boundary, Ann. of Math., 69 (1959), 588-597. (with T. Nagano) The de Rham decomposition, isometries and affine transformations in a Riemannian space, Jap. J . Math., 29 (1959), 173-184. Afine connexions in an almost product space, Kbdai Mail?. Sem. Rep., 11 (1959), 1-24. (with E. T. Davies) On some local properties of fibred spaces, K d a i Math. Sem. Rep., 11 (1959), 158-177. 1960

Conformal transformations in Riemannian and Hermitian spaces, Bull. Amer. Math. SOC., 66 (1960), 369-372. (with T. Takahashi) Some remarks on Einstein spaces and spaces of constant curvature, J . Math. Soc. Japan, 12 (1960), 89-96. Champs de vecteurs dans un espace riemannien ou hermitien, C. R. Acad. Sci. Paris, 251 (1960), 194-195. 1961 (with T. Okubo) Fibred spaces and non-linear connections, Ann. Mat. Pura Appl. 55 (1961), 203-243. O n a structure f satisfying f3 f = 0 , Technical Report. No. 12, June 20, 1961. Department of Mathematics, University of Washington. (with T. Nagano) Les champs des vecteurs gCod6siques sur les espaces symitriques, C . R. Acad. Sci. Paris, 252 (1961), 504-505.

+

XLIII

(with M. S. Knebelman) On homothetic mappings of Riemann spaces, Proc. Amer. Math. SOC., 12 (1961), 300-303. (with Y. C . Wongj Projectively flat spaces with recurrent curvature, Comm. Mail?. Helv., 35 ( 1961 ), 223-232. (with T. Nagano) On geodesic vector fields in a compact orientable Riemannian space, Comm. Math. Helv., 35 (1961), 55-64. (with R. Blum) On imbedding of a Riemannian space in a conformally Euclidean space, K d a i Math. Sem. Rep., 13 (1961), 53-64. (with M. Ako) Almost analytic vectors in almost complex spaces, T6hoku Math. J . , 13 (1961), 24-45. 1962 Eckmann-Frolicher connexions on almost analytic submanifolds, Kiidai Math. Sem. Rep., 14 (1962), 53-58. 1963 (with E. T. Davies) On the tangent bundle of Finder and Riemannian manifolds, Rend. Circ. Mat. Palerrno, 12 (1963), 21 1-228. On a structure defined by a tensor field f of type (1,l) satisfying f3 f = 0, Tensor, N . S., 14 (1963),, 99-109.

+

1964 (with A. J. Ledger) Linear connections on tangent bundles, J . London Math. SOC., 39 (1964), 495-500. (with S. Ishihara) O n integrability conditions of a structurc f satisfying f:’ f = 0, Quart. J . Mathemalics, 15 (1964), 217-222. (with K. Noniizu) O n infinitesimal transformations prcscrving thc curvature tensor field and its covariant differentials, A n n . tnst. Fourier, 14 (1964), 227-236. (with T. Sumitomo) Differential geometry of hypersurfaccs in a Cayley space, Proc. Roy. SOC. Edinburgh, 66 (1964), 216-231. (with K. Nomizu) Une d h o n s t r a t i o n simple d’un thtorhne sur Ic groupe d’holonomie affine d’un espace de Riemann, C . R . Acad. Sci. Paris, 258 (1964), 5334-5335. (with S. Ishihara) On hyperconnections, Bull. Calcutta Math. Soc., 56 (1964), 109-133.

+

1965 (with A. J. Ledger) The tangent bundle of a locally symmetric space, J . London Math. SOC., 40 (1965), 487-492. (with M. Obata) Sur le groupe de transformations conformes d’une variCtC de Riemann dont le scalaire de courbure est constant, C . R . Acud. Sci. Paris, 260 (1965), 2698-2700. (with K. Nomizu) Some results related to the equivalence problem

XLIV

in Riemannian geometry, Proc. United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, 95-100. (with S. Ishihara) Structure defined by f satisfying f:’ f = 0, Proc. United Slates-Japan Seminar in DifJerential Geometry, Kyoto, Japun, 1965, 153-166. Closed hypersurfaces with constant mean curvature in a Riemannian manifold, J . Math. Soc. Japan, 17 (1965), 333-340. (with S. Ishihara) Almost contact structures induced on hypersurfaces in complex and almost complex spaces, Kiirfai Math. Sem. Rep. 17 (1965), 222-249. (with M. Ako) Vector fields in Riemannian and Hermitian manifolds with boundary, Kiidai M a f h . Sem. Rep., 17 (1965), 129-157.

+

1966 (with S. Ishihara) Fibred spaces and projectable tensor fields, Perspectives in Geometry and Relativity (Essays in honor of Valav Hlavatf), Indiana University Press, 1966, 468-48 1. On Riemannian manifolds with constant scalar curvature admitting a conformal transformation group, Proc. Nat. Acad. Sci. U 3 . A., 55 (1966), 472-476. (with Y. Muto) Homogeneous contact structures, Math. A n n . , 167 (1966), 195-213. (with S. Ishihara) The f-structure induced on submanifolds of complex and almost complex spaccs, K d a i Math. Sem. Rep., 18 (1966), 120-1 60. (with S. Ishihara) Differential geometry in tangent bundle, Kiirfui Math. Sen?. Rep., 18 (1966), 271-292. (with S. Kobayashi) Prolongations of tensor fields and connections to tangent bundles, I, General theory, J . Math. SOC.Jupan, 18 ( 1966), 194-2 10. (with S. Kobayashi) Prolongations of tensor fields and connections to tangent bundles, 11. Infinitesimal automorphisms, J . Mafh. SOC. Japan, 18 (1966), 236-246. 1967 (with S. Kobayashi) Prolongations of tensor fields and connections to tangent bundles, 111. Holonomy groups, J . Math. SOC. Japan, 19 (1967), 486-488. (with K. Nomizu) Some results related to the equivalence problem in Riemannian geometry, M a f h . Z . , 97 (1967), 29-37. (with S. Ishihara) Almost complex structures induced in tangent bundles, Kiidai Math. Sem. Rep., 19 (1967), 1-27. (with S . Ishihara) Differential geometry of fibred spaces, K d n i Math. Sem. Rep., 19 (1967), 257-288.

XLV

(with S. Ishihara) Fibred spaces with invariant Riemannian metric, Kbdai Math. Sem. Rep., 19 (1967), 317-360. (with E. M. Patterson) Vertical and complete lifts from a manifold to its cotangent bundle, J . Math. SOC. Japan, 19 (1967), 91-113. (with E. M. Patterson) Horizontal lifts from a manifold to its cotangent bundle, 1. Math. SOC. Japan, 19 (1967), 185-198. Notes on hypersurfaces in a Riemannian manifold, Canad. J . Math., 19 (1967), 439-446. Tensor fields and connections on cross-sections in the tangent bundles of a differentiable manifold, Proc. Roy. SOC. Edinburgh, Section A, 67 (1967), 277-288. Tensor fields and connections on cross-sections in the cotangent bundle, Tbhoku Math. J . , 19 (1967), 32-48. (with S. Ishihara) Fibred spaces with projectable Riemannian metric, J . Differential Geometry, 1 (1967), 71-88. (with S. Ishihara) Horizontal lifts of tensor fields and connections to tangent bundles, J . Math. Mech., 16 (1967), 1015-1030. (with A. J. Ledger) Almost complex structures on tensor bundles, J . Differential Geometry, 1 (1967), 355-368. (with M. Ako) Notes on covariant analytic vector fields, Tbhokli Math. J . , 19 (1967), 187-197.

1968 (with S. Ishihara) Normal circle bundles of complex hypersurfaces, K d a i Math. Sem. Rep., 20 (1968), 29-53. (with S. Sawaki) Riemannian manifolds admitting a conformal transformation group, J . Diflerential Geometry, 2 (1968), 161-184. (with S. Ishihara) Differential geometry of tangent bundles of order 2, Kodai Math. Sem. Rep., 20 (1968), 318-354. (with M. Ako) On certain operators associated with tensor fields, Kodai Math. Sem. Rep., 20 (1968), 414-436. 1969 (with M. Okumura) Integral formulas for submanifolds of codimension 2 and their applications, Kodai Math. Sem. Rep., 21 (1969), 463-47 1. (with Y. Mutd) Homogeneous contact manifolds and almost Finder manifolds, Kodai Math. Sem. Rep., 21 (1969), 16-45. (with S. Ishihara) On a problem of Nomizu-Smyth on a normal contact Riemannian manifold, J . Diflerential Geometry, 3 (1969), 45-58. (with M. Tani) Integral formulas for closed hypersurfaces, Kiidai Math. Sem. Rep., 21 (1969), 335-349. (with S . Ishihara) Pseudo-umbilical submanifolds of codimension 2, Kodai Math. Sern. Rep., 21 (1969), 365-382.

XLVI

Riemannian manifolds admitting a conformal transformation group, Proc. Nut. Acad. Sci. U.S.A., 62 (1969), 314-319. (with S. Ishihara) Invariant submanifolds of an almost contact manifold, K d a i Math. Sem. Rep., 21 (1969), 350-364. Notes on submanifolds in a Riemannian manifold, KcSdai Math. Sem. Rep., 21 (1969), 496-509. Generalizations of the connections of Tzitzeica, K d a i Math. Sem. Rep., 21 (1969), 167-174.

1970 (with S. Sawaki) Riemannian manifolds admitting an infinitesimal conformal transformation, K d a i Math. Sen?. Rep., 22 ( 1970), 272-300. (with S. I. Goldberg) Noninvariant hypersurfaces of almost contact manifolds, J . Math. Soc. Japan, 22 (1970), 25-34. (with S. I. Goldberg) Polynomial structures on manifolds, K d n i Math. Sern. Rep., 22 (1970), 199-218. (with S. I. Goldberg) On normal globally framed f-manifolds, Thhokir Math. J . , 22 (1970), 362-370. (with M . Okumura) O n (f, g , i t , v , ])-structures, K d a i Math. Sern. Rep., 22 (1970), 401-423. On semi-symmetric metric connection, Revue Roumuine dc Math&rnatiques Piires et Appliqudcs, 15 (1970), 1579-1586. Integral formulas for submanifolds and their applications, Canad. J . Math., 22 (1970), 376-388. (with S. I. Goldberg) Affine hypersurfaces of complex spaces, J . London Math. Soc., ( 2 ) 2 (1970). 241-250. (with S. I. Goldberg) Manifolds admitting a non-homothetic conformal transformation, Duke Math. J . , 37 (1970), 655-670. (with M. Tani) Submanifolds of codiinension 2 in a Euclidean space, K6dai Math. S e m . Rep., 22 (1970), 65-76. (with M. Obata) Conformal changes of Riemannian metrics, J . Diflerential Geornetry, 4 (1970), 53-72. (with S. Sawaki) Notes on conformal changes of Riemannian manifolds, K d a i Math. Sern. Rep., 22 (1970), 480-500. O n Riemannian manifolds admitting an infinitesimal conformal transformation, Math. Z . , 113 (1970), 205-214. (with D. E. Blair and G. D. Ludden) Induced structures on submanifolds, Kbdai Math. Sem. Rep., 22 (1970), 188-198. 1971 Submanifolds with parallel mean curvature vector of a Euclidean space or a sphere, KcSdai Math. Sem. Rep., 23 (1971), 144-159. (with M. Okumura) On normal ( j , g, 11, v, ])-structures on submani-

XLVII

folds of codimension 2 in an even-dimensional Euclidean space, Kodai Math. Sem. Rep., 23 (1971), 172-197. (with M. Konishi) Notes on almost isometries, Kodai Math. Sem. Re/)., 23 (1971), 238-247. (with Bang-yen Chen) On the concurrent vector fields of immersed submanifolds, Kodai Math. Sen?. Rep., 23 (1971 ), 343-350. (with D. E. Blair) Affine almost contact manifolds and f-manifolds with affine Killing structure tensors, Kodai Math. Sem. Rep., 23 ( 197 1) , 473-479. (with E. T. Davies) Metrics and connections in the tangent bundle, Kodai Math. Sem. Rep., 23 (1971), 493-504. (with S. Ishihara) Submanifolds with parallel mean curvature vector, J . Diflerential Geometry, 6 (1971), 95-1 18. On (f, g, u, v , A)-structures induced on a hypersurface of an odddimensional sphere, T f h o k u Matli. J . , 23 (1971), 671-679. Conformal transformations in Riemannian manifolds, Differentialgeonietrie im Grossen, Math. Forschungsinstitut, Oberwolfach, Bd., 4 (1971), 339-351. (with D. E. Blair and G. D. Ludden) On the intrinsic geometry of S" x S", Math. Ann. 194 (1971), 68-77. (with Bang-yen Chen) Minimal submanifolds of a higher dimensional sphere, Tensor, N . S., 22 (1971), 370-373. (with Bang-yen Chen) On submanifolds of submanifolds of a Ricmannian manifold, J . Math. SOC.Japan, 23 (1971), 548-554. (with S. I. Goldberg) Globally framed f-manifolds, Zllinois J . Math., 15 (1971), 457-474. (with Bang-yen Chen) Integral formulas for submanifolds and their applications, J . Diflerential Geometry, 5 (1971 ) , 467-477. (with M. Okumura) Invariant hypersurfaccs of a manifold with ( f , g, 1 1 , v, A)-structure, Kodai M a f h . Sem. Rep., 23 (1971), 290-304. (with T. Okubo) On the tangent bundles of generalized spaces of paths, Rend. Mat. Roma, ( V l ) 4 (1971), 327-347. 1972

(with M. Okumura) Invariant submanifolds of manifold with ( f , g, u , v, A)-structure, K6dai Math. Sevn. Rep., 24 (1972), 75-90. (with U-Hang Ki) On quasi-normal ( f , g, u , v, A)-structures, K d a i Math. Sem. Rep., 24 (1972), 106-120. (with U-Hang Ki) Submanifolds of codimension 2 in an evendimensional Euclidean space, K d a i Math. Sem. Rep., 24 (1972), 3 15-330. 12291 (with S. Ishihara) Notes on hypersurfaces of an odd-dimensional sphere, K6dai Math. Sem. Rep., 24 (1972), 422-429. 12301 (with S. S. Eum and U-H. Ki) On transversal hypersurfaces of an

XLVIII

almost contact manifold, Kcdai Math. Sem. Rep., 24 (1972), 459470. (with S. Ishihara) Submanifolds of codimension 2 or 3 with parallel second fundamental tensor, J . Korean Math. Soc., 9 (1972), 1-1 1. (with Bang-yen Chen) Sous-variktis localemcnt conforines h un espace euclidien, C . R . Acad. Sci. Paris, 275 (1972), 123-125. (with M. Ako) Integrability conditions for almost quarternion structures, Hokkaido Math. J . , 1 (1972), 63-86. (with M. Ako) On hypersurfaces of an odd-dimensional sphere as fibred spaces, Diflerential Geonietry in honor of Kentaro Yano, Kinokuniya, Tokyo, 1972, 1-19. (with Bang-yen Chen) Pseudo-umbilical submanifolds in a Riemannian manifold of constant curvature, DifJerential Geometry in honor of Kentaro Yano, Kinoknniya, Tokyo, 1972, 61-7 1. (with C. S. Houh and B. Y. Chen) Structures defined by a tensor field 9 of type ( I , 1 ) satisfying p4 _t (p" = 0, Tensor, N . S., 23 (1972), 81-87. On a special f-structure with complemented frames, Tensor, N . S., 23 (1972), 35-39. (with S. Ishihara) On local fibering and transversed hypersurfaces of contact metric manifolds, Tensor, N . S., 25 (1972), 197-205. Notes on isometries, Colloq. Mafh., 26 (1972), 1-7. (with Bang-yen Chen) Special quasi-umbilical hypersurfaces and locus of spheres, Atti della Accademia Nazionale dei Lincei, 53 (1972), 255-260. 1973 (with Bang-yen Chen) Submanifolds umbilical with respect to a quasi-parallel normal direction, Tensor, N . S., 27 (1973), 41-44. (with Bang-yen Chen) Conformally flat spaces of codimensions 2 in a Euclidean space, Canarl. J . of Math., 15 (1973), 1170-1173. (with Bang-yen Chen) Some results on conformally flat submanifolds, Tanzking J . Math., 4 (1973), 167-174. (with Bang-yen Chen) Special conformally flat spaces and canal hypersurfaces, TAhoku Math. J . , 25 (1973). 177-184. (with S. Ishihara and M. Konishi) Normality of almost contact 3structure, TBlzokzi Math. J . , 25 (1973), 167-175. (with Y. Muto) On alniost cotangent structures, Tbhokii Math. J . , 25 (1973), 111-127. (with D. E. Blair and G . D. Ludden) Differential geometric structures on principal toroidal bundles, Trans. Amer. Math. Soc., 181 (1973), 175-184. (with C. S. Houh and B. Y. Chen) On quasi-umbilical submanifolds of codimension 2 in a space form, Biill. Inst. Math., Academia Sinica,

XLIX

"2491

1 (1973), 125-132. (with Bang-yen Chen) Pseudo-umbilical submanifolds of codimension 3 with constant mean curvature, K d a i Math. Sem. Rep., 25 (1973), 490-501. (with Bang-yen Chen) Umbilical submanifolds with respect to nonparallel normal direction, J . Differential Geometry, 8 (1973), 589597. (with S. S. Eum and U-H. Ki) O n almost contact affine 3-structures, Kodai Math. Sem. Rep., 25 (1973), 120-142.

1974 (with D. E. Blair and G . D. Ludden) Geometry of complex manifolds similar to the Calabi-Eckmann manifolds, J . Differential Geomefry, 9 (1974), 263-274. (with S. Ishihara) Kaehlerian manifolds with constant scalar curvature whose Bochner tensor vanishes, Hokkaido Math. J . , 3 (1974), 297304. (with K. Nomizu) On circles and spheres in Riemannian geometry, Math. Ann., 210 (1974), 163-170. (with F. Brickell) Concurrent vector fields and Minkowski structures, Kodai Math. Sem. Rep., 26 (1974), 22-28. 1975 (with G . D. Ludden and M. Okumura) Totally real submanifolds of complex manifolds, Atti della Accademia Nazionale dei Lincei, 58 ( 1 9 7 9 , 346-353. (with H. Hiramatu) Riemannian manifolds admitting an infinitesimal conformal transformation, J . Diflerential Ceomelry, 10 (1975), 23-38. On complex conformal connections, Kodai Math. Sem. Rep., 26 ( 1 9 7 3 , 137-151. (with T. Imai) O n semi-symmetric metric F-connections, Tensor, N . S., 29 (1975), 134-138. (with E. T. Davies) Differential geometry on almost tangent manifolds, Ann. Mat. Pura Appl., 103 (1975), 131-160. Manifolds and submanifolds with vanishing Weyl or Bochner curvature tensor, Proc. Symposia in Pure Mathematics, 27 (1975), 253262. (with S. Sawaki) Sur le tenseur de courbure de Weyl et celui de Bochner, C. R. Acad. Sci. Paris, 281 (1975), 293-295. (with S. lshihara) Harmonic and relatively f i n e mappings, J . Difflerential Geometry, 10 (1975), 501-509. (with G . D. Ludden and M. Okumura) A totally real surface in CP2 that is not totally geodesic, Proc. Amer. Math. SOC., 53 (1975),

L

186-190. 12651 (with E. T. Davies) The influence of Levi-Civita’s notion of parallelism on differential geometry, Tullio Levi-Civita, Convegno Internazionale Celebrativo del Centenario della Nascita, (Roma, 17-19 dicembre, 1973), Accademia Nazionale dei Lincei, Roma, (1975), 53-76. (with Bang-yen Chen) Manifolds with vanishing Weyl or Bochner curvature tensor, J . Math. SOC. Japan, 27 (1975), 106-112. Differential geometry of anti-invariant submanifolds of a Sasakian manifold, Bolletino U. M . I., (4) 12 (1975), 279-296. 1976 (with D. E. Blair and D. K. Showers) Nearly Sasakian structures, Kddai Math. Sem. Rep., 27 (1976), 175-180. (with H. Hiramatu) On conformal changes of Riemannian metrics, Kbdai Math. Sem. Rep., 27 (1976), 19-41. Note on totally real submanifolds of a Kaehlerian manifold, Tensor, N . S., 30 (1976), 89-91. (with D. E. Blair and G. D. Ludden) Semi-invariant immersion, Kddai Math. Sem. Rep., 27 (1976), 313-319. (with M. Kon) Totally real submanifolds of complex space forms, I . TBhoku Math. J . , 28 (1976), 215-225. (with M. Kon) Totally real submanifolds of complex space forms, 11, Kbdai Math. Sem. Rep., 27 (1976), 385-399. (with M. Kon) Anti-invariant submanifolds of Sasakian space forms 11, J . Korean Math. Soc., 13 (1976), 1-14. Differential geometry of totally real submanifolds. Topics in Diflerential Geometry, Academic Press, 1976, 173-1 84. (with Shun-ichi Tachibana) Sur les dkformations complexes isomktriques des hypersurfaces complexes, C . R . Acad. Sci. Paris, 282 (1976), 1003-1005. (with M. Kon) Infinitesimal variations of anti-invariant submanifolds of a Kaehlerian manifold, Kyunpook Math. J . , 16 (1976), 33-47. On contact conformal connections, Kddui Math. Sem. Rep., 28 (1976), 90-103. 1977 (with M. Kon) Anti-invariant submanifolds of Sasakian space forms, I, TBhoku Math. J . , 29 (1977), 9-23. (with G . D. Ludden and M. Okumura) Anti-invariant submanifolds of almost contact metric manifolds, Math. Ann., 225 (1977), 253261. 12811 (with S . Sawaki) On complex Weyl-Hlavat9 connections, Kddai Math. Sem. Rep., 28 (1977), 372-380.

LI

Infinitesimal variations of hypersurfaces of a Kaehlerian manifold, J . Math. Soc. Japan, 29 (1977), 287-301. (with L. Vanhecke) Almost Hermitian manifolds and the Bochner curvatur tensor, Kodai Math. Sem. Rep., 29 (1977), 10-21. (with H. Hiramatu) Conformal changes of Riemannian metrics satisfying certain conditions, Ann. Mat. Pura Appl., 114 (1977), 155-172. (with T. Imai) On semi-symmetric metric cp-connections in a Sasakian manifold, Kodai Math. Sem. Rep., 28 (1977), 150-158. (with H. Hiramatu) Isometry of Riemannian manifolds to spheres, J . Differential Geometry, 12 (1977), 443-460. Anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor, J . Differential Geometry, 12 (1977), 153-170.

1978 (with Bang-yen Chen) On the theory of normal variations, J . Differential Geometry, 18 (1978), 1-10. (with S . Sawaki) On the Weyl and Bochner curvature tensors, Renrliconti, Accademia Nazionale dei X L , 3 (1977-1978), 31-54. (with M. Kon and I. Ishihara) Anti-invariant submanifolds with flat normal connection, J . Differential Geometry, 13 (1978), 577-588. (with U-H. Ki and J. S. Pak) Infinitesimal variations of the Ricci tensor of a submanifold, K d a i Math. Sem. Rep., 29 (1978), 271284. (with U-H. Ki) On ( f , g, u, v, w, 1, p, v)-structures satisfying 1' p2 u2 = 1, K6rlai Math. Sem. Rep., 22 (1978), 285-307. Infinitesimal variations of submanifolds, Kodai Math. J., 1 (1978), 30-44. (with U-H. Ki and M. Okumura) Infinitesimal variations of invariant submanifolds of a Kaehlerian manifold, Kodai Math. J . , 1 (1978), 89-100. (with U-H. Ki and J. S . Pak) Infinitesimal variations of invariant submanifolds of a Sasakian manifold, Kodai Math. J . , 1 (1978), 2 19-2 36. (with S. Ishihara) Real hypersurfaces of a complex manifold and distributions with complex structure, Kodai Math. J . , 1 (1978), 289-303. (with M. Kon) Infinitesimal variation of submanifolds of an evendimensional sphere, Kodai Math. J., 1 (1978), 362-375. (with H. Hiramatu) Integral inequalities and their applications in Kaehlerian manifolds admitting a holomorphically projective vector field, Bull. Inst. Math., Academia Sinica, 6 (1978), 313-332.

+

+

LII

1979 (with M. Kon) C R sous-variktis d’un espace projectif complexe, C . R. Acad. Sci. Paris, 288 (1979), 515-517. (with U-Hang Ki) Infinitesimal variations of hypcrsurfaces of a Sasakian manifold, Tensor, N . S., 33 (1979), 1-10. (with M. Kon) Submanifolds of a Kaehlerian product manifolds, Atti rtelIa Accademia Nazionale dei Lincei, Memorie, Serie VIIJ, 15 (1979), 267-292. Infinitesimal variations of submanifolds of Riemannian and Kaehlerian manifolds, SEA Bull. Muth., 36 (1979), 292-304. 1980 Notes on infinitesimal variations of submanifolds, J . Math. Soc. Jupan, 32 (1980), 45-53. (with M. Kon) Generic subnianifolds, Ann. Mat. Pirra Appl., 123 ( 1980), 59-92. (with M. Kon) Generic submanifolds of Sasakian manifolds, Kodai Math. J . , 3 (1980), 163-196. (with M. Kon) Infinitesimal variations of submanifolds of a Kaehlerian manifold, J . Korean Math. Soc., 16 (1980), 117-124. 1981 (with M. Kon) Differential geometry of C R submanifolds, Geometriae Dedicata, 18 (198l), 369-391. (with H. Hiramatu) Isometry of Kaehlerian manifolds to complex projective spaces, J . Math. Soc. Japan, 33 (1981), 67-78. (with A. Bejancu and M. Kon) CR submanifolds of a complex space form, J . Diflerential Geometry, 16 ( 198 1 ), 137- 145. (with M. Kon) Generic minimal submanifolds with flat normal connection, E. B. ChristofJel, Birkhiiirser Verlag, Basel, 1981, 592-599.

1982 (with M. Kon) Contact C R submanifolds, to appear in Kodai Math. J . (with M. Kon) CR submanifolds of complex space form, to appear in J . Diflerentiul Geometry. The Hayden connection and its applications, to appear in SEA Bull. Mat11. (with T. Imai) Quarter-symmetric metric connections and their curvature tensors, to appear in Tensor, N . S . (with M. Kon) Infinitesimal variations of submanifolds of a Sasakian manifold, to appear in Tensor, N . S .

LIII

This Page Intentionally Left Blank

LES ESPACES A CONNEXION PROJECTIVE ET LA GEOMETRIE PROJECTIVE DES ,,PATHS" Par

KENTARO YANO

INTRODUCTION.

Pour 6tudier la G6om6trie projective g6n6ralis6eI nous avons actuellement trois points de vue differents: le point de vue de M. E. CARTAN, celui de M. T. Y. THOMAS et celui de M. D. VAN DANTZIG. Mais la relation entre la thborle de M. E, CARTAN et celles des autres n'est pas encore completement 6claircie. Le but de ce MCmoire est de montrer comment on peut traiter la th6orie de MM. T. Y. THOMAS et 0. VEBLEN avec la m6thode du repkre mobile de M. E. CARTAN et dtudier les notions fondamentales dans la th6orie des espaces a connexion projective. Apres avoir expos6 l'aperqu historique de cette th6orie dans le chapitre I et quelques notions preliminaires d a m le chapitre 11, nous consid6rons, dans le chapitre 111, les transformations des repQes projectifs semi naturels. Nous montrons qu'on peut d6composer une transformation de repbre semi-naturel correspondant a u changement de coordonnkes, en deux transformations partielles, l'une transformant un repere seminaturel en un repere semi naturel ayant le mCme hyperplan de I'infini que l'ancien et l'autre transformant simplement l'hyperplan de l'infini : et que la premiere correspond a u changement d e coordonnbes et la deuxieme correspond au changement de la coordonnke surnum6raire de MM. T. Y. THOMAS e t 0.VEBLEN.

-

-

1

KENTARO YANO

396

Dans le chapitre IV, nous montrons qu'une transformation des composantes de la connexion projective correspondant a un changement de l'hyperplan de I'infini donne ce que les geometres amkricains appellent le changemen t projectif de la connexion affine. La chapitre suivant est consacrk a 1'6tude des equations diffkrentielles des geodksiques et du parametre projectif normal. La notion de paramktre projectif normal dans la Gkometrie projective des ,,pufhs" a btk recemment introduite et 6tudike par MM. J. H. C, WHITEHEAD, L. BEKWALD et J. HAANTJES. Nous montrons, dans ce chapitre, qu'on peut facilement arriver, du point de vue d e M. E. CAKTAN, a cette notion et qu'on peut lui donner une interprbtation geomktrique toute naturelle. Dans le chapitre VI, le tenseur de courbure et de torsion de M. E, CARTAY est considk6 et il est montrb que ce tenseur, qui est invariant par rapport a u changement de l'hyperplan de l'iniini s'il n'y a pas de torsion, devient le tenseur projectif de courbure trouvk par M. H. WEYLquand la connexion est normale. Dans le chapiire VII, nous montrons que les composantes ne sont autre de la connexion projective de M. T. Y. THOMAS choses que les composantes de la connexion par rapport au et, de plus, nous ktudions la repere nature1 de M. E. CARTAN relation entre le parametre projectif de M. T. Y. THOMAS et le parametre projectif normal. Dans le dernier chapitre, nous nous occupons du probleme de la reprksentation des espaces a connexion projective, c'esta -dire du probleme qui consiste a chercher un espace a con1 dimensions qui peut representer l'espace nexion affine a R donne a connexion projective a n dimensions, une gbodesique correspondant a un point et une surface totalement g6odCsique a une gkodksique et nous montrons que les espaces a connexion affine employks par MM. J. H. C. WHITEHEAD et J. HAANTJES sont caractbrisks par les m6mes propribt6s intrinseques. Qu'il nous soit permis d'exprimer ici notre respectueuse dont les conseils et gratitude i notre maitre, M. E. CARTAN, I'encouragemeni ont 6tk extremement prkcieux pour nous.

+

2

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

397

Chapitre I APERCU HISTORIQUE.

On doit la thkorie des espaces a connexion affine a M. H. WEYL[(125)]'1 qui a gkneralisk la notion de parallklisme de M. T. LEVI-CWITA [(6111dans les espaces de Riemann, en prenant

+

n r (n 1)/2 fonctions Ilf, symktriques par rapport aux indices infkrieurs comme composantes de la connexion A la place des symboles de Christoffel. Dans une variktk a connexion a f h e , les courbes autoparalldes qui correspondent aux courbes gkodksiques dans l'espace riemannien sont donnkes par les kquations . dui d d d'ui _ _ 4-11: __ - = o . ds ds d s

Les courbes dkfinies par les equations differentielles (1.1) sont appelees ,,paths" par les gkombtres amkricains, MM. L. P. EISENHART [(271,(28)'(291, (301, (311,(331, (341, (361,(3711, 0. VEBLEN

(1031, (1041, (10% et

(114),(1151,(11611,J. M. THOMAS "911, (9211

T. Y. THOMAS [(loll],

Les savants amkricains ont considkrb que ce qui est essentiel, c'est I'existence du systkme de paths, tandis que M. H. WEYLa pris comme notion fondamentale celle d e parall6lisme. La thkorie des invariants diffkrentiels des kquations (1.1) a k t k ktudike par les mathkmaticiens d e l'Ecole d e Princeton et nommke par eux la Gkomktrie des paths. [(124)] a montre que deux conneEn 1921, M. H. WEYL xions affines IIf, et ]If,, satisfaisant aux relations

ou @; sont n fonctions arbitraires, donnent le m2me systbme de paths, Dans le sens que cetie transformation des composanies de la connexion affine ne change pas le syisteme de paths qui correspondent aux droites de l'espace euclidien, ce changement s'appelle changement projectif de la connexion affine. La thkorie 11 Les noinbres figurant entre puenthescs renvoient B la Bibliographie plache i la fin du Mh-oire.

3

KENTARO YANO

398

des invariants differentiels qui ne dkpendent pas de ce changement s'appelle la Gkometrie projective des paths. Cette gkomktrie a CtC ktudike aussi par M. L. P. EISENHART [(34),(35)l et ses collbgues, MM. 0. VEBLEN(1031, (114), (115)], J. M. THOMAS [(90)], T. Y. THOMAS [(94), (961, (971,(IOl)]. La Gkomktrie projective est la theorie des propriktes des lignes droites qui sont indkpendantes des conditions poskes dans la Gkometrie affine. Ce fait se traduit par celui que la Gkomktrie des paths est la thkorie des lignes droites qui sont dkfinies par les equations diffkrentielles (1.1) et qui ne dkpendent du choix particulier du parambtre. MM. L. P. EISENHART [(27)]et 0. VEBLEN[(103)]ont montrk qu'a un changement de parametre correspond un changement de la connexion affine de la forme (1,2), Cette question de parametrlsation a ete etudike aussi par M. J. DOUGLAS[(261] qui a choisi comme equations differentielles dkfinissant les

pafhs :

p' = dui

OU

dt-'

et les HI ( u , p ) sont des fonctions homogenes du second ordre par rapport aux p i , Dans le cas consider6 par M. J. DOUGLAS, 2

les composantes de la connexion sont donnees par

par conskquent, les llj, sont des fonctions non seulement des variables u' , mais aussi des diffkrentielles dzr' , M. J. DOUGLAS a montr6 que si l'on effectue un changement arbitraire du parametre t, il correspond toujours un changement des composantes de la connexion de la forme (1.2) ou les l v k et (D, sont des fonctions des variables zz' et des differentielles du' , Pour bien faire comprendre que les propriktks ktudikes dans la Geomdtrie projective des paths sont indkpendantes du choix du parambtre, il l'appelle la Gkometrie descriptive des paths.

4

CONNEXION PROJECTIVE ET GkOMETRIE DES PATHS

399

On appelle actuellement la g6omCtrie de M. J. DOUGLAS la GCometrie projective gCnerale des paths, tandis que la Ckom6trie etudiee par MM. L. P. EISENHART, 0. VEBLEN,J. M. THOMAS et T. Y. THOMAS est appelee la Geom6trie projective restreinte des paths. D'autre part, M. E. CAKTAN [(9),(lo), (111, (18), (19), (20), (2111 a introduit, dans 1'Ctude des variktbs, la mkthode du repere mobile et avec cette m6thode ClCgante il a rbussi a etablir tout naturellement une belle theorie des varidibs a connexion projective. Dans cette thborie, M. E. CARTAN a defini une variete a connexion projective comme une varihte numerique qui presente, au voisinage immbdiat d e cbaque point, tous les caracteres d'un espace projectif ordinaire et doue, de plus, d'une loi permettant de raccorder en un seul espace projectif les deux petits morceaux qui entourent deux points infiniment voisins. En d'autres termes, il a associ6 a chaque point A,, de la vari6tb numerique un espace projectif tangent, rapport6 a un repere form6 avec n 1 points analytiques independants A,,,A,, . . , A , et il a dbfini la loi de raccord des espaces attaches aux deux points infiniment voisins A, et A, idA, , c'est - a -dire la connexion projective, au moyen d'bquations diffkrentielles de la forme

+

1

(1.4) i,

1

+

dA,, = w:: A,, f- w,: A, . . . -1dA, = A,,-1- cO; A , t- . . . . . -I-

(0::

An, A,, ,

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

dA,, = w:, A,, 4- w;, A, 4-. . . , .

+ w:: A,, ,

ou les I L sont des formes de Pfaff par rapport aux coordonn6es de la vari6te initiale et par rapport aux parametres arbitraires. I1 faut bien remarquer que les points A,,, A,, , A,, sont analytiques, que les formules (1.4) n'ont de sens que dans l'espace projectif tangent et, de plus, que ces formules ne servent que de moyen analytique pour developper une courbe de la vari6tC a connexion projective sur l'espace projectif tangent en un point de cette courbe. M. E. CARTAN a, de plus, 6tudie un probleme qui consiste a attribuer une connexion projective a une equation diffbentielle de la forme y"= f ( x , y, y'J , c'est-i-dire le probleme de trouver une connexion projective par rapport a laquelle 1'Qquation differentielle des geodesiques est celle (1)

...

5

400

KENTARO YANO

qui prkcbde. Cette etude l'a conduit a la notion de variktk d'6lkments a connexion projective. Ce probkme a Btd tout rhcemment generalis6 par M. M. HACHTROUDI [(43)] dans sa Thbse, a u cas d'un complexe de surfaces F ( x , y , w b , )= 0 , integrale gknkrale d'un systbme d'bquations aux d e r i d e s partielles complbtement intkgrable du second ordre,

06

La notion de connexion projective &ant une fois introduite, les gkometres americains, surtout ceux de 1'6cole de Princeton, ont commenck a Btudier les variktks a connexion projective par la mbthode du Calcul tensoriel. En partant du changenent de composantes de la conne[(94)] a obtenu les fonctions xion (1,2), M. T. Y. THOMAS

qui ne changent pas quand on effectue un changement d e lli,, de la forme (1,2). I1 est a remarquer que M. E. CARTAN a, deja en 1924, consid6rB ces composantes de la connexion projective en prenant comme repbre projectif le repere nature1 [(lo)]. M. T. Y. THOMAS a choisi les *I[;, comme composantes de la connexion projective associke au systeme d e pafhs par (1.1). Les coaposantzs de la cmnexion projective * l l ~ne~ se transforment pas comme composantes d'une connexion affine, mais si I'on ne considere que les transformations de coordonnes dont le jacobien est egal a I'unitb, c'est-i-dire Ies transformations de coordonnees qui ne chan8ent pas les volumes, les *I];,,.

6

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

40 1

se transforment comme composantes d'une connexion affine. M. T. Y. THOMAS [(94), (96)] a nommb Gbombtrie bqui-projective des paths la gbombtrie des paths vis-A-vis d e ce groupe special de transformations. Dans son Mbmoire paru en 1926 dans ,,Mathematkche Zeitschrift", M. Y. THOMAS [(95)), (98)] a montrb qu'il est possible de rkduire la Gkombtrie projective d'une varibtb a n dimensions a la Gbombtrie affine d'une autre vari6tb associ6e, a n 1 dimensions. I1 a associk a une transformation de coordonnbes u" de la varibtb initiale une transformation de la forme

+

u" =

{ u'

(1.6)

LZO

i-log 1,

= u'(n),

OU

e t il a introduit les csmposintes d'une c3nnzxiozl affix2 dan; la varibtk associbe a n 4- 1 dimensions

1

En utilisant le fait-que les fonctionq *llli.J sz transforment h

en *llpt, lors des transformstions de la forin? (1.5) de la msniere suivante :

M. T. Y. THOMAS 1(93)] a etudib les caordonnbzs normsles projectives, les tenseurs normaux projxtifs, le tenseur de c3urburz projectif, etc...

7

KENTARO YANO

402

L'introduction d'une telle coordonnke surnumeraire uo a et6, en 1929, adopthe par M, 0. VEBLEN [(llo), (1111, (11211. I1 a reussi, par un emploi heureux de cette coordonnee surnumdraire, a ktablir une th6orie projective contenant une Gkomktrie projective qui peut 6tre regardde comme une generalisation d a la GQometrie non euclidienne d e Cayley. MM. 0. VEBLEN et B. HOFFMANN [(113)]ont trouvb une belle application de cette thkorie projective a la thkorie unitaire "551, (56)]a trouve mQme des champs physiques; M. B. HOFFMANN une relation dtroite entre cette th6orie projective de la relatilivitC et la th6orie unitaire des champs d'EINSTEIN et MAYER. D'autre part, en utilisant les coordonn6es homogbnes introduites en 1932 par M. D, VAN DANTZIG[(23),(241,(25)],MM.

J. A. SCHOUTEN "761, (771,(781,(791,(801,(811,(821,(83),(84)],J. HAANTJES"4211, D. VAN DANTZIC[(231,(241, (2511,et ST.GOLAB

[(41)]ont ddvelopp6 une thkorie des varidtes a connexion projective. M. D. VAN DANTZIG emploie n 1 coordonnees homogenes Z). pour d6crire la varibt6 numerique P, initialement donnke et suppose que les relations entre les coordonn6es ordinaires u' et ces coordonnees homogbnes Z h soient donn6es par

+

ul

==

u' (Z",Z' , . . , 2 )

ou u' (Z) sont des fonctions homogenes d'ordre zero par rapport aux n 1 variables Z). , Par consequent, si l'on regarde Zk comme les coordonn6es ordinaires dans une vari6tC X,,,, a n 1 dimensions, a une courbe Z l = C i t dans cette vari6te oh les C i sont constantes correspond un point de la varibte P,, , Donc, l'emploi de telles coordonn6es signifie un choix d'un systeme de courbes sphciales. A notre avis, ces courbes correspondent aux ,,rays" 6tudies par M. J. H, C. WHITEHFAD [(127),(128),129)].MM. J. A. SCHOUTEN, J. HAANTJES et D. VAV DANTZIC ont aussi trouvQ m e application de leur thkorie a la relativit6 [(77j]. Comme nous l'avons expos6 dans les pages pr6ckientesl la Geometrie projective gbnbralisbe a et6 considerablement d6veloppde dans ces derniers temps. Mais malheureusement la relation entre la theorie de M. E. CARTAN et les aufresthkories n'est pas encore completement eclalrcie, malgrk qu'on ait actuellement les MBmoires de MM. E. CARTAN [(t2),(131,(151,(16)],

+

+

8

CONNEXION PROJECTIVE ET GgOMETRIE DES PATHS

403

J. A. SCHOUTEN [(69), (70), (72)], 0. VEBLEN [(106), 108)] et H. WEYL[(126)] a ce propos. M. J. A. SCHOUTEN a montrb dans son Memoire [(72)] comment on peut expliquer la th6orie de M. E. CARTAN par la theoG rie de la connexion de K O N ~[(60)]. I1 s'agit de construire une th6orie gbnerale de la connexion de KONIGqui contient comme cas speciaux la theorie des variktks a connexion projective ou conforme de M. E. CARTAN. La connexion de KONIGa 6te dtudiCe specialement par les gbometres japonais, mais cette thkorie n'est pas encore suffisamment developpbe pour qu'on puisse l'appliquer a la theorie de M. E. CARTAN, comme l'a dkja dit M.J. H C. WHITEHEAD [(129)]. Dans la conference faite au congres international d e Bologne en 1928, M. 0. VEBLEN [(lOS)] a expose la relation entre 1'Erlanger Programm et le point de vue pris par les gbometres americains, mais il n'a rien dit sur la relation entre les thboet de M. E. CARTAN et sa propre ries de M. J. A. SCHOUTEN theorie. Pour ktudier les geometries modernes, presque tous les gkombtres ont adopte la notion de parallblisme de M. T. LEVI-CIVITA comme notion fondamentale, tandis que pour M, E. CARTAN [voir par exemple (9), (lo)], la notion de transport parallble n'est pas la notion fondamentale; ce qui est fondamental pour M. E. CARTAN, c'est une loi qui nous permet d e raccorder en un seul et mbme espace deux morceaux infiniment voisins de la vari6te consideree, ces morceaux pouvant etre regardes comme espaces de Klein a groupe fondamental donne. Quand il s'agit de la theorie des variktes a connexion affine, on peut bien l'etudier soit au point de vue de MM. J. A. SCHOUTEN et 0, VEBLEN, soit a u point de vue de M. E. CARTAN, Mais quand il s'agit des vari6tes a connexion projective ou conforme, il faut bien des modifications dans la thCorie d e M. J. A. SCHOUTEN et celle de M. 0.VEBLEN.11s ont quand m6me bien r6ussi a 6tablir une thkorie des varietes a connexion projective et a connexion conforme, d'une manibre elegante, mais assez artificielle. Si l'on part de l'idee fondamentale de M. E. CARTAN, la gidralisation n'offre ni difficulte, ni inconvenient. Ce problbme a kt6 aussi examine par M. H. WEYL[(126)] qui a pris l'idCe de M. E. CARTAN comme fondamentale et a

9

KENTAROYANO

404

bien precise la relation entre la variete et les espaces tangents associbs a chaque point de la varibt6 initiale. Nous allons, dans le present Mbmoire, examiner en dQtail la relation entre la thkorie des variBtks a connexion projective de M. E. CARTAN et celles des autres, surtout celle de MM. L. P. EISENHART, 0. VEBLEN, T. Y. THOMAS, en nous appuyant sur les travaux citds ci - dessus. Chapitre I1 LES ESPACES PROJECTIFS TANGENTS.

Comme nous avons l'intention d'Qtudier les relations entre fa theorie des variktbs a connexion projective d e M. E. CARTAN e t celle des gQometres americains, nous allons exposer tout d'abord quelques BlBments essentiels de la theorie de M. E. CARTAN, d'une mani&re aussi approchbe que possible de celle des auteurs americains. Imaginons une vari6t6 numkrique a n dimensions, c'esta -dire une varibte dont chaque point est dCfini par un systkme d e coordonnees u', u 2 , . un, Le voisinage immbdiat de cette vari6tQ peut Ctre regarde comme un espace ordinaire affine, projectif ou conforme. Dans ce MQmoire, nous allons regarder le voisinape immkdiat de chaque point de la vari6tC comme un espace projectif ordinaire. Si l'on se donne une loi qui nous permet de raccorder les espaces projectifs ordinaires atttachhs a deux points infiniment voisins de la varihth, on obtient une vari6t6 a connexion projective. Pour d6crire les espaces projectifs ordinaires attaches a chaque point de la vari6te (nous les appellerons les espaces projectifs tangents), donnons- nous dans chaque espace projectif tangent, n 1 points analytiques AorA, , ,,, A,, indbpendants, fonctions des variables u', u 2 , , ,un, 11 est bien nature1 que l'on suppose que A, coincide avec Ie point (u', u 2 , . .. , t2' 1 de la vari6t6 auquel est attach6 l'espace projectif tangent, ce que nous ferons toujours dans la suite. Cela posh, un point quelconque XI situk dans un des espaces projectifs tangents, peut Ctre reprbente comme combinaison linQaire des n 1 points indhpendants A,, A,, . . . , A,, de l'espace considQr6:

.

+

.

+

10

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

X=X"Ao+X'A,

(2.1)

405

+ ...+X A , , .

Nous appellerons repbre projectif cet ensemble de points A,,, A , , . . . , Arret sommets du (n 1 ) - e d r e de reference, ces points respectivement. I1 va cans dire qu'il ne s'agit que des rapports mutuels des composantes X",X', , , . , P . Nous appelons X", A', . . . , X" les coordonnees homogenes d u point X. Les coordonnkes non - homogenes du point X sont definies par :

+

(2.2) Ici l'indice i ne prend que les valeurs 1 , 2 , .. ., n. En gen6ral nous emploierons les lettres latines i, j , k , . . . pour les indices qui ne prennent que les valeurs 1, 2,.. ., n, tandis que les lettres grecques seront conservt5es pour les indices qui parcourent les valeurs 0, 1,2, . . . n. Le choix des n 1 points formant un (n 1)- edre de reference ktant pour le moment tout a fait arbitraire, on peut prendre n 1 points independants Bn,B, , . .,B,, a la place des n 1 points A,, , A,, . . , A,, , En dbsignant par b i , b; , . . . ,6?. respectivement les coordonnbes homogbnes des points Bi, par rapport a u (n -/- l ) - & d r e de reference form6 avec les points A ; . , on a les formules

+

+

+

+

+

+

B,)= b: A,, b:,A , . . + b: A B, = by A , t-bt A , 1- . . I-b': A,, , *

11 9

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

B,, = b::At,-t b:,A, t . . L- bj: A,,, le dbterminant ibyi n'etant pas nul. D'apres la convention de sommation pour les indices rbpbtbs deux fois, on peut Ccrire les formules (2.3) sous la f orme plus condensee,

+ +

1 points B,, B,, B,, . . . B,, pouvant etre regardes Les n comme les n 1 sommets d'un nouveau (n 1)-edre de rkfb-

+

11

KENTARO YANO

406

rence, on a les formules suivantes, qui sont les reciproques de (2.4):

A;, = a; B,,,

(2.5)

oii les al;: satisfont a

Comme nous avons suppos6 que le premier sommet du (n 1)-edre de reference coincide toujours avec le point de la variCt6 auquel est attach6 l'espace projectif tangent, les formules (2.3) peuvent etre reduites a la forme suivante:

+

.

.

.

.

.

.

.

.

.

.

Les formules (2.7) ne doivent pas avoir nkcessairement cette forme, mais on peut supposer qu'on les ait r6duites a cette forme en multipliant au besoin tous les Bi par un meme facteur. Alors les formules (2.5) s'ecrivent

A, = B,, A, = a: B,)

+ a: B, + . . + a; B,,

(2.8) .

a

,

.

.

A,, = a;, B,, Les matrices I/ a:l formes nuivantes :

.

.

.

.

.

+ a:, B, 4-. + a:: B,,.

et 1) b i 11 ont donc respectivement les l,O,

by, b : ,

. . . . . .

*

12

*

,o

. . , b:'

. . . . b:,, b:., . . . , b::

a a ! /

. .

,

.

.

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

407

ces deux matrices 6tant rbiproques. On remarquera dans les formules (2.9) que

a; = h;

et

h,; = h i ,

et que, par conskquent, on a a'! - ! - a(I b"' E 0, I

a; b!:= hi, ,

a)'bj -=0,

ai, b; = hi, .

b'! I -I

-

Cela pose, voyons comment les coordonnees homogenes a ce chan-

Xi d'un point X se transforment par rapport dement du (n 4- 1)-edre. D'apres (2,1), on a :

x

X?.A. . 1, 1

en designant par X" les coordonnkes homogenes du point X par rapport au (n - ! - l)-6dre de rkfkrence form6 avec B,,,B, , . . B,, on a X = xi.Bib don::

X'. A). = X'. BA . En substituant (2.5) dans cette equation, on obtient

X~ a! B,, -- XI: BI'

(X'. a?

-

211)BI'

:

I

z

0,

Les n -I- 1 points B,, B,, . . . , B,, etant independants, les equations preckdentes donnent (2.10) Nous avons ainsi obtenu la loi de transformation des coordonnees homogenes d'un point defini dans un espace tangent, par rapport aux changements des (n 4-1 ) -6dres d e reference, I1 est a remarquer que les kquations (2.10) peuvent s'ecrire encore sous la forme (2.11) en vertu des identitks (2.6).

13

KENTARO YANO

408

Cela etant, cherchons la loi de transformation des coordonnees non-homogenes x i du point X quelconque. En dksignant par X i les coordonnees non-homogenes du point X par rapport au (n -!- 1)-edre d e rkference form6 avec B,,, B,, , .. , B,,, on a

D'autre part, on a d'apres (2.10) 12.12)

I-

+ a: X ' ,

=

X'-

a! X I , I

clonc on a finalement

(2.13)

la forme reciproque htant (2.14) Les equations (2.12) nous montrent que les X i ne dhpendent que des X'. et des coefficients qui dbfinissent le changement du (n -I- l)-&drede reference, donc on peut dire que les Xh sont les composantes d'un 6tre gkomktrique d'aprbs la definition de M. E. CARTAN.Comme les dkpendent en outre lineairement des a:, nous appelons le point X I vecteur contrevariant projectif et X).ses composantes. M. E. CARTANl'appelle vecteur contrevariant analytique [(20), (2111. Les deuxiemes equations de (2.12) nous montrent que dependent lineairement des X et des a ; ; donc les x' sont aussi regardees comme composantes d'un t t r e geomktrique : nous l'appellerons simplement vecteur contrevariant et X' ses composantes. Quand les coordonnees non-homogenes x' du point X

x'

XI

14

CONNEXION PROJECTIVE ET GEOMETIIIE DES PATHS

409

sont infinitesimales du premier ordre, les formules (2.13) peuvent s'ecrire aux infinitesimales du second ordre pres, 7'= a; -r/,


oh

X' sont

aussi des quantites infinitesimales du premier ordre. Donc, si les x' sont infinitesirnales, elles sont aussi regardees comme etant les composantes d'un vecteur contrevariant. Cela pose, considerons maintenant d'autres figures qui se presentent dans l'espace projectif tangent. Premierement, un hyperplan analytique dans un espace projectif tangent est represente par n -t 1 quantites T;. telles que 1'6quation lineaire

T i XI. = 0 ,

(2.15)

oh XI- sont des coordonnees homogenes du point courant, reprhsente I'hyperplan. En substituant (2.1 1) dans (2.15) o n trouve que (2.16)

peut 6tre adoptee comme loi d e transformation des Ti,, Nous appelons cet hyperplan analytique vecteur covariant projectif et TA ses composantes. Les equations (2.16) se decomposent comme il suit: (2.17)

Or, on voit que T,, ne change pas pendant la transformation d u (n 1) -edre de reference; nous l'appelons scalaire projectif. Si T,, = O par rapport a un (n 1)-edre de rCference, cela sera vrai par rapport a tous les (n 1)-edres de reference. Dans ce cas, l'hyperplan passe par le point A,, et on a

+

+

+

~~

(2.18)

TI= b! Ti ,

donc T, representent un 6tre gkometrique, que nous appelons vecteur covariant et T, ses composantes. Comme dernier exemple, prenons une quadrique analytiaue (2.19)

G. Xi- XP /.;I

= 0.

Si l'on effectue un changement de (n -1- 11-edre de ref&rence, Gj+ se transforment en Gill de la maniere suivante: ~

15

KENTARO YANO

410

(2.20)

A=

d'ou on a en posant i L = O et

11"

0,

-

Go, = b;

GUT

'

~-

G",, = Goo

*

Nous appelons cette quadrique tenseur covariant projectif du second degrk et G A !ses ~ composantes. I1 est a remarquer que Go, sont les composantes d'un vecteur covariant projectif et Go, un scalaire projectif. Les Go, nous offrent l'interprktation suivante : Considkrons le premier sommet A, d u (n 1) -8dre de reference : 11 a (1.0, , ,0) comme coordonnkes homogenes et l'kquation de son hyperplan polaire par rapport a la quadrique (2.19) est

+

GolL XIJ = 0 .

Si l'on pose (2.21)

l'kquation de la quadrique s'kcrit, avec les coordonnkes nonhomogenes x' : (2.22)

xi

xi

-!- 2 , ( ) ; x i + 1 = 0 ,

L'kquation de l'hyperplan polaire du point A,, par rapport a cette quadrique est (2.23)

cpj x "

+ 1 = c,

Ibrivons encore l'kquation du cbne d e sommet A,, circonscrit a la mQme quadrique. De l'equation (2.23) on tire ' p i 'pj xi xi 2 ' p i xi -1 1 = 0,

+

En retrancbant cette kquation de l'kquation (2.22) on trouve 1'6quation d u c6ne: .

.

(2.24)

8..x' x'

OU

g,j = ' ' I. - T i 'pi -1

16

=- 0 ,

.

CONNEXION PROJECTIVE ET GBOMETRIE DES PATH

41 1

La consideration des quadriques de cette sorte dans la theorie des varietes a connexion projective a 6th premibrement donnee par M. E. CARTAN [(13)] et developpee par M. 0. VEBLEN [(llo), (1111, (112)], qui a identifie le cbne (2.24) avec le cBne de lumikre dans la theorie projective de la Relativite. La notion de vecteurs et tenseurs projectifs 6tant une generalisation d e la notion d e vecteurs et tenseurs ordinaires, les operations bien connues pour les vecteurs ou les tenseurs ordinaires, addition, multiplication, contraction, peuvent s'appliquer aux vecteurs et tenseurs projectifs.

Dans le Calcul tensoriel projectif, il y a une operation caract Bristique. ..\, Prenons par exemple un tenseur projectif Fj+ contrevariant par rapport a l'indice v e t covariant par rapport aux indices ?, et pa Si l'on pose ),= 0 , les Fi: sont les composantes d'un tenseur contrevariant par raport a Y et covariant par rapport A [I, Si I'on pose encore = 0 , les F:,,:" sont les composantes d'un vecfeur contrevariant projectif. Si l'on pose v = i on ob~

tient un autre tenseur projectif, Fii,L. Sur ces questions, on peut consulter le Memoire [(20)1 s u r le Calcul tensoriel projectif ou le livre [(21)] sur les espaces a connexion projective, de M. E. CARTAN.

Chapitre

III

LES TRANSFORMATIONS DES REPI~RESPROJECTIFS.

Passons maintenant a la loi qui nous permet de raccorder e n un seul les espaces projectifs tangents attaches a deux points infiniment voisins. Prenons un point ( u l , u', . . . , u " ) de la variet6 et un ( n + 1)-edre de reference attache a ce point, forme avec n f l sommets A,, A, , . . . , A,, , le premier sommet A,, co'incidant

17

KENTARO YANO

412

.

avec ce point. A un point ( u'-1 du', u'+du". , u" -1- du" ) infiniment voisin du point ( u l , .. if') est associe aussi U R ( n + l ) Qdre de reference form6 avec les r14-1 points A,, 4-dA,,.....

Ail

+ dAu .

Si l'on fait le raccord des deux espaces projectifs tangents, on aura des formules de la forme

sont des formes differentielles linkaires par rapport ou les aux differentielles du',du?,... . , d u " , (1)

1. I1

= 1'2. 1;

du" ,

Le formules (3.1) peuvent s'kcrire

Dans les formules (3.2), multiplier A,, , A , , . . , A , , par un mdme facteur revient a ajouter une mCme diffbentielle exacte aux elements de la diagonale principale d e la matrice 11 ( oj.l r I C Mais comme les formules (3.2) ne servent qu'au dbveloppement d'une ligne de la variete sur I'espace projectif tangent, on peut ajouter aux elements de la diagonale principale une m6me forme d e Piaff quelconque, cette forme de Pfaff etant toujours une difkrentieile exacte le long de la courbz qu'on developpe. Donc on peut dire que la loi de raccord est compktement dbterminke par la connaissance des formes I) (I)

?. I'

-d

k !'

(IY;

.

Nous appelons, avec M. E. CARTAY, m:, , composantes de la connexion projective, 1) Voir

E. CARTAN [lo), (21).

18

et to'I -- h;

(11;;

le 3

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

Cela pose, voyons comment Ies formes

+

413

se transforment 1)-edre de reference dCit):

pendant la transformation de (n finie par (2.5) Aj. = a: B,, Oli

Designons par W A les composantes de la connexion projec;I tive par rapport au (n 1)-edre de refhrence formi avec les n $- 1 points B,,, B, , . . . . B,, ; alors on aura:

+

(3.3)

ou sous une forme plus condensee :

dBi

(3.4)

BI, .

= G:

Des formules (2.5), l'on t i r e :

-+

dA>.= ( d a i ) B l ,

0;

dBI, ,

donc on a (3.5)

'I

11

arl(o>. = d a :

11 -'I

f a>.(ol,.

Posons dans les formules (3.5) i. = 0 , alors on aura, puisque o!,'= dj,',

(3.6)

-Y

n:I- (0;; = ( t l , , ,

En posant encore v = i dans les formules (3,6',-on a , en vertu des identitbs ad = 0,

"I

.

-

w,: = w,I,

19

.

KENTAROYANO

414

Commeo: sont des formes de Pfaff linbairement ind6pendantes, on peut choisir a; de maniere a avoir _.

tot

= du"

.

Les reperes realisant cette condition sont appelbs seminaturels par M. E. CARTAN. Nous supposerons dans la suite que les reperes soient toujours semi-naturels, c'est-a - dire que w,: = du"

.

Cherchons la loi de transformation des reperes seminaturels, c'est a - dire une transformation d e la forme (2.5) qui permet de passer d'un repere semi-naturel a un autre repere semi-naturel. A cet effet, posons dans les formules (3.5)

-

to:

= du' , w i= d u ' ,

alors, on obtiendra des (3.5) en posant a,:to::

. .

-1-

a', I w / = dai (1

;. = 0 ,

Y

=i

4-a::Wi i-a,!,:,

aj dui =- du'

Donc: Afin que les formules (2.5) fassent passer d'un repere semi-naturel a un repere semi-naturel, il faut et il suffit que la matrice a; arbitraires ; soit

11

1

11

ait les 61ements a:

= hf,

a; =

I

et

En d'autres termes: Si les points analytiques A,, , A i dbiinissent un repere semi naturel, le plus gbnbral repere semi naturel est dbfini par les points

-

-

11 Les 2; (symboles de Kronecker) s'emploieront dans la suite touiours dam ce sens.

20

CONNEXION PROJECTIVE ET GPOMETRIE DES PATHS

415

-

-

A,, = A , , A ; = A ; 4-(I);/!,) avec (I'i arbitraires. I1 s'ensuit qu'un repbre semi-naturel est determine par la donnee des points A; sur les tangentes des lignes parametriques issues du point A,, ou, ce qui revient a u insme, par son hyperplan de l'infini. Nous allons maintenant prouver qu'il y a un repere semi naturel et un seul, vkrifiant la condition 0 ; - d': w" = Mi. - n to'' =o.

-

. A. = j

I

En effet, posons

'I

0

=I ,

dans les formules (3.5); on a

+ a!:w;, , dai + a; 6,; 4-a; G;;. ,

a'I' dr I =dalI a:,to;

+ a;,

(0: ==

o(= a'!dui I

en posant i \ = v = O

/

+ W( ;

(1)::

+ a; duJ. =-to::

donc on obtient -. (0:.

.I

dans (3.5) on a ab mi,' =da::-t- a GR

- n W::=~j - n w:: - (n

+ 1) a:;d d ,

ce qui nous montre que pour annuler Wj - n W::, on n'a qu'h prendre a; de telle manibre qu'on ait (0::n o : : = ( n + 1)aydu'. Le repere projectif realisant cette condition est appele par M. E. CARTAN, repdre naturel. Dans la theorie de M. E. CARTAN que nous avons exposbe, la connexion projective btant completement determinbe par les A A formes wp - h!, w:), la forme de Pfaff w i ne joue aucun rdle, Mais, dans ce qui suit, nous voudrions presenter les choses cl'une autre manibre en faisant jouer un r61e a la forme parasite w::, Nous avons vu que, pour un sysibme de coordonnees (uL,u2 ,, .. u" ) , on a une classe de repbres semi-naturels et un seul repere naturel dans un espace projectif tangent a la varibte i connexion projective. Les coordonnbes (u' , u ' , .. . , u" ) qui dksignent des points de la varibtb btant tout a fait arbilraires, on peut passer d'un systeme de coordonnbes un autre systeme par une transformation analytique . -. (3-7) u' = u' (u.').

21

KENTARO YANO

416

Les reperes semi-naturels relatifs a l'ancien systeme d e coordonnbes ne peuvent plus bvidemment &tre semi-naturels par rapport au nouveau systbme. I1 se pose alors la question d'btudier le passage d'un repere semi naturel relatif a l'ancien systbme, a un repere semi - naturel attach6 a u nouveau systeme. Ce passage peut &re effectub en deux btapes : 1'' on change les coordonnbes ( u ) et on conserve I'hyperplan de l'infini d u repere, et 2" on conserve les coordonnbes (U) et on change l'hyperplan de l'infini du repere. 1 0 Dans le systeme de coordonnbes (u', u ' , . . . , u" ) on a, pour un repbre semi - naturel, la relation caractbristique

-

dA,,= ou

(0:;

A,,

+ du' A,.,

est une forme de Pfaff: posons

(3.8) alors on a

((:

= p , du' ,

dA,,=p , du"A,,

(3.9)

+ du' A , .

Effectuons la transformation de coordonnbes (3.7) ; on tire alors des formules (3.9)

Cela nous fait voir que le repere dbfini par les points analytiques (3.10)

I A,, = A,,

est un repere semi-naturel. Car en posant (3.11)

-

P.1 =

all,

P'

on a la relation caractbristique

dZ,, =pi diii A,, 4-dii'

A; .

De plus, comme les points A; dbpendent linbairement des points A;, ce repere a le m$me hyperplan de l'infini que le repere initial, et cela, comme nous I'avons remarqub, le dhtermine completement.

22

CONNEXlON PROJECTIVE ET GkOMETRIE DES PATHS

417

Nous appelons simplemen t (,transformation d e coordonnees" cette transformation qui ne change pas l'hyperplan d e 1 'infini. Cela etant, voyons comment se transforment les coefficients des autres formes de Pfaff 11); et LO: pendant cette transformation. Posons (3.12)

Des equations (3.13), l'on deduit

d'Oli

(3,141 (3.15)

Donc on a le thborkme: Pendant la transformation des coordonnees (qui ne change pas l'hyperplan de l'infini) les fonctions coyL. se transformenf comme les composantes d'un tenseur covariant affine et les fonctions wiL, comme les composantes d'une connexion affine. 2O. Maintenant, nous allons considker la transformation entre repkres semi-naturels relatifs a un meme systkme de coordonn&es,c'est a dire la transformation qui change l'hyperplan de L'infini. Nous avons vu plus haut que ce changement peut ktre reprksente par

--

23

KENTARO YANO

418

(3.16) ou

sont des fonctions de u' . En substituant (3.16)dans

(Ill

+ du' A,

dA,,=w",X(l on voit que

-

(3.17)

w: = w:

,

+ 4)&du' .

Ici la forme d e Pfaff w:: joue un r61e assez important, mais il faut remarquer que la considkration de cette forme n'a aucun rapport avec la thkorie de M. E. CARTAN. Pour apporter plus de syrn6trie dans la formule

dA,,=p , du' A,, 4-du' A , , nous Ccrirons

(3.18)

du"=

=p , du',

(1):;

alors, nous aurons

(3.19)

dA,,=-- duoA,,

+ du' A, ,

La forme de Pfaff w: = p , du' n'ktant pas en genkral une diffkrentielle exacte, uo n'est pas une variable vkritable ; nous appellerons uo, avec M. J. A. SCHOUTEN [(75),(86)1, la variable

-

non holonome. La variable non holonome u0 Ctant ainsi ddfinie, la transformation (3.17)peut 6tre reprksentde comme une transformation de variable non holonome :

-

-

(3.20)

diP= duo

+ 9,du' .

Nous appelons simplement ,,la transformation de la variable non holonome u" cette transformation qui changz l'hyperplan de I'infini en transformant les repbres semi-naturels en repBres semi-naturels. Voyons maintenant comment les fonctions myk et wik se transforment quand on effectue cette transformation de la variable non holonome u". En dksignant les fonctions transformdes de w;k et wjk par to;!, et G;,< respectivement, on a

-

-

24

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

419

d'Oh

Oh

Donc on a le theorbme:

Pendant la transformation de la variable non - holonome uo (transformation de l'hyperplan de l'infini), les fonctions w;:, et wjl: se transforment respectivement d'apris Ies formules suivantes :

Soulignons, en terminant ce chapitre, qu'une classe d e reperes semi-naturels par rapport a u nouveau systbme de coordonnees ( 2 )correspondant a une classe de reperes seminaturels par rapport a l'ancien systbme de coordonnhes ( u i ) , nous avons dbcompos6 le changement de repkre semi - naturel en deux operations partielles :

- premikre

operation : passer d'un ancien repere semi - nature1 a u nouveau repbre semi-naturel qui a le mbme hyperplan de l'infini que l'ancien : - deuxieme opkration : passer d'un repere semi-naturel a un autre repbre semi-naturel dans un meme systbme de coordonn6es en changeant l'hype-plan de l'infini; e t que nous avons convenu d'appeler la premiere operation la transformation des coordonnt5es ui et la deuxibme la transformation d e la variable non -holonome u".

25

420

KENTARO YANO

Chapitre IV LES TRANSFORMATIONS DES COMPOSANTES DE LA CONNEXION PROJECTIVE.

Considerons un contour ferme infiniment petit, partan t d'un point P de la varikte et y revenant, Les coordonnees hom o g h e s Xk d'un point dans un espace projectif tangent attache a un point P subiront une variation infinitbimale AX;.et on aura les formules [(lo), (2111

Oil

(4.2)

I1 faut remarquer que (4.1), comme (3.21, definissant un L A &placement projectif infinitesimal, seules les formes 5iIL- hi, ont un sens gkometrique. Le point P regarde comme etant celui de l'espace projectif tangent a comme coordonn6es homogknes (1,0,0, ,. .0). Donc la variation de ce point satisfait a

I I

I xo- i!;; = 0 , Ip-i!'=O,

. . . . . 1 x"

- $2" =. 0

.

Si le point P revient a sa premiere position apres un tow-, on dit que la connexion est sans torsion. Une condition necessaire et suffisante pour que la connexion soit sans tor.ion, est donnee par les equations suivantes 14.3)

!i' = 0 , 51;

z=z

26

0,

a

. . , ti,; = 0 .

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

421

D'apres les deuxiemes kquations de (4.21, ces conditions p euven t s'kcrire -

((t()'

-!.

[o:: (o:)] I [w;,

En substituant (o;, =du'

41 = 0 ,

,

w : : = p , d u l. , w!1 = Wji. du"i ,

on obtient

[p,,duL, dui] ((oj;: -

+ [dd,

w;!:

d d =] : O ,

"ppi,)[ d d d d ] =: O ,

En posant I]!/ 1: = C d / ; - h;,pi; ,

(4.4)

on a, comme condition pour que la connexion s oit sans torsion,

ll!,I =

(4.5)

JL

Cela dit, considerons un point

X=dx'lA,

X'A,

. . . -,-

2 1

A,,

dans un espace projectif attach4 a un point P(d)d e la variete ; ce point viendra coi'ncider avec le point correspondant quand on fait le raccord des espaces tangents, si l'on a d X = h X, ou

dX= dX- A;,

+ X" d A ,

L=(&. 4-XI' '. 1 A,. , 'I):,

c'est - a - d i r e si l'on a

dX'

+ XI' yti drz' =h X'..

Les premiers membres de ces equations s'ecrivent encore sous la forme suivante

Posons (4.7)

27

KENTARO YANO

422

alors, en remarquant que

duo = p i du’ , on obtient de (4.6)

dX”

(4.8)

+ XILllivdu”,

Des gquations (4.7), on tire les suivantes: 11”. =0”. - p I. =0 , (I/ #I/ (4.9)

IIiI k = 0,iI k - hIi pI: * Les premibres et troisibmes relations nous assurent que

donc on a (4.10)

Nous avons d6ja vu que, pendant la transformation de la variable non-holonome uo

dsio= duo

-

+

du‘

c‘est - a dire pendant la transformation (4.11)

-

pi=pi

+@it

les fonctions wyk et wjlc se transforment en i$/:et G;k respectivement de la manibre suivante: (4.12) (4.13)

w‘!i k - Q i ,/; - (Dj (Dk -. w;/; =w;,< - Qi ,

+ (Pi

Wjk

- qji PI
q6

En outre, nous avons aussi vu que les fonctions et wjl; se transforment en i$k et l;j, respectivement pendant la transformation de coordonn6es ui d e la vari6t6, de la manibre suivante : (4.14)

28

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

423

1

Cela Ctant, voyons comment les fonctions 11, se transfoment en pendant les deux transformations. En tenant compte des formules (4.9), (4.12) et (4.13), on voit que I'effet de la transformation (4.11) est

$,,

a/.- (4',,

- Ql I l f k ) ,

(4.16)

Les quatriemes equations representent le changement des composantes de la connexion projective, obtenu tout d'abord L. P. EISENHART et 0. VEBLEN [voir (1241, (271, par MM. H. WEYL, (103, (133) I. Pour un changement des coordonnCes u' de la varietC a connexion projective, on a la loi suivante de transformation :

I -11"' = II" =o, 'I

(4.17)

I

)

I"."I

01

= II'.=:h ; , "I

Enfin, pour un changement des coordonnbes ui suivi d u n changement de la coordonnee u", (4.18)

on a Ies formules suivantes :

-

= I I::i = 0 ,

29

KENTARO YANO

424

I1 est a remarquer que la symbtrie de lly,z par rapport aux indices infbrieurs ne se conserve pas en gbnbral, tandis que la symdtrie d e par rapport aux indices 1 et k se conserve toujours. Remarque. Pour obtenir la condition 1

11p0 =

i,

( M econstante)

;i

l l o p = Mh,$ 1

posee par M. 0. VEHLEN [(112)] et les autres [(I), (531,(9811, on n’a qu’a poser

alors on aura pour les fonctions

II;,.,

et pour la transformation d e la variable non-holonome

Chapitre V LES EQUATIONS DES ,,PATHS’ ET LE PARAMETRE PROJECTIF NORMAL.

Dans ce Chapitre, nous allons consid6rer les gkodbsiques et quelques parambtres spbciaux sur ces courbes, en partant de la ddfinition des gbodbsiques de M. E. CARTAN [(lo), (21)]. Prenons une courbe issue d‘un point d e la vari6tb. Cette courbe s‘appelle gbodbsique ou path si son dbveloppement sur l‘espace projectif tangent attache a ce point est une ligne droite. Ce fait peut dtre exprime par

d”Ao =?. dA,, f i t A,, , OU

+ + = (dw::+ w::

dA,,= w:: A(, du‘ A, dA,,= dw::A,$ 0):;dA,,

+ d‘u‘ A ,

0:: -1u)‘d d ) A,,

1- du’ dA,

+ (u::d d -I-

@u‘ 4-~f du’JA, ,

donc, comme bquations diffdrentielles ,dbfinissant les gbodbsiques, on obtient les suivantes:

30

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

d u ' + w;dul: du'

---=---

425

dJui I o;dul:

dui

--.

Mais, en choisissant convenablement un parametre f et une fonction Q de t sur la gdodksique, on peut dcrire l'equation de la gdodesique sous la forme suivante: 65.11

La ligne developpee sur l'espace projectif tangent dtant une droite, le rapport anharmonjque des quatre points sur cette droite correspondant a quaire valeurs de t, est 6gal a u rapport anharmonique des quatre valeurs de t. Nous allons chercher 1'6quation qui dkfinit le parametre t. En employant un paramktre arbitraire r sur la g6odesique, on a

donc

P o u r que cette courbe soit gdodhsique, on doit avoir

(5.4)

.1. du' - - =d2u' dr dP L"i'

dujduk - 2 - duo du' d r dr dr dr

Alors on aura

dLA, . . - . dA,

&'-'. ~

&

31

$.

A,,

.

KENTARO YANO

426

Cela pose, calculons

On a d'abord

ou la l'accent indique la dbivation par rapport au para-

m&tre t.En dkrivant encore par rapport a t , on a

Donc on a

comme condition pour que t soit le param6tre cherch6. En 6Hminant Q, on nbtient

ce qui, en posant (5.7)

s'bcrit ( 5.8)

,

I f l ,I

-;'-

.

1 .* 2/\--21?.

Ivoir E. CARTAN (21) p. 51.

;,.

L'expression bien connue { t s'appelle la derivbe schwarzienne de la fonction t ( r ) par rapport 'a la variable r.

32

CONNEXION PROJECTIVE ET GEOMQTRIE DES PATHS

427

Le parametre f etant ainsi determine, nous allons introduire un autre parametre s qui nous permettra d'ecrire les equations des gBodBsiques sous une forme plus simple. Des Bquations (5.41, l'on tire

donc, si l'on prend le parametre s de maniere a avoir s" 4-2 -dun s'l - j , sf ~

-

_1

0

ds

c'est - a dire S"

;.=->+2

(5.9)

S

duo , dr

les Bquations des geodesiques s'ecrivent

En substituant (5.5) et (5.9) dans (5.81, on trouve

donc

D'apres la formule bien connue (5.10)

on a enfin

33

KENTARO YANO

428

Donc, on peut knoncer le thkoreme [(134)]: Le systBme des ghodesiques dans une variite a connexion projective htant donne par

(5.11) le parametre t qui donne la forme (5.1)a l'equalion d'une geodesique est determine par

(5.12) sur chaque ghodesique. Nous avons ainsi dkcompose les equations des gkodksiques en deux systbmes d'kquations (5.11)et (5.12).Si l'on se donne arbitrairement, un point et une direction en ce point, c'est - a dire ( u' et (duli ds),,, les kquations (5.11) dkterminen t un path. La thkorie des kquations diffkrentielles d e la forme (5.11)a 6tk surtout ktudike par les gkometres de 1'Ecole d e Princeton, C'est la ,,Geometry o f paths" de MM. L. P. EISENHART "271, (281,(291,(301,(311,(331,(34)1,0. VEBLEN [(103),(1051,(1141, [(loll,(11611,J. M. THOMAS [(91),(92), (115),(116)],T. Y. THOMAS (1141,(115)l. Les kquations diffkrentielles (5,111 dbterminant ainsi les gbodksiques, l'kquation (5.12)dktermine une fonction t(s)le long de ces courbes. Comme l'on ne connait que la dbrivke schwarzienne de la fonction t (s), f (s) n'est dbtermink qu'a une substitution homographique pres, ce qui est kvident d'apres l'jnterprktation gkomktrique de t donnee au debut de ce Chapitre, Ce paramktre t, premierement introduit par M, J. H. C. WHITEHEAD [(129)][voir aussi L. BERWALD fl)], s'appelle paramGtre projectif normal. La thborie des equations diffkrentielles (5.11) par rapport au groupe de transformations

-

),)

U'= U' ( u ) ktant la Gkombtrie des paths, la thkorie des kquations diffkrentielles (5.11) et (5.12)par rapport a u groupe de transformations duo= du" 4-(P, du'

34

CONNEXlON PROJECTIVE ET GQOMBTRIE DES PATHS

429

c'est-a-dire par rapport a u groupe de transformations des coordonnees ui _. u' =U' (u), et a u groupe de transformations de la variable non- holonome uo

dii" = du" + cIji du' , qui entrainent les transformations des composantes

- a); (I)/. - a);, + (f), I ]./A.1' ll,;.k=li;/., - (rp,- b p ; , = 11,;i.

/.,

et

?

est appelee par les gkometres americains la Geornetrie projective des paths [(voir (11, (341, (42), (90), (94, (961, (101), (10311. Comme- les fonctions et lJii, se transforment respectivement en ll.yk et i,;,~; d'apres les formules (4.17) quand on effectue m e transformation des coordonn6esZi=u" ( u ) , il est bvident que les parametres s et t restent invariants pendant cette transformation, Si I'on effectue une transformation de la variable nonholonome u", c'est a dire un changement de l'hyperplan de l'infini, qui entraine la transformation (4.16) des ll;k et lI;h, les 6quations (5.11) prennenl la forme suivante :

- -

(5.13)

d2u1 ds'

+

--

du' d d

du' dul - - =o. ds ds

11;i ds d s T 2 d , , - -

Pour mettre ces equations sous la forme de (5,111, effectuons un changement de parambtre s. On definit une fonction S (s) par

d2S (5.14)

ds

- -

c'est a dire par (5.15) Alors, les equations (5.14) se rkduisent aux equations

d 2 d -i dul d d + 11,k- ds- ds-= = 0, ds:!

~

35

KENTARO YANO

430

ou

S est le paramktre affine relatif aux composantes ll,jL,d e la

connexion affine. En ce qui concerne le lparametre projectif normal t, il est evident, d'aprbs la signification g6om6trique de t, que t ne changera pas pendant un changement de l'hyperplan de l'infjni. En effet, on peut montrer par un calcul facile que (5.16) Les ui et t Btant d6termines sur chaque g6odesique comme fonctions du parambtre affine s, nous allons chercher la valeur de la variable non holonome u" sur chaque g6od6sique. En substituant (5.9) dans la premi6re 6quation de (5.61, on trouve

-

(I'

2"+, t'

2 log 1 t logs'

s" duo t" + 2 dr - t l ' - O , s

+ 2 uo - log t'=

constante,

donc : on a, a une constante additive prds, le long de la geodkique, (5.17)

On sait que le paramktre projectif normal t etant defini par une dkrivee schwarzienne, t peut subir une transformation homographique (5.18)

ou l'on peut supposer sans restreindre la generalit6 (5.19)

ad- bc=l.

iio etant determinee sur chaque gkodksique, on voit que la fonction Q subit, pendant la transformation homographique (5.18) de t, la transformation suivante

(5.20)

I, =

'C

c t t d '

Dans son Mkmoire intitule ,,On the projective Geometry of paths" M. L. BERWALD [(I)], essayant d'expliquer, uniquement

36

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

43 1

du point d e vue de la Gbometrie des paths, la theorie des espaces projectifs d e I'ficole d e Princeton et l'introduction d e la coordonnee surnumeraire u", est parvenu a la notion d e parambtre projectif normal. I1 part d'un systbme d e paths determine par (5.2 1)

et iI definit, sur chaque geod6sique, le parametre projectif normal f par (5.22)

et la coordonnee surnumeraire u", sur chaque geodesique aussi, Par ds u" - - -1 (5.23) 2 log dt ' ou les Il.yket lI,iL,sont symetriques par rapport aux icdices j et R, et il pose les deux conditions suivantes: lo, t reste invariant quand on effectue une transformation d e coordonnees _.

u'-

U'(U),

et 2O, t reste invariant quand on effectue un changement des composantes d e la connexion affine (5.24)

-

Ilj, = Ilj, - hi 6,- hi (4)'

.

Le point de vue de M. BERWALD est donc different d u precedent, pufsqu'il n'introduit pas tout d e suite u", mais il 1

clefinit la variable u"sur chaque geodesique par - 2 log

ds I

ce qui veut dire qu'il a choisi s et t sur cette courbe. De la premiere condition, on conclut que les sont des composantes d'un tenseur affine, tandis que de la deuxikme on obtient la loi de transformation des composantes d u tenseur affine II;, vis-a-vis d'une transformation (5.24) :

31

KENTARO YANO

432

Ces formules coincident avec les formules classiques si l'on n'utilise que les repkres naturels et qu'on soit dans un espace normal (voir les Chap. VI et VII). Sinon elles ne sont susceptibles d'aucune interprbtation gbombtrique simple, bien que la thborie soit cohbrente. Si l'on effectue le changement des composantes (5,241, le paramBtre affine s se transforme en S de la maniBre suivante : (5.25)

donc, f restant invariant, on obtient la loi de transformation de la variable u"

(5.27)

WALD

ii "

= u')

duk .

I1 est trQs intbressant d'examiner la thkorie de M.L. BERde notre point de vue. Dans la Gbometrie des paths on a les equations diffbren-

tielles

d?u'

du duL' dSL -t II'J'ds ds- = o ,

dbfinissant le systeme de paths, et l'introduction d'un tenseur affine lJ;k veut dire que l'on considere une varibtb A connexion projective dont les composantes sont Il;, et IIf. et dont le systbme de gbodbsiques coincide avec celui de paths. Comme le systbme de gkodbsiques d'une varibte B connexion projective est completement dbterminb par les fonctions , on peut choisir arbitrairement les fonctions lIyk, Alors le parametre projectif normal de M. L. BERWALD coincide avec notre paramBtre t, et le changement (5.24) correspond a u changement de notre variable non holomone uo. Mais les (5.25) ne coincident pas tout a fait avec nos Cquations

-

WALD,

Ce fait revient B ce que, dans la thborie de M. L. BERon n'a besoin que de la parfie symbtrique des fonc-

tions lIJ"k. La definition de la coordonnke surnumBrah-e de M. L.

BERWALD,

38

CONNEXION PROJECTIVE ET GEOMETRIE DES PATH

u"=

1 2

- - log

ds dt

--

433

'

et notre r6sultat

ne coi'ncident pas non plus en gdndral. Quand on effectue une transformation homographique sur t, la coordonn6e surnumdraire de M. L. BERWALD change en g6ndral tandis que notre '1 reste invariant grdce a la pr6sence d e L)'.

Chapitre VI LE TENSEUR DE COURBURE ET DE TORSION DE M. E. CARTAN.

Rappelons -nous les kquations de structure de la vari6t6 a connexion projective

Les formes bilindaires difkrentielles i!; -- b; $2,; avec !:, et de courbure et d e torsion de M. E. CARTAY. Les identitks correspondant a celles de BIANCHI peuvent Ctre obtenues en dbrivant extkrieurement les (6'1) et en tenant compte des (6.1) elles-mCmes, !!; definissent completement le tenseur

Nous allons d'abord calculer explicitement les composantes du tenseur de courbure et de torsion de M. E. CARTAN. A cet effet, posons:

39

KENTARO YANO

434

de

Des deuxiemes equations de (6,1),on tire, en tenant compte = p i du' et de w: = dui,

... It

1 O ' , , [duiduk] 2 "U//. -=pi [ dui du' ]

+ wjk [ dui d d ]

= (wjI:4-hi1pi ) [ du'

donc : !I:,jk =Oil,

+ q:pi -

du" ] ,

WLi

--

hip,

,

et on a, en tenant compte des bquations (6.4

Nous avons d6ja vu qu'une condition necessalre et suffisante pour que la connexion projective soit sans torsion est i!i, =0 , par consdquent

II!/I2 = II'.. It/ Des troisikmes dqations (6,1),on tire

' yi/<

0" " i =2

[ dui

- w:.) pl; [ du' -

, du'] - o/!W"Ilk [ dui du" ] I/

Donc on a finalement

40

a

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

435

Le calcul pour les dernibres formules (6.3) est le plus compliquk. Calculons d'abord I!:: et 52; skparkment,

on obtient :

En posant

Toutes les composantes du tenseur de courbure et d e torsion 6tant calculkes, voyons comment se transformed ces composantes lors des transformations de variables. Nous atlons d'abord considCrer l'effet des transformations des variables u', c'est-a-dire des changements de repbre seminature1 avec conservation de l'hyperplan de l'infini. Nous savons que les IJilL et llyk se transforment respectivement comme composantes d'une connexion affine et d'un tenseur affine quand on effectue une transformation de coordonn6s u' Par conskquent, il est bien Cvident que S.):B,,l S2yl/L et Si;hl, sont des composantes des affineurs par rapport aux transformations de variables u'

.

.

41

KENTARO YANO

436

Considkrons ensuite la transformation d e la variable nonholonome u", c'est-a-dire changement de l'hyperplan de l'infini,

dii'

= du"

+
On sait qu'alors les fonctions II;; et iiiL se transforment respectivement en It;/, et l I f k de la manikre suivante:

-i

En dksignant par i!oi/; les composantes transformkes dc on a :

Les composantes i!& sont donc invariantes par rapport L: cette transformation. Calculons ensuite les composantes !$/;. En substituant (6.9) clans les formules analogues de (6.5)

et tenant compte des relations (4.16) on obtient

-

(6.12)

-1-

I!;./
w;,/; -

-t @p/J q/ -(ai,//

--+,,/

q-r il,/ll/l)p

- y @ k , / / - Cb//,J - cbj(l[;(// -- ll;,J Les

l$k,, _.

!i,:,

ktant obtenues, en substituant ces valeurs dans -.

= l&,

-

.

-

.

. _

-

4- Il;/2ii;/ - IlJ// 8;s - b; (Hh/- I L ) ,

42

CONNEXION PROJECTIVE ET GEOMCTRIE DES PATHS

437

on obtiendra aprbs un calcul facile

(6.13)

-

s!;/l/, = "j/
.

a,,, I!:;,/,

- (Di I!;/
que seules les composantes 12:,i/: restent invariantes, dans le cas gknbral, par rapport a u changement de l'hyperplan de I ' h fini. Dans le cas de la connexion sans torsion, on voit queles composantes Qjkl, sont aussi invariantes par rapport a cette transformation. Supposons dans la suite que la connexion soit sans torsion. Comme S2;A,l, est un tenseur invariant, le tenseur d6fini par

(6.14)

I!j/<

= i!;;,,

,

est aussi invariant. En contraciant dans (6.8)par rapport aux indices i et h, on obtient o =:I[. .jk -k I I '/ ,:/~ -- I kI i ~ (6.15) -j/; t

OU

(6.16)

1I, =

Les tenseurs I!jn,,, et I!,

q/, .

etant invariants, le tenseur defini par

c//;h = s!;/zlt + i-$ i!/;h

(6.17)

,

est aussi invariant.

Le tenseur Cj/c,L dtant invariant, le tenseur contract6

c//;= c;/;/,

(6.18) est aussi invariant. De (6.17),on tire

(6.19)

+ .

Cj,<= I!j/;

S!/;j

En substituant (6.15)dans (6.191,on obtieni (6.20)

c/<j = (lIi/<+ lI,J

+ ( n - 1) (IIYh + II$

.

Ainsi a-t-on obtenu les tenseurs suivants qui sont invariants par rapport a la transformation de la variable non-holonome u0, =

+

q,

H;/< b;, hi; - b; (11;,/, - il;,/:), I!i/< = lrj, -I- n IIYh - Hii CjlC/,= llikl, hj II,,, -t11;/: q, - Iy/, ". (n - 1) ll;,/) , = (11.. I!. irIgj)-t (n - 1) (u;,, t i i y .

+

+

-+

43

KENTARO YANO

438

Nous pouvons annuler it,,2 = 1,

+n1

-1

en choisjssant convenablement les f onctlons llylz : (6.22)

i!,k === 0,

La connexion projective sans torsion realisant ces conditions la connexion projective normale est appelbe par M. E. CARTAN [(lo),(2111. De I'kquation ll.,.,, 4- n

IIyk

-

ll;,j

~-

10,

on a

[I,, 5 n ll;, - [I;,;

1

10,

donc (6.23)

II'(//:

(6.24)

ll;/d -

f Il''.=k/

1 -n-1

(llj,,.

1

Illi== - _ _ (Ili, 11 ' 71

f Il/;;), - lIh,).

Par condquent, on obtient une propri6t6 du tenseur dans le cas de la connexion projective normale: (6.25)

C;)
ci/;= 0.

Des (6.23) el (6.24)' on obtient

Substituons cette expression de I[;/< dans (6.8); alors on aura

dans ce cas C//{,, coincide avec i!ik/,, Le tenseur $*,, (6.27) obtenu tout d'abord par M. H. WEYL [(124)] s'appelle le tenseur projectif de WEYL. Dans le cas d e la connexion projective normale les 6quatfons des geodesiques prennent la forme suivante, en vertu des equations (6.23):

44

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

439

Le paramktre f dCfini ici est le ,,preferred projecfiue normal parameter” de M. L. BERWALD [(l)].

Chapitre VII RELATIONS ENTRE LA THBORIE DE M. E. CARTAN ET CELLE DE M. T. Y. THOMAS.

Jusqu‘a present, nous n‘avons employ6 que les repbres que M. E. CARTAN appelle les repbres semi-naturels. Ces repbres 6tant adopt&, on a

M. T. Y. THOMAS a employ6 un repkre qui a, en outre, la propri6tC * II:, = 0,

(7.2) OU

*]I!i k z Z * w /I,i , - hl’p J I;’ * q,- p i du‘, , I _ *

* w i. --* wil( du”. I Ce repire est le rep8re nafurel de M. E. CARTAN [(lo), (21)]. II est bien facile de choisir un repkre nature1 parmi les re-

-

peres semi naturels. En effet, faisons un changement de la forme pidu‘,soit:

* p . == p . f @; , alors on aura

* Ir!’ Jfi

= I I ! - i-j!+,( - b;k+i, Ill

donc, pour avoir

* I[:, = 0,

45

I

KENTARO YANO

440

nous n'avons qu'a prendre .= -1-- 1" (7.3) I n - ; 1 Ji

et par consequent

sont les composantes de la connexion projective de M. T. Y. THOMAS "941, (lol)]. Le rephre nature1 etant ainsi choisi, voyons comment se transforment les fonctions *Ilyk et *lijl: pendant une transformation de coordonnees

Les

(7.6)

2'r= SI ( u ' ,. . , , u" ) .

On sait que les fonctions p i et IIj, se transforment respectivement e n pi et

Uj,.d'apres

la loi d e transformation,

Des dquations (7.8),on tire la loi de transformation des fonctions I:, , (7.9) Oii

(7.10) donc on a la loi suivante de transformation des fonctions (Pi

;

(7.11) Or, nous avons les equations definissant la connexion projective dans les deux systbmes de coordonnkes :

46

CONNEXION PROJECTIVE ET GEONETRIE DES PATHS

441

I

(7.12)

1

et

(7,13)

1

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

Comme on obtient des kquations (7.7) et (7.11) 1

-

on a, en remarquant que * A , ,= *A,,I I

d‘ou resultent les formules :

1 *&=*AO

I

(7.15)

qui nous donnent la loi de transformation du repere nature1 quand nous effectuons une transformation de coordonnees (7.6). Les formules (7.15) ktant obtenues, on peut trouver sans difficult6 la loi de transformation des fonctions *w;~, et *mi,{ : en substituant les formules (7.15) dans (7.13) et en comparant les coefficients de *A, , * A , *A,, on a 3 . .

47

KENTARO YANO

442

1

I

I

1

Donc on peut dire que : Pour Qtudier le cas de M. T. Y. THOMAS, il faut considerer toujours la transformation de coordonnhes non holonome e t holonomes, 1 dr" = du" + d log A " (7.20) u' =U' (u', u?, . . . , u " ) ,

-

I-

+

I,

qui entraine line transformation de repdre nature1 (7.15) et de fonctions *II,"k et * I l i k (7.18) et (7.19): d log A 2 1 6tant toujours une diffkrentielle exacte, la loi de transformation peut s'kcrire

(7.21)

5;o = u"

I U'

= i'

+ n +1l

log 1 ,

(u', u",, * , , u" ) ,

et on obtient le groupe de transformations

THOMAS [(95),(98),(101)].

48

*G de M. T. Y.

CONNEXION PROJECTIVE ET GBOMkTRIE DES PATHS

443

Nous supposerons dans la suite que les fonctions lIyk et et par suite *n;/(et *11$ soient symetriques par rapport a u x indices inferieurs et la variable u0 soit holonome; alors on voit facilement, d'apres (7.18) et (7.191, que cette propriete se conserve toujours pendant la transformation (7.21). Cela pose, considerons les quantites *Qijk, *B:j, et *i!;/rl, f ormees avec les grandeurs asterisees, et correspondant respeciivement aux blfjk, s!~,~: et nj, ,

111;.

* ( ) I .

--o,/L

(7.24)

-*I]'. - *I]',. /k 4.1

*6!;k/l

= *Ii;/:/,

Or, les quantites

7

t *II;I/:b;! - *ll;/Jhk. .

et *Cjh, corespondant respectivement aux grandeurs Sij/: et Ci/:ilsont donnees par *Sij/;

(7.25)

*I!jk

(7.26)

*c/k/,=*1!;/:/,+

*IIj/i

+ ( n - 1)*II;/:~ *
,

*OU I1

(7.26)

*Hj/;

Supposons que (7.21) c'est - a dire (7.28)

-

= *IIj/
= 0,

*i!jk= 0,

c e qui suppose que la connexion soit normale ; on a alors d'apres la formule (7.25)

49

KENTARO YANO

444

(7.29) En substituant ces 6quations dans (7.24), on obtfent : (7.30)

* l l ; L / l t..,A,,--

* ( ) I

1

I1

*

- 1 ( 1 I,

~~

Cela pos6, calculons les *II;rl,,

q; - * lIJk q

d

.

En substituant (7.5) dans

on trouve

Oli

En contractant dam (7.32) par rapport aux indices i et h, on obtient encore (7.34)

n Ilj, 4-II,, *I[.=-/It n f 1

i- (n- 1) A;,,,.

En substituant enfin (7.32) et (7.34) dans (7,301, on a 1

*qiql=qh/) + n-1

(7.35)

1

( n I I, (nI$/, f II,,,,) h; -__ nA-1

ainsi a-t-on retrouve le tenseur d e M . H. WEYL. Remarque. Pour obtenir la transformation (7.36) e t les composantes de la connexion projective

50

+ I I L j ) hi, - I -

CONNEXION PROJECTIVE ET GEOMkTRIE DES PATHS

445

(7.37)

introduites par M. T. variable uo par

Y. THOMAS, nous

n'avons qu'B definir la

du" = - (n f 1 ) *p,( du" , alors la transformation de la forme *o( 1

*P'{du"i -+ *p/: -td log peut 6tre represenfee par

d-U u -duu-dlogl, -

u" = u" - log 1 .

Si l'on ddfinit

*IIg

et *IIi/: par

"II'.' / '<-=- ( n f l)*w;/: , * I I f ./ ; = *W)'{ - 0; *PI{ ,

les Cquations (7.24) et (7.25) prennent la forme suivante

-

donc : quand la connexion projective est normale, c'est - a dire *Gi/: = 0,

on trouve les formules:

initiulement introduites par M,

T. Y. THOMAS [(95), (981,(101)l.

Revenons a notre cas et considbrons les equations des gCodCsiques

51

KENTARO YANO

446

(7.38) en substituant (73) dans (7,381, on obtient

Effectuons miintenant une transformation de param8tre dkfinie par (7.40)

- -

c'est a dire par (7.41) alors les Bquations (7.39) deviennent : (7.42)

Le paramdtre p introduit ici est le pararndtre projectif de

M. T. Y. THOMAS [ (94), (96)1,

En dkrivant le premier membre de l'kquation (7.40) par rapport au paramktre s, on a

d'ou, en tenant compte de

on obtient

52

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

447

D'autre part, on a d'apres (7.40)

(7.44) donc on obtient finalement, en ajoutant les Cquations (7.43)et

(7.44) et tenant compte de (7.33), (7.45) Les fonctions Ajn n'6tant pas en general les composantes d'un tenseur, le deuxikme membre de 1'6quation (7.45) n'est pas invariant par rapport a la transformation de coordonnCes, donc on voit que Ie paramitre projectif de M, T. Y. THOMAS n'est pas invariant par rapport a la transformation des coordonnees. Cherchons cette loi de transformation. Supposons que les equations -des geodksiques dans le systkme de coordonnees 3' , U' , ,, u" , soient

.

(7.46) Alors, les equations (7.42)peuvent s'kcrire

d'ou on a, en tenant compte de la loi de transformation des fonctions *llh (7.19),

53

KENTARO YANO

448

Donc on a, en tenant compte de (7,461, I

d'oU

(7.47)

M. T. Y. THOMAS [(94), (9611 a

remarque que si l'on considkre la classe de transformations (7.21) qui satisfait a la condition A = 1, " = _u", . (7.48) u' = u' ( u ) ,

{ I.

les Bquations (7.18) et (7.19) se rhduisent aux suivantes :

ce qui dit que les fonctions *ll$ se transformed comme composantes d'une connexion affine et les fonctions *U;k comme les composantes d'un tenseur a f f h e du second ordre ; et il a nommB la gBomBtrie des paths vis-a-vis de ces transformations la Geomitrie equi-projective des paths. Cela Btant, nous allons montrer que Ze paramitre project i f normal est aussi defini au moyen des *Ilj/&et p , par (7.50) [voir L. BERWALD (l)]. En effet, on a d'apres les Bquations (7.4) et (7.33) *lI'.', I 1. = I I;/; - A,

,

donc on a

en raison de (5.12),(7.45) et de la formule connue (5.10) sur l a la dBrivBe schwarzienne.

54

CONNEXION PROJECTIVE ET GBOM~TRIEDES PATHS

449

Chapitre VIII REPRESENTATION DES ESPACES A CONNEXION PROJECTIVE.

On doit B M. T. Y. THOMAS [(95), (981, (loo], (101)] l'introduc tion d'une variable surnumdraire t10 pour ramener I'Ctude des espaces a connexion projective a n dimensions, a l'ktude 1 dimensions. des espaces a connexion affine B n D'autre part, M. D. VAN DANTZIG [(23), (24), (25)] a introduit n 4-1 coordonnkes homogknes pour d6crire I'espace projec1)" fonctions Hit, d e ces tif genkralisk B n dimensions et ( n coordonnkes homogknes pour dbfinir la d6rivee covariante. M. J. HAANTJES [(42)] a tout recemment montrC que l'on peut aussi la Geobien ktudier, d u point d e vue d e M. D. VAN DANTZIG, mdtrie projective des paths de I'Ecole de Princeton. Nous allons, dans ce dernier Chapitre, examiner les propri6tQ dzs espaces B connexion affine employes dans ces deux theories pour representer les espaces B connexion projective et montrer que Ies espaces a connexion affine de MM. T. Y. THOMAS et J. HAANTJES ont les mimes proprietes caractiristigues. Considerons une vari6tC a connexion affine B n 1 dimensions, rapportbe a un systeme d e coordonn6es u?-dont les comh posantes de la connexion sont ,JI , et un champ de vecteur contrevariant ih et supposons que cette variete connexion affine vtrifie les trois conditions suivanfes :

+

+

+

oh le point-virgule reprksente la d6rivke covariante et I I :\*,,,, les composantes d u tenseur d e courbure, soit A y,.,u, = 1 I!:*/,,*, 1.

-

Il,;w,v

Oil

La premiere condition veut dire que la connexion est sans torsion. La deuxieme signifie qu'il y a un point invariant par rapport a u groupe d'holonomie.

55

KENTARO YANO

450

En effet, en attachant, B chaque point

M

de la variCte

-b

le repere nature1 e i , on a

*+ = du '*ej. - du i.-' ej. = 0 .

On peut donner une autre interpretation B la deuxieme condition. Considkrons une courbe engendrbe par le champ de vecteur , c'est-a-dire dkfinie par

-

cette courbe est auto parallele. En effet,

Nous appelons ces courbes rayons, avec M, J. H. C. WHITEHEAD [( 129)]. La troisieme condition avec la premiere et la deuxieme expriment que cette variktk a connexion affine admet une collinkation affine. En effet, une condition nkcessaire et suffisante pour que cette varikte admette une collindation dans la direction ck est

Mais le premier terme dans le premier membre est nu1 cause de la premiere condition et de la deuxibme. La considkration gbnkrale 6tant faite, examinons mainteet de M. J. HAANTJES. nant les cas de M. T. Y, THOMAS Prenons d'abord le cas de M. T. Y. THOMAS et J. H. C.

a

WHITEHEAD.

On choisit u t systkme d e coordonn6es par rapport auquel le vecteur
= h; ;

alors les rayons peuvent btre represent& par

56

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

45 1

u" = arbitraire, u' = constantes. Pour rester toujours dans le systeme de coordonnees realisant la condition (8,1J, nous ne devons considerer que les transformations de la forme, -

(8.3)

u" = u"

{-,u

t 11) ( u ' ,. , ,, LZJ'),

- ( u ' ,, , ,, Uj').

== u L

Le systeme de coordonnees etant ainsi choisi, les trois conditions prennent la forme suivante (8.4)

ll;.,

= II;!:,

(8.5)

Ill,,, = h, ,

i,

h

A

I I,,,,,,

(8.6)

= 0.

Ce sont les conditions posees par M. T. Y. THOMAS et J. H, C. WHITEHEAD sur l'espace affine a n t 1 dimensions qui repr6sente un espace projectif a n dimensions. Cela dit, prenons dans la vari6t6 a connexion affine, une gkod6sique definie par

ou par (8-8)

et

et considerom la surface formee par les rayons qui rencontrent cette geod6sique. Si l'on effectue sur cette surface le dkplacement de points (8.10)

U" -+ ti"

u'

+ constante,

-+ U l ,

les 6quations (8.8) et (8.9) ne changent pas, donc les gbod6siques se transforment en gkodksiques par le d6placement (8.10)

57

KENTARO YANO

452

et le parametre affine reste invariant pendant ce ddplacement. Cela 6tant, nous allons montrer que l'on peut ddterminer une fonction o ( t ) qui n'est pas constante, telle que le ddplacement (8.11) transporte les geodesiques en g6odSsiques.

En eifet, on a de (8.11) u i =:* l l A -

(8.12)

q

ci

(f);

le point u' decrivant une gboddsique, portons (8.12) dans (8.7), alors on obtiendra (S.13)

d2*"Uh -

dt'

-

-1 11". d*ui'. d*UV - 2 dn d*u" !?'I df dt dt dt2 .-

dt'

Pour que cette courbe soit aussi gbodesique, il faut e t il suffit que le dernier terme du premier membre ait la forme

d'0U (8.14)

La solution gdndrale de cette dquation diffdrentielle est (8.15)

6

=-

log ( x t 4-p),

Donc, si l'on se donne deux points sur la surface, on peut determiner les constantes CI et P de manikre que l'on ait une et une seule geoddsique se trouvant entikrement sur cette surface et passant par ces deux points. En portant (8.15) dans (&13), on obtient

Le parametre affine *t pour la gkodbsique (8.16) est dkfini par

58

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

453

ou par

dt d'ou

donc on voit que le parametre affine t subit, dans ce cas, une transformation homographique, D'aprks les considkrations faites ci-dessus, on peut dire que la surface engendrke par Ies rayons qui rencontrent une ghodksique est totalement gkodksique. Si l'on prend l'espaoe a connexion affine de cette sorte et fait correspondre, a un rayon de cet espace affine, un point de la varikih a connexion projective, alors la gkodksique de la variktk projective peut 6tre reprksenthe par une surface totalement gkodhsique. Alors, le parametre t correspond a notre param6tre projectif normal. Car, comme M. J. H. C. WHITEHEAD I'a deja remarqu6 [(129), p, 3451, on volt de la condition (11) que tous les rayons se rencontrent en un m6me point qui peut etre considhrk comme ,,pointideal" de l'espace a connexion affine, donc Ie rapport anharmonique des quatre valeurs de t reprbsentele rapport anharmonique des quatre rayons qui se rencontrent en un meme point, par consequent le rapport anharmonique des quatre valeurs de t reprhsente celui des quatre points de la variktk a connexion projective qui sont reprksentks par les rayons. Passons ensuite au cas de MM. D. VAN DANTZIG et J.

HAANTJES, On prend un systbme de coordonnkes par rapport auquel on a (8.17)

;" - _ ,Ii.

59

;

KENTARO YANO

454

alors les rayons peuvent 6tre reprksentks par (8.18)

ti'

= C' r ,

ou les Ch sont constantes est r est parametre. Pour rester toujours d a m le systeme d e coordonn6es r6alisant la condition (8.17), on ne doit consid6rer que les transformations de la forme --A u -u

-2.

(8.19)

(u),

( u ) sont des fonctions homogenes de degrd 1, parce ou les que des equations -I. 6 = U' , -- A

-A

c = u

,

l'on tire (8.20)

Cela ktant, les trois conditions (I), (11) et (111) prennent la forme sutvante : 1

(8.21)

lll?./ = I I

h '/I\

I

1

Ill,.>/ u" =0 ,

(8,22) (8.23)

A

(8.23) exprimant que les fonciions II,,, sont homogenes de degrk 1. Ce sont les conditions posdes par M. J. HAANTJES [(42)] pour ktudier, du poSnt de vue de M. D. VAN DANTZIG, la Gkom6trie projective des paths. Dans ce systeme de coordonn6es, les dquations des paths etant donndes par

dPU;. I. dulL du' _ -- 3: u)' drz -k ll;('/p dr dr

--

60

-1-

13 duh __ ' dr

CONNEXION PROJECTIVE

ET GhOMhTRlE DES PATHS

455

on peut choisir un paramktre f et une fonction 0 ( t ) tels que I'on puisse Ccrire les 6quations sous la forme

Le parambtre t correspond a notre paramktre projectif normal.

Remarque. Dam leur thdorie des espaces a connexion D. VAN DANTZIG, J. HAANTJES projective, MM. J. A. SCHOUTEN, et ST. GOLAB,en ernployant les coordonn6es hornogenes, ont h suppose seulement que les composantes de la connexion 111,,, sont des fonctions homogbnes de degrk 1, c'est-&-dire la condition (8.23). Comme ils ne considkrent que les transformations de coordonn6es = (u)

u'.

ou les i'. sont homogbnes de degr6 1, les uh sont les composantes d'un vecteur contrevariant, donc de 1 -tu" ll,t, A u';*,- u,:/

i.

= h,)

4-u'

'L

h

ll!t.,

on conclut que (8.24)

1.

1.

UI'

1.

lIlV/ = u : . / - h,

sont des composantes d'un affineur. Pour mettre (8.23) sous une forme tensorielle, formons

et substituons, dans ces kquations, (8.23) et

qui s'obtiennent de (8.24). Alors on aura (8.25)

61

456

KENTARO YANO

Donc, on voit que l’espace a connexion affine employe par M. D. VAN DAYTZIG est un peu plus general que celui de

M. T. Y. THOMAS.

Mais, si l’on pose les conditions suivantes dans la thdorie de M. D. VAN DANTZIG

pour etudier la gbom6trie projective des paths, ces deux espaces sont caracterises par les msmes proprietes (J), (11) et (111). BIBLIOGRAPHIE. BERWALDL,

-

1. On the projective geometry of paths, Ann. of Math. 37, 879 898, 1936,

BOKTOLOTTI E.

-

2. Connessioni proiettive. BoLI. Unione Mat. ltal. 9, 288 294, 1930 ; 10, 28-34 e t 83-90, 1931.

3. Differrntial invariants of direction and point displacement. Ann. of

-

Math. 32, 361 377, 1931. 4. Spazi proiettivainente piani, Ann. Mat. pura appl. 11, 111 134, 1932.

-

-

5. Sidle connessioni proietlive. Rend. Circ, Mat. Pa’ermo 56, 1 57, 1932,

BORTOLOTTIE. et HLAVATYV, 6. Cont ibuti alla tzoria delle connessioni I. Connessioni proiettiva. Ann, Mat. pura appl. 15, 1-45, 129 154, 1936,

-

CARTANE. 7. Levons sur les invariants intbgraux. Paris, Hermann, 1922. 8. Stir les espaces gbnbralisbs e t la thborie d e la relativitb. C. R. Ac. Paris 174, 134 731, 1922. 9, Stir les varietbs a connexion affine et la theorie d e la relativite genbralisee. Ann. kc. Norm. Sup. 40, 325 412, 1923. 10. Sur les varibtbs i connexion projective. Bull. SOC.Math. do F l a n c e 52,

-

-

-

205 241, 1924.

-

11. Sur la connexion projective des surfaces. C. R. Ac. Paris 178, 750 752, 1924. 12. Les rbcentes gbnbralisations d e la notion d’espace. Bull. Sc, Math. 48, 294 320, 1924.

-

62

CONNEXION PROJECTIVE ET GEOMETRIE DES PATHS

457

13. Lz thkorie das groupes et les recherches recentes d e geometrie differentielle; L'Enseignement Math. 24, 1 - 18, 1925, 14. La geometrie des espaces d e Riemann. Memorial des Sc. Math, 1925, 15. La theorie des groupes e t la geometrie. L'Enseignement Math, 26, 200 225, 1927. 16. Rapport s u r le MPmoire d e J. A. Schouten intitule ,,Erlanger Programrn und ifbertragungslehre. N e u e Gesichtspunkte zur Grundlegung d e r Geometrie" Bull. SOC.Phys Math. Kazan 2,

-

71 -76, 1927,

17. Levons sur la geometrie des espaces d e Riemann, Paris, Gauthier-Villars, 1928. 18. Le calcul tensoriel en geometric projective. C. Ac. Paris. 198, 2033 2037, 1934. 19. La mkthode du repere mobile, la theorie des groupes continus et les espaces g6nkraIis6s. Paris, Herman, 1935. 20. Le calcul tensoriel projectif. Recueil m a t h h a t i q u e SOC.Math. Moscou 42, 131 - 147, 1935. 21. Lecons sur la thkorie des espaces a connexion projective. Paris Gauthier - Villars, 1937. C H U R C H A. 22. On the form of differential equations of a system of paths. Ann. of Math. 28, 629 630, 1927.

-

DANTZIGD. VAN 23. Theorie d e s proiektiven Zusammenhangs n - dimensionaler Rauine. Math. Ann. 106, 400 - 454, 1932, 24 Zur allgemeinen projektiven Dif:erentiaIgeometrie I e t 11. Proc. Akad. Amsterdam. 35, 524 - 534, 535 542, 1932, 25. On the general projective differential geometry, 111. Proc. Akad. Amsterdam, 37, 150 - 155, 1934.

-

DOUGLASJ, 26. The general geometry of paths. Ann, of Math. 29, 143 - 168, 1928. EISENHARTL. P, 27. Spaces with corresponding paths. Proc. Nat. Acad, Sc. U. S. A . 8, 233 - 238, 1922. 23. Fields of parallel vectors in the geometry of paths. Proc. Nat. Acad. SC. U. So A. 8, 207 - 212, 1922. 29. Affine geometries of paths possessing a n invariant integral, Proc. Nal. Acad. Sc. U. S. A. 9, 4-7,1923, 30. The geometry of paths and general relativity. Ann. of Math. 241 367 - 392,1923. 31. Geometries of paths for which the equations of the paths admit a quadratic first integral, Trans. Amer, Math. SOC. 26, 378 - 384, 1924,

63

KENTARO YANO

458

32. Riemannian Geometry. Princeton University Press, 1926. 33. Geometries of paths for which the equations of the paths admit n (n $- 1)/2 independent linear first integrals. Trans. Amer. Math. SOC,28, 330-338, 1926. 34, Non Riemannian Geometry. Amer. Math, SOC,Coll. Public. VIlI 1927. 35. Projective normal coordinates. Proc. Nat, Acad. Sc. U. S. A. 16, 731740, 1930.

EISENHART L. P. et KNEBELMANM, S. 36. Displacement in a geometry of paths which carries pats into paths. Proc. Nat. Acad. Sc. U. S. A. 13, 38-42, 1927. E~SENHART L. P, e t VEBLEN0. 37. The Riemannian geometry and its generalisation. Proc. Nat. Acad. Sc. U. S. A. 8, 19-23, 1922,

EYRAUDH. 38. Sur l e caractere riemannien projectif du champ gravifique electromagnetique. C. R. Ac. Paris, 180, 127- 130, 1925.

FRIESECKE H. 39. Vektoriibertragung, Richtungsubertragung, Metrik. 101 118, 1925. FUBINI G. et CECH E.

Math.

-

94,

Ann.

-

40, Gdometrie projective differentielle. Paris, Gauthier Villars, 1931.

GOLABST.

-

41. Uber verallgemeinerte projektive Geometrie. Prace mat. fiz. Warszawa. 37, 91 153, 1930.

-

42,

HAANTJESJ. On the projective geometry of path.. Proc. Edinburgh Math. SOC.

-

5, 103 115, 1937.

HACHTROUDI M. 43. Les espaces d'klements 1 connexion projective normale (These). Paris. Hermann, 1937.

HESSENBERG G, 44. Beispiele zur Richtungsubertragung. Jahresbericht der D. M. V. 33, 93-95. 1924.

HLAWATYV.

-

45, Sur les deplacements isohodoiques. L'Enseignement Math. 26, 84 97, 1927. 46. Bernerkung zur Arbeit von Herrn T. Y. Thomas ,,A projective theory of affinely connected manifolds". Math. Zeitschr. 28. 142 146, 1928. 47. Projektiven Invarianten einer Kurvenkongruenz und einer Kurve. Math. Zeitschr. 34, 58-73, 1931. 48. Invariants projectifs differentiels d'une courbe dans I'espace projectif Pti-I Accad. Naz. Lincei Rend. Roma, 16, 109- 114, 206 211, 299 304, 1932.

-

-

-

64

CONNEXION PROJECTIVE ET GEOM6TRIE DES PATHS

459

49. Connexion projective et deplacement projectif. Ann. Mat, pura appl.

-

12, 217 294, 1933. 50. Invariants projectifs d'une hypersurface. Rend. Circ. Mat. Palermo. 57.

-

402 430, 1933. 51. Uber eine Art der Punktkonnexion. Math. Zeitschr. 38, 135 145, 1934, 52. Espaces abstraits courbes de Konig. Rend, Circ. Mat, Palermo, 59, 1-39, 1933. 53. Systkme complet des invariants differentiels projectifs d'une courbe dans un espace projectif courbe. Abhandlungen des Moskauer Seminar fur Tensoranalysis 2 3, 13 - 50, 1935.

-

-

HLAVATYV , e t GOLABST. 54. Zur Theorie der Vektor - und Punktkonnexion. Praces mat. fiz. Warszawa, 39, 119 129, 1932.

-

HOFFMANNB. 55. Projective relativity and the quantum field. Phys. Review 37, 8 8 - 8 9 , 1931.

-

56, Projective relativity and the Einstein Mayer unified field Phys. Review. 43, 615 619, 1933.

-

theory.

KANITANIJ. 57. GBometrie differentielle projective. Ryojun Coll. eng. 1931,

KNEBELMAN M. S. 58. Groups of collineations in a space of paths. Proc. Nat. Acad. Sc. U. S. A. 13, 396 - 400, 1927. 59. Collineations of projectively related affine connections. Ann. of Math. 29, 389 -394, 1928.

KON~GR. 60. Beitrage zu einer allgemeinen Mannigfaltigkeitslehre. Jahresbericht der D. W. V. 28, 213-228, 1920.

LEVI- CIVITAT.

0

61. Der Absolute Differentialkalkul. Berlin, Julius Springer, 1928.

LEVY H. 62. Normal coordinates in the geometry of paths. Proc. Nat. Acad. Sc. U. S. A, 16 492 496, 1930.

-

MICHALA. D. 63. Projective integral invariants attached to the trajectories of differential systems. Bull. Amer. SOC.37, 441 454, 1931.

-

64,

NEWMANM. H. A. A gauge -invariant tensor calculus. Proc. Royal SOC. 116,603 - 623, 1927. PAULIW.

65, u b e r die Formulierung der Naturgesetze mit fiinf homogenen Koordinaten. Ann. der Physik 18,305 336, 1933.

-

65

KENTARO YANO

460

ROBERTSON H. P. 66, Note on projective coordinates. Proc. Nat. Acad. Sc. 154, 1928.

U. S. A. 14, 153-

,ROBERTSON H.P. et WEIL H. 67. On a problem in the theory of groups arising in the foundations of infinitesimal geometry. Bull. Amer. Math. SOC.35, 686 - 690, 1929.

SCHOUTEN J. A.

-

68. Der Ricci Kalkiil. Berlin Springer, 192!. 69. Sur les connexions conformes et projectives de M, Cartan et la con-

nexion lineaire generale d e M. Konig. C. R, Ac. Paris 178, 2044 2046, 1924. 70, On the place of conformal and projective deometry i n the theory of linear d splacements. Proc. Acad, Amsterdam 27,407 424, 1924. 71. On the condition of integrability of covariant differential equations. Trans. Amer. Math, SOC.27, 441 473, 1925. 72. Erlanger Programm und Ubertragungslehre. Neue Gesichtspunkte zur Grundlegung der Geometrie. Rend. Circolo Mat, Paler1110 50, 142 169, 1926.

-

-

-

-

73, Uber die Projektivkriimmung und Konformkriimmiing halbsymmetrischer Ubertragungen. Bull, SOC. Phys, Math. Kazan. 2, 90 -98, 1924. 74. Projective and conformal invariants of half symmetrical connections. Proc. Akad. Amsterdam. 29, 334 - 336, 1926.

-

75. Uber nicht holonome ifbertragungen in einer 1.9 172, 1929.

-

L w , Math. Zeitschr 30,

-

76. Zur generellen Feldtheorie. Zeitschr, fur Physik. 81, 129 138 , 1931 ( 0 .F. IV), 405 417 (G. F. V.); 84,92 111, 1933 (0.F. V11).

-

-

77. La theorie projective d e la relativite. Annales Inst. 1935. SCHOUTEN

J. A. e t DANTZIGD.

H.Poincare

5, 51 - 88,

VAN.

78. Uber eine vierdimensionale Deutung der neusten Feidtheorie. Proc, Akad. Amsterdam 34, 1398 1407, 1931.

-

79. Zum Unifizierungsproblem der Physik, Skizze einer generellen Feldtheorie. Proc. Akad. Amsterdam. 35, 642 - 655, 1932. 80. Zur generellen Feldtheorie 1932.

( 0 ,F. 111). Zeitschr. fur Physik. 78,639 - 667,

81. On projective connections and their application to the general field theory [G. F, VIJ, Ann. of Math. 34, 271 -312, 1933. S C H O U T t N J. A. e t GOLAB ST.

82. Uber projective ubertragungen und AbleitungeG I, Math. Zeitschr. 30, 149 172,1929 ; 11, Ann. Mat. pura appl. .8, 141 157, 1931

-

-

66

CONNEXION PROJECTIVE ET GEOMETRfE DES PATHS

461

J. A. e t HAANTJES J. 83. Autogeodatische Linien und Weltlinien (G. F. VIII) Zeitschr. fiir Physik. SCHOUTEN

-

89, 357 363, 1934. 84. Zur allgemeinen projektiven Differentialgeometrie. Comp. Math. 3, 1 51, 1936.

-

SCHOUTEN J. A. e t HLAVATYV. 85. Zur Theorie der allgemeinen linearen ubertragung. Math. Zeitschr, 30, 414 432, 1929.

-

SCHOUTEN J. A. et K A M P E N E. R

VAN.

-

86. Zur Einbetlung - und Kriimmungstheorie nicht holonobner Gebilde. Math. Ann, 103, 752 783, 1930.

-

SCHOUTEN J. A . e t STRUIK D. J 87. Einfuhrung in die neueren Methoden der Differentialgeometrie. Groningen, Noordhoff, 1935.

STRUIK D. J. 88. On the theory of linear displacement. Bull. Amer. Math. SOC.33, 523564, 1927. 89, The theory of linear connections. Berlin, Julius Springer, 1934.

THOMASJ. M. 90. Note on theprojective geometry of paths. Proc. Nat, Acad. Sc. U. S. A. 1 I , 207 - 200, 1925. 91. On normal coordinates in the geometry of paths. Proc. Nat. Acad. Sc. U. S. A. 11, 58 63, 1926. 92. First integrals in the geometry of paths. Proc. Nat. Acad. Sc: U. S. A. 12. 117 124, 1926. 93. Asymmetric displacement of a vector. Trans. Amer. Math. SOC.28, 658 670, 1926.

-

-

-

THOMAS T. Y. 94. On the projective and equiprojective geometries of paths, Proc. Nat. Acad. Sc. U. S. A. 11,199 - 203, 1925. 9 j, Announcement of a projective theory of affinely connected manifolds. Proc. Nat. Acad. Sc. U,S. A. 1 1 , 588 589, 1925. 96. On the equiprojective geometry of paths Proc. Nat. Acad. Sc. U. S. A. 1 I , 59% 594, 1925.

-

-

97. Note on the projective geometry of paths. Bull. Amer. Math. SOC. 31, 318 322, 1925. 98. A projective theory of affinely connected manifolds. Math. Zeitschr. 25, 723 733, 1926. 99. The replacement theorem and related questions i n the projective geometry of paths. Ann. of Math. 28, 549 561, 1927. 100. Concerning the *G group of transformations. Proc. Nat. Acad. Sc. U. S, A. 14, 728 - 734, 1928.

-

-

67

KENTARO YANO

462

101. The differential invariants of generalized spaces. Cambridge University Press, 1935. 102. On normal coordinates. Proc. Nat. Acad. Sc. U. S. A. 22, 309 312, 1'336.

-

VEBLEN0. 103. Projective and affine geometry of paths. Proc. Nat. Acad.Sc, U. S. A. 8, 347-350. 1922, 104, Normal coordinates for the geometxy of paths. Proc. Nat. Acad. Sc. U. S. A. 8, 192 197, 1922, 105. Equi affine geometry of paths. Proc. Nat. Acad. Sc. U. S. A. 9, 3 4, 1923. 106. Remarks on the foundation of geometry. Bull. Amer. Math, SOC.31, 121 141, 1925. 107. Invariants of quadratic differential forms. London Cambridge Tract. 1927. 108. Differential invariants and geometry, Atti Congress0 Internazionale Bologna 1, 181 189. 109. Projective tensors and connections. Proc, Nat. Acad. Sc. U. S. A. 14, 154 166, 1928. 110. Generalized projective geometry. Journal of London Math. SOC.4, 140160, 1929, 111. A generalisation of the quadratic differential form, The Quarterly Journal of Math. 1, 60 76, 1930. 112. Projektive Relativitatstheorie. Berlin Springer, 1933.

-

-

-

-

-

-

-

VEBLEN0. et HOFFMANNB. 113. Projective relativity. Physical Review, 36, 810 822, 1930.

-

VEBLEN0. et THOMAS J. M. 114. Projective normal coordinates for the geometry of paths. Proc. Nat. Acad, Sc, U. S. A. 11, 2n4 207, 1925. 115. Projective invariants of affine geometry of paths. Ann. of Math. 27, 279 296. 1926.

-

-

VEBLEN0. et THOMAST. Y.

-

116. The geometry of paths. Trans. Amer. Math. SOC.25, 551 608, 1923. 117. Extensions of relative tensors, Trans. Amer, Math. SOC. 26, 373 377, 1924.

-

VEBLEN 0. et WHITEHEAD J. H. C. 118. A set of axioms for differential geometry. Proc. Nat, Acad. Sc. 17, 551 561, 1931. 119. The foundations of differential geometry. London Cabridge Tract, 1932.

-

VRANCCANU G. 120. Les espaces non holonomes. Memorial des Sc. Math. 1936. 121. Sur une theorie unitaire non holonome des champs physiques, Journal d e Physique et le Radium 7, 514-526, 1936.

68

CONNEXION PROJECTIVE ET GkOMETRIE D E S PATHS

463

WEITZENROCK R. 122. u b e r projectiven Differentialinvarianten. VII, Proc. Akad. Amsterdam 35, 462 468, 1932. 123. Ube: den Reduk tionssatz bei affinem und projektivem Zusammenhang Proc. Acad. Amsterdam. 35, 1220 1229, 1932,

-

-

WEYL H. 124, Zur Infinitesimalgeometrie. Einordnung der projektiven und der konformen Auffassung. Gott. Nachr. 99 112, 1921. 125. Temps, Espace, Maticre. Paris Blanchard, 1922. 126. On the foundations of general inf;nitesimal geometry. Bull. Amer. Math. SOC.35. 716 -725, 1929.

-

J.H. c.

WHITEHEAD

$27. A method of obtaining normal representations for a projective connection. Proc. Nat. Acad. Sc. U. S. A. 16, 754 - 760, 1930. 128. On a class of projectively fiat affine connections. Proc. London Math. SOC.32, 93 - 114, 1931. 129. The representation of ptoiective spaces. Ann. of Math. 32, 327-360. 1931. 130. On linear connections. Trans. Amer. Math. SOC.33. 191 -209, 1931. 131. Convex regions in the geometry of paths. Quart. Journal Math. 3, 3342, 1932. 132. Affine spaces of paths. which a r e symmetricabout each point. Math. Zeitschr. 35, 644 659, 1932.

-

YANO

K.

133. S u r le changement des coefiicients d'une connexion projective. C. K. Acad. Paris 203, 637 639, 1937. 134, S u r les equations des geodesiques dans une variete B connexion projective. C. R. Ac. Paris 205, 829 - 831, 1937. 135. Sur la theorie unitaire non holonome des champs I, 11, Proc. Phys Math. S x . Japon. 19, 867 - 896, 945 976, 1937. 136. The non holonomic representation of projective spaces (sous presse).

-

-

69

464

KENTARO

YANO

TABLE DES MATI~RES.

.

Introduction

I. Aperqu historique

Page

.

.

I

.

11. Les espaces projectifs tangents

.

111. Les transformations des repkres projectifs

4

395 397 404 411

IV. Les transformations des composantes de la connexion projective 420

V, VI. VII, VIII.

Les Bquations des ,,paths" et le parambtre projectif normal

.

424

Le tenseur de courbure et de torsion de M. E. Cartan.

,

433

Relation entre la theorie de M.E. Cartan et celle de M. T. Y. Thomas 439 ReprBsentation des espaces B connexion projective

Bibliographie

.

I

,

449 456

70

Sur la thiorie des espaces a connexion conforme Par Kentaro YANO. Introduction.

La notion d’espace B connexion conforme a kt6 premierement introduite, en 1923, par M. Elie Cartan” en meme temps que celle d’espaces & connexion affine et projective. I1 considgre une variktk numkrique dont le voisinage de chaque point prksente toutes les propriktks d’un espace conforme proprement dit, et il l’appelle un espace B connexion conforme quand on se donne une loi qui nous permet de reccorder deux voisinages (deux espaces conformes) infiniment voisins. En employant sa belle mkthode du repgre mobile, M . E . Cartan a Qtabli sa thbrie des espaces A connexion conforme. D’autre part, en 1918, M. H. Weyl” a considkrk une transformation de la mktrique (0.1)

Sij = Pgij

d’un espace riemannien dont la forme fondamentale est ds2=gijdu‘duj. La transformation (0.1) change la dbfinition de longueur des vecteurs mais ne change pas celle d’angle entre deux vecteurs, de sorte que nous appelons (0.1) transformation conforme de la mktrique. M. H. Weyl a trouvk un tenseur qui reste invariant quand on effectue une transformation conforme (O.l), un tenseur qu’ on appelle actuellement tenseur conforme de courbure de M. Weyl. En remarquant que les symboles de Christoffel { / k } formks avec les gij se transforment, par la transformation (0.1)’ en (jik): (0.2) 1) E. Cartan (3). Voir la Bibliographie plac6e Ila fin de ce MBmoire. 2) H. Weyl (11, (2).

71

Kentaro Yano.

2

Oii

M. J. M. Thomas” a trouv6, en 1925, les quantiths

qui sont invariantes par rapport & la transformation (O.l), et en partant de ces symboles invariants Kjk, il a retrouve le tenseur conforme de courbure de M. H. Weyl. Pour trouver les invariants conformes d’un espace riemannien c’est&-dire les invariants d’un espaces de Riemann par rapport & la transformation (0.1)’ M. T. Y. Thomas” a introduit, en 1925, une densit6 tensorielle du poids - 2 ~

n

(0.4)

1 -

G.%J.=g.23./gn

oii g est le determinant form6 avec les gij. Comme g se transforme, lors de la transformation (0.1)’ en 8 d’aprgs ij = P V

il est evident que les Gij restent invariants pendant cette transformation. M. T. Y. Thomas a 6tudi6 les invariants de la forme relative quadratique Gdjduidd. Les symboles Kjk de M. J. M. Thomas coincident avec les symboles de Christoffel form& avec les invariants Gij de M.T.Y. Thomas. MM. J. A. Schouten et J. Haantjes3) ont recemment 6tudi6 la g k mktrie diffhrentielle conforme avec la methode projective & l’aide des coordonn6es homoghes de M. D. van Dantzig. Le but de ce MQmoire est de developper l’id6e fondamentale de M. &. Cartan de maniere qu’on puisse expliquer la th6orie analytique de 1’Ecole de Princeton de ce point de vue geometrique et d’ktudier la thkorie des courbes dans les espaces & connexion conforme. Dans la premiere partie de ce Memoire, nous allons 6tudier la relation entre la thbrie des espaces & connexion conforme de M. E. Cartan 1) J.M. Thomas (l),(2). 2) T. Y. Thomas (1). 3) J.A. Schouten et J. Haantjes (I), (2).

72

Sur la thborie des espaces 5 connexion conforme.

3

et la th6orie des invariants des espaces riemanniens de 1’Ecole de Princeton. Dans la deuxiBme partie, nous Btudierons les courbes et en particulier celles que nous appellerons les circonf6rences gBnBralis6es.l) Dans la th6orie des espaces B connexion affine et projective,” on a les courbes qu’on peut regarder comme une gkn6ralisation des lignes droites, mais dans la gkom6trie conforme la notion de lignes droites n’existe pas. Ce sont les circonfkrences qui leur correspondent, En partant de l’idke fondamentale de M. Cartan, on peut facilement parvenir B cette notion : on peut appeler circonfkrences g6n6ralisks les courbes qui deviennent les circonfkrences ordinaires quand on les d6veloppe sur l’espace conforme tangent attach6 B un point sur la courbe. Nous partirons cependant, dans le prBsent M6moireJ d’un autre point de vue. Nous allons d’abord chercher les formules de Frenet pour les courbes d’un espace B connexion conforme et en annulant une certaine courbure, nous obtiendrons les circonfhrences g6nQralisGes. En terminant, nous tenons B exprimer notre respectueuse gratitude B notre maitre M. Elie Cartan pour les conseils et les encouragements qui nous ont 6t6 pr6cieux au cows de ce travail.

a.

1. Espaces conformes.

ConsidBrons un espace euclidien B n dimensions En rapport6 a un systhme de coordonn6es obliques E l , E2, ..., En, la forme quadratique fontammentale Btant (1.1)

€2

=gijpp

.

( i , j , k ,...= 1,2, ..., n)

Alors, dans cet espace euclidien, une hypersphhre peut Btre reprbsentk par une Bquation de la forme: Xog,j€F- 2gijxEj42X” = 0 .

(1.2)

Nous dirons que les coefficients X o ,X’, ..., X”, X m (non tous nuls) sont des coordonn6es homoghes de cette hypersphgre. Pour trouver le 1) K. Yano. (2). 2) Voir, E. Cartan: Leqons sur la thhorie des espaces 5 connexion projective.

et

Paris, Gauthier-Villars, 1937. K. Yano: Les espaces 1 connexion projective et la gbm6trie projective des “paths”. Thbe, Paris, 1938.

73

Kentaro Yano.

4

centre et le rayon de cette hypersph&e, nous 6crivons (1.2) sous la forme suivante :

et on voit bien que les coordonnkes du centre sont est

Jg,xLxJ~~~Ox-=

xXOi

et le rayon

(xo)2

La condition pour que cette hypersphhre se rkduise 8 un point, c'est-&dire & une hypersph2re du rayon nu1 est (1.4)

g$3. . ~ X j - 2 X O X " = O .

Cela &ant, considbrons deux hypersphhres : (1.5)

et (1.6)

yOgijpEj- zgijy i c j + 2 y - = 0 ,

et cherchons l'angle d'intersection de ces deux hypersphhres. (1.5) reprksente une hypersphhre dont le centre est

xl XO

(+O)

et dont le rayon est 9

et (1.6) reprksente une hypersphhre dont Ie centre se trouve ii y" YO et dont le rayon est -

Donc en dksignant par 8 l'angle d'intersection, on a

c'est-&-dire

Par conskquent, la condition pour que deux hypersphares (XO,F', Xm)et (YO,Y",Y") se coupent orthogonalement est

74

Sur la thhorie des espaces A connexion conforme.

(1.8)

5

g ..x,xj- xo Y” - X” yo = 0 .

Dans ce raisonnement, nons avons supposQ que les rayons de deux hyperspheres ne soient pas nuls. Considkrons le cas oil le rayon de la premigre hypersphere est nul, celui de la deuxi6me ne 1’Qtant pas. Dans ce cas, la premiere sph6re se rQduit & un point. Nous allons chercher la condition pour que ce point se trouve sur la deuxi6me sphere. Sous la condition g , x i x j - 2X”X” = 0 ,

(1.9) (1.5) reprksente le point

Xi

xo

*

En substituant ces valeurs dans (1.6), on trouve

d’oil, on tire, en tenant compte de (1.9),

2YOX” XO

-

2 g i j x Y ’ +2Y“=O, XO

c’est-&-dire (1.10)

- p y a- x”y0 = 0 ,

gijx’yj

c’est la condition cherchbe. X u ,XI, . .., X , X” Qtant des coordonnkes homogenes de spheres, les substitutions 1inQaires portant sur les X” ( A ,B, C, ... =O, 1,2, ..., n, a), qui laissent invariante la forme gijxixj-

2XOX”

et, par consQquent, laissent aussi invariante la forme g i j x . yJ- xo y” - X“ yo

sont appelQes les transformations conformes. La gQom6trie conforme est la thQorie des invariants par rapport aux transformations de cette sorte. Pour la simplicith, nous introduisons, dans la suite, le symbole X pour reprbsenter la sph6re dont les coordonnQeshomoggnes sont X A , et Qcrivons comme il suit :

75

Kentaro Yano.

6

(1.11) (1.12)

x.x=g,,X'XJ - 2XOX" , x*Y = g L , x Y ' - x o Y " - x " Y o .

Alors, X . X = O signifie que la sphere X est au fond un point, X . X = O , Y .Y + 0, X . Y=O signifient que le point X se trouve sur la sphere Y et X . X + O , Y.Y+O, X . Y = O montrent que deux spheres X et Y se coupent orthogonalement. Cela ktant, prenons n + 2 spheres (en rkalitk un point, n plans et un point 2 l'infini) A0 (1, 0, 0, ...... 0, 0) A1 (0, 1, 0 , ....... 0, 0 ) , .............................. (1.13) A , (0, 0, 0, ...... 1, 0) , A , (0, 0, 0, ...... 0, 1)

1

1

9

.

9

alors on a, comme on peut facilement vkrifier, (1.14)

AoAO= AmA,= AoA A,A,=O , A , A , = g , , AoAm=-1. &=

Une sphi?re quelconque X dont les coordonnkes homogenes sont X A est reprksentke comme une combinaison lineaire des Ap (P,Q, R,... = 0, 1,2, .... n, 00) : (1.15)

X=XpA18.

Cela ktant, prenons un autre systeme de n S 2 spheres telles que (1.16)

AoAo= A,A,

= AoAj = A,Ai = 0

,

A,Aj=oij, AoA,= - 1 ,

si les A, sont linkairement indkpendants, X peut ktre reprksentk cornme une combinaison linkaire des A,:

x=X P A , .

x"

La transformation de X A en est une transformation conforme parce que AI, ayant la forme Ap= U$AU on a XQ=U$xz'e t

X.X=XPApXVAv=gijX"X'-2XoX",

x.x=XPA,,X"A,, =&,X"J

- -

- 2XOX"

,

Les formules (1.16) nous disent que A, et A, sont des points tels que AoA, = - 1 et A, sont n hypersphgres passant par ces deux points.

76

Sur la thhorie des espaces A connexion conforme.

7

Dans la suite, nous ne considkrerons que les systemes de rkfkrence form& par n f 2 hyperspheres satisfaisant 2 la condition exprim& par (1.16). 2. Les repsres mobiles conformes pour les espaces i connexion comforme.

Considerons une vari6tk arithmbtique & n dimensions, c’est-$-dire une varikt6 num6rique d6crit par un systeme de coordonnkes (ui)(i,j, k , ... = 1,2, ..., n) et supposons que le voisinage imm6diat de chaque point de la vari6t6 pr6sente toutes les propri6t6s d’un espace conforme ordinaire. Si l’on se donne, de plus, une loi qui nous permet d’accorder les espaces conformes ordinaires consid6r6s 5 deux points infiniment voisins de la variet6, on obtient ce que M. E. Cartan appelle la vari6te ZL connexion conforme. Cela &ant, prenons une vari6te b connexion conforme. Dans chaque espace conforme attach6 B un point courrant Ao, (nous l’appellerons l’espace conforme tangent) prenons un repere mobile (n+ 2)-sph&ique form6 avec deux points analytiques A. et A , (spheres dont les rayons sont nuls) et n spheres analytiques A i passant par A. et A,, alors on aura (2.1)

A:= A%= AoA;= A ,Ai = 0 .

Comme les spheres AP (P, &, R, ... =0, 1, 2 ..., n, m) sont d6termin6es ii un facteur pres, nous fixons ce facteur de telle maniere qu’on ait (2.2) AoA,= - 1, et posons (2.3) Un repere 6tant choisi en chaque point de la varikt6, une sphere

X dans un espace conforme tangent peut &re reprksentke comme une combinaison linkaire des spheres AP : (2.4)

+

+

X=XoAo+ X’A1+ *...*. X”A, X”A, ,

de la sphere X. Comme le choix de n t 2 spheres AoAl... A,A, est tout & fait arbitraire pour le moment & condition que le point A, coincide avec le point de contact, (c’est-&-dire le point auquel est attach6 l’espace con-

oh X p sont des coordonnkes (n+2)-sph6riques

77

Kentaro Yano.

8

forme tangent), on peut prendre un autre repere (n+2)-sphBrique form6 avec deux points A, et 2 , e t n spheres Ai passant par 2,et A,. En posant Ug les coordonn6es (n+ 2)-sph6riques de n+ 2 sphkres AQ par rapport au repere form6 avec Ap, on a

(2.5)

I

A0 = GAo+ U;AI+ ......+ UtA,+ U,"A,, A1 = U?Ao+ U:A1+ ...... UrA, + UTA, ...................................................... A, = ULAo+ UkA1+ . . * . - .+ UzA,+ UZA, , A,= UO,Ao+ UL41+*.*..*+ U2A,+ UZA,,

.

+

oh le determinant form6 avec les U s est suppos6 non nul. Comme nous avons suppos6 que le premier sphere-point coincide toujours avec le point de contact, A0 et A0 doivent coincider g6ometriquement, donc nous devons avoir

Ao=U:Ao et

u;+o, u;=u;=...... = U"0-

Des conditions on tire

U"0 - 0 .

AoAi= o et AoA,= - 1 purn...... = u,-=o 12 u:u2=1.

Si l'on pose on en tire de plus

A a.A3. = g .w. ?

u: u; gijwauc- u:u;=o, g ..ULu j,- 2 uo,UZ= 0 , Si.,= g a b

9

23

et on en conclut que la matrice (17;) a la forme : (2.6)

u,", 0 , uj, uj,

1% &I

(U$=

0

UL,

Oii

u;u:=1, (2.7)

giju;uj,- u:u2=0, gij ULUL - 2 uo,u 2=0 .

1) Comparer avec les rbsultats de M.O. Veb!en (3).

78

l)

Sur la th6orie des espaces

ti

connexion conformc.

9

Le repere ittant transform6 par

Ap= U$AQ,

(2.8)

oii les Ug satisfont B (2.6) et B (2.7), les coordonn6es (n+2)-sph&iques X p d6finissant une sphsre X :

X = X"A, se transforment d'apres

En tenant comptes des (2.6), (2.7) et (2.10), on trouve les relations auxquelles doivent satisfaire les Vg :

v:, v;, VL (V$ =

(2.11)

(2.12)

En comparant (2.7) avec (2.12), on trouve les relations entre les 616ments de la matrice (2.11):

I

(2.13) Oii

(2.14)

v:v:=1, g..v;vi- v:v:=o, g,T/wvi- 2 VL v: = 0 , %'

gij=&bu;u:.

Cela pos6, passons B la loi qui nous permet de raccorder en un seul espace les espaces conformes tangents attach& B deux points inu2,..., zP), finiment voisins de la vari6t6. Si l'on prend un point A . (d, un repere (n 2)-sphitrique form6 avec les Ao,Al, A*, ..., A,, A,, y est

+

79

Kentaro Yano.

10

attachk, et en un point infiniment voisin de A. est attach6 un autre repere (n+2)-sphkrique. I1 est nature1 de supposer que la transformation conforme de l’espace conforme tangent en A, en espace conforme tangent en un point infiniment voisin de A. soit celle infiniment voisine de la transformation identique. Done, si l’on fait le raccord des deux espaces conformes tangents, on aura des formules de la forme suivante:

(2.15)

Ao+dAo=(l+(*~)Ao+to~A1+...... +&A,$tt$’A,, A1 dAl= tofA0 (1 (0:) A1 - * . * * * coyA, wPA, ,

+

+ +

+

+ + +

+

............................................................... A, +dA,, = &A0 + (&A1 ...... (1 w:) A, + &Am , A,+ dAm=toO,Ao+ w L A ~ + &A,+ (1+to:) A, ,

+ +

. a * . * *

oti les to$ sont des formes de Pfaff par rapport aux coordonnkes ZG’, d,..., u”. Nous kcrivons (2.15) symboliquement comme il suit :

(2.16)

Ici, les formes de Pfaff (2.17)

I

tog

ne sont pas arbitraires, des relations

A;=A%=AoAi=A,Ai=O AoAm=-l, et AiA.=g.. 3 . L J ?

,

on tire les formules: 1fJ; = 0 , (2.18) ,*O,=o, (2.19) (2.20) 1o;gij - toy = 0 , (2.21) coQgij- to: = 0 , (2.22) to: t o 2 =0 , (2.23) tflggik+ coig;j= dgj,; . RepBres mobiles semi-natzcrels. La connexion conforme etant dkfinie par les formes de Pfaff cog, nous allons chercher la loi de transformations des tog par rapport aux transformations (2.5) des rep&res mobiles.

+

80

Sur la thkorie des espaces A connexion conforme.

11

E n dksignant par GG les formes de Pfaff qui dhfinissent la connexion par rapport au repgre (n+2)-sphkrique form6 avec les A,,, Ai,Am, nous avons dAp= ZPAO. En substituant Ap=@ A R , dans ces kquations, on trouve

(dUjt)AR+ UgdA,=G$U$A,, donc (2.24)

(d Ug)AR

+ U$W$A,= KJ$UgA, , - u0 +d U;:,

UR-0ao~p

p ~Rc )

d'oh on obtient les relations suivantes, grace & la forme spkciale de la matrice (U$) et aux kquations (2.18), . . .... (2.23) :

(2.25)

La deuxiGme hquation de (2.25) nous assure qu'on peut choisir un rep&re (n+2)--sphkrique pour lequel on a (2.26)

c&=

dtbi,

c'est-$-dire que l'on a, pour le dkplacement de A,,, (2.27)

dAO=to;Ao+dulAl+ . . . . . . d t ~ " ,A ,

et (2.28)

(dAo)2= gijduidd.

Nous allons appeler rep&re mobile semi-naturel un repere mobile realisant la condition (2.26) et par suite (2.28). Les klkments de la matrice ( U g ) transformant un repGre seminature1 en un autre rep&resemi-naturel doivent satisfaire & la condition : (2.29)

uj= u;$j.

Pour une telle transformation, (2.25) prend la forme suivante :

81

Kentaro Yano.

12

(2.30)

’:=’!I

Nous dkcomposerons,les transformations des repBres semi-naturels en transformations de deux catbgories suivantes : Ao=dAo,

(2.31) oh 1. et

(I)

A,=

A,,

pi sont des fonctions de

(11) 26‘

[

Ao=A0 , Ai=$id o+Ai A, - = 1 gijqi4jAo @Ai A , , 2 7

+

+

et

Nous appellerons la premiBre la transformation du tenseur fondamental et la deuxikme la transformation du point de l’infini A,. I1 est bien entendu que pour la transformation du tenseur fondamental (I) nous avons Pour cette transformation, la matrice (Ug)a la forme

82

Sur la th6orie des espaces ii connexion conforme.

13

ou

(2.34)

Alors, on obtient de (2.34)

(2.36)

Oh

(2.37)

R'( =

(2.38)

(0:

a log I adL '

et = gij

9

en vertu de (2.20) et de w6=dui. Cela btant, nous allons considher comment les formes et t o g j se transforment par les transformations du point de l'infini donnkes par (2.31) (11). Pour les transformations du point de l'infini, la matrice (27;) a la forme : 1, 0, 0 (u;)=[ $3, Jj, 0 pgij@p,

donc, on obtient de (2.25)

83

#,

1

oii le virgule d6signe la d6rivke partielle par rapport B uk. Nous avons vu que quand on se donne un systeme de coordonnkes (uk)de la varikt6 B connexion conforme, il existe un groupe de rephres semi-naturels par rapport B ce systhme de coordonnkes, et que les transformations entre ces rephres semi-naturels se dkomposent en des transformations spkciales de deux catkgories, l’une donnke par (2.31) (I) et appelke transformation du tenseur fondamental et l’autre par (2.31) (11) et appelke transformation du point de l’infini. Cela dit, qu’est-ce qui se passe quand on change le systhme de coordonnkes? A un autre systkme de coordonnkes, il correspond un autre groupe de reperes semi-naturels. Pour passer d’un repere seminature1 par rapport B l’ancien systhme de coordonn6es B un pareil

84

Sur la thdorie des espaces i3 connexion conforme.

16

repere par rapport au nouveau systeme, on peut proceder de la manihre suivante : on fait correspondre d’abord un rep6re semi-naturel par rapport ii l’ancien systcme au repkre semi-naturel par rapport au nouveau systkme ayant les mkmes A, et A4, que l’ancien, et apr& cela on change le tenseur fondamental en multipliant par un facteur et on change encore le point de l’infini de maniGre qu’on parvienne enfin au repere seminature1 donne a priori dans le nouveau systkme. Quand on effectue une transformation des coordonnkes (2.41)

ti?

= iii

(211,

d,.. , u”), ,

les formules (2.42)

nous donnent une transformation des reperes semi-naturels et cette transformation ne change ni le tenseur fondamental ni le point de l’infini, parce que le tenseur fondamental gfi a les composantes suivantes dans le nouveau systkme de coordonnees (2.43)

ou bien

Donc, (2.42) est une transformation qui fait correspondre & un repkre semi-naturel par rapport B l’ancien systkme de cooronnhes un repere pareil par rapport au nouveau systcme ayant les memes A. et A , que l’ancien. Comme la matrice (27;) de la transformation (2.42) a la forme:

(2.45)

on a, de (2.25)’

85

16

d'oh on obtient

(2.46)

en remarquant que Gj=

i&cdG".

Nous appelons cette transformation, transformation des coordonnkes (ui) pour la simplicit& 3. Les coefficients de la connexion conforme par rapport

aux repiires semi-naturels.

Comme nous avons, pour un rephre semi-naturel, (3.1)

+

dAo = to?jAo d?.b'Al+ * .. * *

+dun&

,

posons, pour apporter plus de symktrie dans la notation, 1fJ;

= dti,'

,

pkdu" =du" ,

86

Sur la theorie des espaces & connexion conforme.

17

de maniere qu’on ait

dAo=duOAo+du’Al+

(3.4)

.*-..

+du”A,.

Ici, nous introduisons les indices grew 1, p , U, ... qui prennent les valeurs 0, 1,2, ..., n, de sorte qu’on peut kcrire (3.4) comme il suit :

-

dAo= du’Al

(3.5)

.

Cela etant, considkrons une sphere X dans un espace conforme tangent, alors

X=XPAp.

(3.6)

Si les X p sont des fonctions de dC,on a toujours une sphere dans l’espace conforme tangent en chaque point de la vari6tk. Pour un dkplacement infiniment petit, on a de (3.6)

dX=(dXp+XVio,P)Ap. Si l’on appelle X p les composantes d’un vecteur contrevariant conforme, dXP+X‘’(O$ sont des composantes de sa dhrivhe covariante. Pour mettre

c;xp= dXP+ x%Jg ,

(3.7)

sous la forme

JXP=Xr,.duY,

(3.8)

on transforme (3.7) de la manikre suivante:

a X p = X p ,kdttk+Xo(t,~kdulC = X P , ~ d ~ ~ o + X p , ~ ~ d ~ ~ 1 C + X V d ~ d ~ ~, o + X ( 3 ( ~ ~ ~ ~ - ~ Oti

(3.9)

xp,o=o.

En posant (3.10)

on trouve (3.8) oti (3.11)

XP;”=xp,

+XQrIg,.

Cela &ant, si Yon considere un contour ferm6 infiniment petit et qu’on re@re les espaces conformes attaches aux points du contour par

87

18

Kentaro Yano.

rapport 5 l’espace conforme attach6 ii un point fixe sur ce contour, l’intkgrale SdAo ktendue 5 ce contour fermk partant de ce point fixe donne

Donc, pour que le point revienne ii sa premi6re place apr6s un

Nous dirons qu’une variktk 5 connexion conforme qui rkalise la condition (3.12) n’a pas de torsion. Les / I & ( I , p, u, ... =0, 1 , 2 , ... n) &ant dkfinies par (3.10) nous avons

(3.14)

et (3.15)

D’autre part, on a, en tenant compte de (2.23), gljdk

+

giL“‘;k=

donc, en substituant 1) E. Cartan (3).

88

gijvk

9

Sur la th6orie des espaces A connexion conforme. Wjl<

19

+

= Iljk J;pk ,

dans cette kquation, on trouve (3.16)

g t.j.*/c - y, 1,111 rle - giL//jk = 2y.ij~Ic*

E n supposant que la connexion soit sans torsion, on a (3.17)

11.iJk = {.tki } -81. JPk-

g i a p a g 3k .

I

Oii

Nous supposerons, dans la suite, que notre connexion soit toujours sans torsion. Les I l i a ktant dPfinies par (3.14) et (3.15), nous pouvons facilement dkduire la loi de transformation des I / & de celle des w&. Voici la loi de transformation des Pour la transformation du tenseur fondamental,

pour la transformation du point de l'infini,

introduits ici sont les coefficients d'une con1) 11 est bien entendu que les nexion de M. Weyl. 11s peuvent se trawformer en E l k d'apr8s (3.19). mais ils restent invariants par rapport A la transformation conforme du tenseur fondamental gtJ= A%J.

Voir, sup ce sujet, V. Hlavatjr (l), (2), (3,(4).

89

Kentaro Yano.

20

(3.20)

4. Les repiires mobiles naturels.

Nous allons tout d'abord montrer qu'on peut choisir, parmis les repbres mobiles semi-naturels, un repere qui satisfait aux deux conditions supplhmentaires

En effet, si Yon effectue, d'un seul coup, une transformation du tenseur fondamental

et une transformation du point de l'infini

on obtient la loi de transformation de

90

pk

et f l & :

Sur la thkorie des espaces d connexion conforme.

En posant dans ces hquations

on trouve

mais, d'autre part, on a de (3.17) = { j k } - npl, ,

donc, on a

Les lk htant definis par on peut choisir, comme I.,

(4.4) et comme

(4.5)

1 ' = f 2 n

Sk, 1

a log g - 2nQlC =p , + ~ - - _ _ _ _ _. 2Uk

91

21

Kentaro Yano.

22

Donc, on voit bien qu'en prenant 1 et @k donnks par (4.4) et (4.5) respectivement et en effectuant, avec ces 1 et Sk,la transformation du tenseur fondamental et la transformation du point de l'infini, on a, par rapport ii ce repere semi-naturel, * I/:/c= * I/:/< = 0 , ?I& = 0 .

Si l'on se donne un systeme de coordonnkes de la varibtb, un repere semi-naturel de cette sorte est uniquement dbtermink. Nous l'appellerons donc repere mobile naturel, e t dbsignerons par *I/$, les coefficients de la connexion conforme par rapport au rep6re naturel. En substituant les valeurs de 1 et Qk dans (4.3), on trouve "115k=

1 ;;G

I*I& = *I&

iu

(Guj,/~+Ga/~.j-Gjk,u) ,l)

=*

=0 ,

= :v

*/I". oJ - , Gi,,"116, = *

,

o t ~nous avons posh 1

G.. g .. / g 23

(4.7)

ZJ

(4.8)

Gij =

1

1

rrgij ,

alors on peut facilement vhrifier les identitbs : Gij, 1;- Glj*fl:k- Gil"//gk= 0 .

(4.9)

Les Ao,Al, ..., An, A , formant un repere mobile semi-naturel, les *AO,*Al, ..., *A,, " A , qui forment le repere naturel sont donnks par 1 -_

(*Ao=g

212

Ag ,

I 1) Les *U$, coi'ncident avec les coefficients conformes de MM. J. M. Thomas et T. Y. Thomas. Voir J. M. Thomas (1) ; T. Y. Thomas (l),(X), (5).

92

Sur la th6orie des espaces B connexion conforme.

23

Cela btant, nous allons ehercher comment les * A , se transforment lors des transformations de coordonnbes. Le dbterminant g form6 avec les g i j se transforme d’apr&s

2 = J2g, oh J est le jacobien de la transformation de coordonn6es

ui=ui (ZL’,

2, . .. u79 )

)

done, 1

a log g - j L _- a log$? ~ a2 auj

1

~~

~~

ad

aui

+

a log J-; a2

1 9

par consbquent, on a, comme loi de transformation des * A p ,

et en posant (4.11)

(4.13)

93

Kentaro Yano.

24

I1 est facile de verifier que les *fl& se transforment d'aprss les formules suivantes quand on effectue une transformation de coordonn6es

(4.14)

Supposons qu'on ait une sphere X dans un espace tangent en un point ( d ,3, .. , u")de la variete. [Ao,Al, ...,A,, A,] &ant un repere semi-naturel dans cet espace tangent, nous avons

X=XPAp.

(5.1)

Si Yon effectue une transformation du tenseur fondamental donn6e par (2.31) (I), (5.1) devient

x=P

A P ,

Oil

Pour une transformation du point de l'infini (2.31) (II), (5.1) devient x=X'A, , Otl

1 XO=P- $ixi+-gab$a$bX-, 2

x" - p.x, , x- ,

x- .

94

Sur la thkorie des espaces Sr connexion conforme.

25

Enfin, quand on effectue une transformation de coordonnkes (2.421, (5.1) devient X=XPA,, Oil

(5.4)

Nous appelons, les quantitks comme X p les composantes d’un vecteur contrevariant conforme par rapport ii un repitre semi-naturel. Nous pouvons aussi dkfinir les composantes d’un vecteur covariant conforme comme des quantitks qui se transforment de la m$me manikre que les Ape Les quantitks X p sont donc des composantes d’un vecteur covariant conforme si elles se transforment de la maniBre suivante: Pour une transformation du tenseur fondamental (2.31) (I), 2x0,

1 xo=-xo, I

xi=;ixi,

xi= 1A. xi,

xo=

(5.5)

1

x,=

1 x,, R

X w = Z .

Pour une transformation du point de l’infini (2.31) (II),

Pour une transformation de coordonnees (2.42),

(5.7)

Cela dit, consid6rons la mkme chose pour les repBres naturels. X ktant une sphere et [*AO, *Al,..., * A , , *A,] &ant le repere naturel, nous avons X = *XP”Ap.

95

26

Kentaro Yano.

Les *XI' sont des composantes d'un vecteur contrevariant conforme par rapport au repere naturel. Quand on effectue une transformation de coordonnkes, *AlJse transformant en * A p d'aprks (4.12), et on a

x=* X ' * A p, par conskquent les * X p se transforment en

* Z p suivant les formules:

De mbme m a n h e , on peut dkfinir les composantes covariantes d'un vecteur conforme comme il suit :

respectivement, nous pouvons dkfinir les tenseurs ghnkraux, par exemple,

96

27

Sur la thhorie des espaces ti connexion conforme. >v

(5.11)

-

U Q* Tr’. Q . TA. B- VA P B

I1 est A remarquer que les U$ et V i satisfont aux relations

u;v;=1, U2V2=1, uy;+up,”=o, u“,v:‘+ uiv:+U~V!L=O, flJf=aj,

u~v:+u,-v~=o, 1 U$l=l

V”,=l.

On voit que si la (.n+2)-iGme composante *X” d’un vecteur contrevariant conforme * X A est nulle ou la premiere composante *Xo d’un vecteur covariant conforme :’XA est nulle, il en est toujours ainsi dans tous les systsme de coordonn6es. Ces composantes se transforment toutes les deux d’apres la loi de transformation de la forme 1

“;T=il--

(5.12)

~ ‘kz T .

En langage affine c’est une densit6 du poids -

1 ~ . Nous

n

l’appel-

lerons scalaire conforme. En d6rivant (5.12)’ on a

On peut done dire que * T et a*T sont les nS1 composantes d’un 2Ui

vecteur covariant conforme. Les (,n,+l) composantes 6tant (5.13)

pour trouver la dernisre composante *T,, on peut employer la condition invariante

GAB*T,A*T,B = 0”

(5.14) Otl

GAB=*AA*AB.

GARGBC=a?, ~~

1) 0. Veblen (1).

97

Kentaro Yano.

28

De (5.13) et (5.14)' on obtient (5.15)

On a ainsi obtenu un vecteur covariant conforme

en partant d'un scalaire conforme "T. Le mkme prockdk s'applique pour la formation d'un tenseur conforme en partant d'un vecteur conforme par diffkrentiation. Prenons par exemple un vecteur contrevariant conforme

X = * X A* A A .

(5.17)

Alors, on a 8X=(d*XA+ */l&*XBdui) *AA,

Donc, les quantitks dkfinies par (5.18)

+

xA

- a*xA +*I 1 ~ A * .VxA?O x* ~ ~A

;l-

9

ad

se transforment comme les composantes d'un vecteur contrevariant conforme par rapport B l'indice A et coinme les (nt-1) premieres composantes d'un vecteur covariant conforme par rapport B 1. Si l'on emploie le langage de M. 0. Veblen, * X A sont des composantes d'un vecteur conforme du poids

'.

n

En effet, en posant

N=- 1 dans les formules de M.O. Veblen pour la dkrivke covariante n

d'un vecteur conforme (5.19)

T A :i-

?lTA

ad

o"f+NTAJ;+K&TB ):

et en remarquant que

Kjk = * l l j k , on voit que

T~;~=O. ~

1) 0. Veblen (l),p. 745.

98

Sur la thdorie des espaces A connexion conforme.

29

La definition de M.O. Veblen coincide done avec notre definition de * X A i l . Pour le tenseur fondamental conforme, on a

+

GAB;A =aGAn - GCB*IlzA- G ~ c * l l & 2GAR& = 0 . ad

(5.20)

6. Tenseurs de courbure.

Considkrons un contour fermh infiniment petit de la varietk et reperons les espaces conformes attaches aux points du contour infiniment petit par rapport B l’espace conforme tangent en un point fixe P

.c

infiniment voisin de ce contour, alors les integrales dAI:’ ont un sens geometrique et l’on a

..

S~A,,=SS{(‘.PY-[(UF.::]}AU= - JJQ~A,,

(6.1) Oti

-2: = - ((0;)’

i

J2$=

(6.2)

+

- (mi)’+

Qj = - (o,:)’+

[togtoo,] [(0:(0;]

+ + + +

[(09053

+

-2; = - (Us>’ [(O,”(#J:]

+

J2Z = - ((42)’ “OZ(03

,

’

[&0:]

-2: = [tIJZ~4],

.Or=[w;&‘J ,

[(0:t0,3

, $7 = - ((07)’+[ O y ( O : ]

[OJ,”tOL]

[OJy(Oi]

,

’

-@c

+

= - (0%)’

+ +

[ W % J ~ ]

[(0,”10:]

,

[CO~OJ$]

.

Les coefficients des formes bilinhaires -25 donnent ce qu’on appelle les tenseurs de courbure de la varibth B connexion conforme. Calculons d’abord les composantes des tenseurs de courbure par rapport B un repsre semi-naturel. En remarquant que (0: = p/LdtblC , (0; = du*, Yon obtient de (6.2)

c

+[cEuJlI,”&k] = - PI;,[du3duk] + Ldu’du”]

Q: = - (p&uk)’

flyk

99

Kentaro Yano.

30

9:=

2

JZ&lk[duk,duh] ,

on obtient

De m&me maniGre, en posant

100

Sur la t h h r i e des espaces ii connexion conforme.

31

on trouve fizkh=

-fi&ch

fi!okh=n&=O

RLkh= gi'fijkh

9

.

fi&=o

Cela &ant, on voit que les composantes fi:kh, Q;kh et fiFkh sont invariantes par rapport B une transformation de la forme (2.31) (I) et on peut de plus facilement verifier qu' elles forment les composantes des tenseurs affines par rapport aux transformations de coordonnkes (2.42). I1 nous reste B trouver la loi de transformation de ces composantes par rapport B une transformation du point de l'infini (2.31) (11). Les pi,f&, 17jlc et f k k 6tant transformees suivant les formules

101

on trouve

102

Sur la thkorie des espaces it connexion conforme.

103

33

34

Kentaro Yano.

104

Sur la thkorie des espaces A connexion conforme.

done (6.24)

36

( n - 1 ) ( / / ~ / < + P J / ~ ) - ( / / ~ < ~ + ~ D ~ -R3kS<J)= g'k-R 2(n- 1 )

.

Le deuxikme membre de cette kquation est, comme on le sait bien, doit etre symktrique par rapport 8 j et k , done le tenseur /I,Ok+pjDjlC aussi symktrique par rapport aux indices j et k , c'est-8-dire (//yk

done

a,,=

+

PI/,)

(//?.h

-

(Ilk+iDkj) = 0

9

+

- l/?l,J (P/,h - P / J = 0

9

par conskquent, on en conclut que: Pour une connexion conforme normale, on a (6.25)

B&h

en meme temps que

=0

,

ci . = o . 3kz

Cela ktant, on a de (6.24) (n-2)(/1,0k+pik)= -Rjk+- g -j k R~ 2(n- 1)

~ ,

d'oh (6.26)

donc, en substituant (6.18) et (6.26) dans (6.12), on a finalement (6.27)

CZ3kh = RJ k h-

n-2

-~

+ (n-l)(n-2) OG

j

+

-gj/LRc)

g jhk R h ~

(gj/
di)

9

R',, =gikRkh,

c'est le tenseur conforme de courbure de M. H. Weyl." Les Cjkh &ant obtenues, nous allons ensuite calculer Q,OklL. En substituant

~~

1) H. Weyl (1).

105

36

dans (6.5)' on obtient apr6s quelque calcul

(6.29)

On voit que le deuxi&me terme du deuxieme membre de (6.28) est

picjkh.

Pour trouver une forme plus simple du premier terme, dkrivons covariantement l'kquation (6.27) par rapport aux symboles de Christoffel {jk}7 c j k h l l =R:khll-

n-2

-

- RjiLllo";

(R~kli&

-k g 3 k R h i i - g J i L R k I i )

En contractant par rapport aux indices i et I, on trouve

Mais, d'autre part, on a les identitks de Bianchi: Rjkhii

+

R $ i Ik

+

R j i k ~h= 0

d'oii en contractant par rapport B i et 1 R j k h l i = R j k Ih

d'oti en contractant encore gik, on a

106

-R j h Ik

9

Sur la th6orie des espaces A connexion conforme.

37

donc

+

gjhRlk

2 ( n - 1) (n- 2 ) et l'on a finalement

I'

(6.31)

Les (6.12), (6.25) et (6.31) nous donnent le Thhr6me : U n e condition nbcessaire et a i m a n t e pour qu'u,n espace ?IL connexion conforme normale soit tin espace conforme proprement dit est que le tenseur de M. W e y l s'annule (n>3). Quand on effectue une transformation donn6e par (2.31) (11)) les / I & se transforment suivant les formules : I

-. 11"03 ii&=II&=O, - 113 = 3; , IIo,i=IIo,i=o, I-i~=/l~=O, /I".= =a -2pi= - 2 ( p , - + J , 1 ll!k= I l g k + (9;.k - g i l l j k ) - @j$k - g a b P g b g i k 2 I T j k = /I:, $$k &+; - s i g j / c

+

+

v

+

._

11%= '

~

~

=g j k ~

=

f

9

~

~

k

+

donc, en prenant pi comme

(

~

,

Oi,l'on

/

~

+

obtient

107

~

1 i 2

/

~

~

,

9~

)

-

~

~

/

~

+

~

~

p

~

Kentaro Yano.

38

Mais, on a vu qu’on a, pour un espace

connexion conforme normale,

donc, (6.32) devient

Ce sont les composantes de la connexion conforme considbrbes par M. 0. Veblen.” Cela Qtant, considbrons les composantes des tenseurs de courbure par rapport au repere conforme naturel. On tire des (6.3), (6.4) et (6.5)

parce que *pi = 0

par rapport au repare conforme naturel. On sait que nous pouvons choisir le repare naturel parmi les rephres 1) 0. Veblen (3).

108

Sur la thhorie des spaces 5. connexion conforme.

39

semi-naturels en effectuant les transformations de la forme (2.31) et de plus que les composantes de courbure GrCtL et Gfkh sont invariantes par rapport aux transformations (2.31) (I) et (11). On en conclut que (6.38)

* Q!kh = Q t k h

(6.39)

*a< =Q< 3kh 3kh *

9

Par conskquent, si la connexion est normale *J& donnent aussi les composantes du tenseur conforme de courbure de M. Weyl. Dans ce cas, on a, de (6.351,

donc

(6.40)

*

11 est remarquer que -(n-2) * 1 1 ? k coincident avec les Q j k de M. T. Y. Thomas. La connexion &ant normale et le repere ktant naturel, nous avons, comme composantes de la connexion conforme :

109

Kentaro Yano.

40

I"

I/$=

Gjk

*jl&=Q.j"ll;k.

Ces composantes coincident avec celles de M. T. Y. Thomas" ii des constantes pr6s. 7. Les formules de Frenet et les circonfkrences gknkralis6es.

Consid6rons une courbe ui =ui(r) dans notre variktk a connexion conforme. Le long de cette courbe, on peut developper notre varietk sur un espace conforme tangent attach6 a un point fixe de la courbe d'apres les formules

oa nous avons choisi le repere naturel.

Cela ktant, consid6rons une sphere

S=p*Ao

(7.2)

(0)

dans chaque espace conforme tangent attach6 ii un point de la courbe ; c'est une sphere dont le rayon est nul, S2=(~*A0)2=0.

(0)

S est donc un point qui coincide avec le point de contact. Si Yon (0)

change le systeme de coordonnkes, on obtient

1) T.Y. Thomas. (l),(Z), (5).

110

Sur la th6orie des espaces Iconnexion conforme.

41

donc, p se transforme suivant 1

(7.3)

p=Axp.

Ce fait peut &re aussi interprktk comme il suit : S=p*Ao etant (0)

une sphkre, (p, 0, 0, ... 0,O) sont les composantes d’un vecteur contrevariant conforme par rapport au repere naturel, donc si l’on effectue une transformation de coordonnkes, les nouvelles composantes sont donnees 1

par les formules (5.8) : la premiere composante est donc ATp, les autres etant nulles. DQrivons (7.2) le long de la courbe

Le parametre r &ant invariant par rapport aux transformations d de coordonnbes, --S donne une sphere dans l’espace conforme tangent, dr dp

(0)

dui sont les n+1 premikres composantes d’un par consbquent -, p dr dr vecteur contrevariant conforme ; on peut le verifier facilement : ~

Ce fait nous sugg6re une methode de formation d’un vecteur contrevariant conforme en partant d’une densi tb donnee. De (7.4)’ Yon tire

nous allons choisir p de manikre qu’on ait (7.7)

Pour cet effet, posons (7.8)

p=- d r

ds’ oh s est un paramktre satisfaisant 5 (7.9) Nous appelons s parametre conforme. p se transformant d’apr6s

111

Kentaro Yano.

42

(7.3) et r Qtant invariant par rapport A une transformation de la forme Qi=Ui(u),on a comme loi de transformation de s : (7.10)

1

dS =A - x d s ,

Posons maintenant (7.11)

alors

(7.12)

DQrivons encore (7.12) le long de la courbe ui=ui(r), (7.13)

112

Sur la th6orie des espaces B connexion conforme.

43

Ce fait nous suggBre une mbthode de formation d'un vecteur contrevariant conforme par diffbrentiation en partant d'un vecteur contrevariant conforme dont la dernikre composante est nulle. Cela dit, considkrons le carrb scalaire de la sphkre d S: d r (1) -

et calculons le deuxiBme membre de (7.15) comme fonction de s. En remarquant que d2r d3r d r d2r dr dp - ds2 d2p - d?- d y - ( d k ? ) P=--' ds dr dr ' dr2' ds ( -

~

d ui du'_ -- dsdr dr ' ds -

d2ui dr d2ui~-ds2 ds dr2-

et

on tire de (7.15)

(7.17)

113

d'r dui ds2 ds9

Kentaro Yano.

44

r Qtant un paramhtre gBnBral sur la courbe ui=ui(r),nous allons choisir un parametre t qui rend nu1 le carre scalaire de la sphare obtenue en dbrivant deux fois le sphbre-point mouvant S. (0)

DBfinissons un nouveau paramhtre t sur la courbe par

duj duk

1

{ t, s} = -GikajarC - " I l ~ ,2 ds ds

(7.19)

,

(7.20)

t dtant dBfini par une dBrivBe schwarzienne, t est dhtermine comme fonction de s 2i une substitution homographique pras. I1 est facile de verifier que t reste invariant, B une transformation homographique pr&, pendant une transformation de coordonn6es. En effet, en tenant compte des relations

I

d2S

on a

d'oti en comparant avec les relations

114

Sur la th6orie des espaces 1 connexion conforme.

45

on obtient

Cette kquation et la formule

nous donnent

{f, 5 ) = {t, S } ,

donc (7.21)

{f, t>=o,

et on voit que t reste invariant, ii une transformation homographique pr&s, pendant la transformation de coordonnkes. Le parametre t ayant cette propriktk gkomktrique peut btre appelk parametre projectif sur cette courbe. Le parametre projectif 6tant ainsi dkfini, nous allons prendre t au lieu de parametre r et posons de nouveau

alors, on a facilement (7.23)

Pour trouver le produit scalaire de S et S, il suffit de dkriver (0)

S S= 0 le long de la courbe ; alors on trouve

(0) (1)

115

(2)

46

Kentaro Yano.

donc (7.24)

s s=-1.

((1)

(2)

En tenant compte des relations (7.23) et (7.24), on voit que la d dkrivbe de S le long de la courbe S passe par deux points (0) S et (2)S (2) d t (2) et elle est orthogonale B S. Donc, on peut trouver une sphere S telle que ~

(3)

(1)

s s=s s=s s=o, (1)

(7.25)

(0) (3)

S'=l,

(2) (3)

(3)

(3)

d 3 -S=aS.

(7.26)

dt

3

(2)

(3)

3

En posant a = - n , on a d dt

(7.27)

__

s=

3

-nS.

(3)

(2)

La sphere S 6tant ainsi dkfinie, (3)

dt

S est orthogonale ii S et S et (1)

(3)

(3)

3 passe par le point S, donc en remarquant que S - d S-H=O, on peut (0) (2) d t (3 ) choisir une sphere unitaire S telle que (4)

S S=S S=S S=S S = Q ,

(7.28)

(0) (4)

(1) (4)

(2) (4)

~

donc en posant

4

4

/3=w,

S2=1

1

(4)

d S+uS=PS, 3 4 dt (3) (0) (4)

on a d dt

(7.29)

(3) (4)

__

s=- H S + H S , 3

4

(0)

(3)

(4)

d Cela etant, considbrons la dbrivbe S de S. Cette sphere est dt (4) (4) orthogonale B S et S et passe par les points S et S. Comme on a (1)

dS S *+H= (3) dt

4

(2)

(0)

(4)

0, on peut trouver une sphere unitaire S telle que (6)

116

Sur la th6orie des espaces ii connexion conforme. 6

47

6

donc en posant y=x, on a (7.30)

~

d dt

s=-wS+nS. 4

(4)

5

(3)

(6)

En continuant de cette mani8re. on trouve successivement

s=-nS+wS, 6

-

dt

(7.31) I

(6)

1

(5)

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

(7)

oii les S, S, ..., S S satisfont aux relations (0) (1)

(n). (-1

I

(i+j=~ 1,2, , ..., n, 0 0 , sauf i=o, j = 2 et i = 2 , j = o )

Les relations (7.22), (7.27), (7.29), (7.30) et (7.31) nous donnent les formules de Frenet pour une courbe d’un espace B connexion conforme. Les formules de Frenet 6tant obtenues, nous allons considbrer des courbes dont la premi&re courbure est nulle. Nous avons

mais, d’autre part,

donc on a (7.33)

-p*Ao=O. d3 df‘

1)

C’est l’bquation differentielle de la courbe cherchk. Prenons un point fixe (tJ sur la courbe et d6veloppons la courbe sur l’espace conforme tangent attache B ce point, alors on aura 1) K. Yano: (2).

117

48

Kentaro Yano.

c’est-8dire (7.34)

donc

ce qui exprime que ?*Ao reprksente un point qui se trouve toujours. sur n- 1 spheres diffkrentes (orthogonales) [Sltu... [Slt,. (-) (3)

On peut donc regarder cette courbe comme une gknbralisation de la circonf krence d‘espace conf orme ordinaire?) Cela dit, cherchons les kquations diffkrentielles qui donneront la reprksentation paramktrique de cette courbe. De l’kquation d3

dt3

p*Ao=O,

on tire

d3*A0+3pr d2*Ao+3p1/ d*Ao +p”’*Ao=O, di! at 2 dt

(7.36)

oii les primes indiquent la dkrivke par rapport au parametre projectif t. Les dbrivkes de *Aole long de la courbe sont donnbes par I

d*Ao = $Ui * A i , dt dt

dzAo =B*Ao+biKAAi+C*Am, dt2 (7.37)

4

d3* -- Ao = (dB- + * / p k b i 4 C ) * A ~ dt3 dt dt dbi +*zi$J-- duk +B-du’- + C ’ / l & d g ) * A ,

+(

~

dt

dt

1) K. Yano: (2).

118

dt

dt

Sur la theorie des espaces B connexion conforme.

B=*/l0 duj jk dt i- dzui bd t2 c=* I / ; du3 dt -

(7.38)

dtbk dt ’ . dui dUk ~, dt dt duk-=G,k- dui- duk dt d t dt

+

49

.

En substituant (7.37) dans (7.36) et en considbrant le coefficient de *A,, l’on obtient

%’‘ +“c) dt +3p’C= 0 ,

P (Gjkb’

mais, on a, d’autre part,

donc 3 2

-p

d C +3p‘C=O, dt

~

1 dC+2.dE=O* C dt p dt

(7.39)

~-

Cette 6quation est identiquement satisfaite, parce que l’on a Gjk-

dui du” = , ds ds ~

G. dui duk ( d t ) 2 , 1 , 3k d t dt ds cp2=1,

et en d6rivant cette 6quation logarithmiquement, on obtient (7.39). Considbrons cette fois les coefficients de * A i :

+3p’bi+3p”- dui

-

dt

En substituant p=-- d t

ds ’

d ‘t p’= d s2 dt ’ ds

d3t

fill =

- --

~~

119

-

=o.

60

Kentaro Yano.

dans (7.40). on trouve

dui 3- ds

d2t

-( d 3 ) ds

+

3 -dui -d3t ds d 2

( ciYJ2

=0,

ds

d'oti

{t, s} Qtant donne par (7.19)' on tire de ces Qquations

120

Sur la thikrie des espaces A connexion conforme.

61

Ce sont les Qquations diffbrentielles qui donnent une reprbsentation parambtrique de notre circonfbrence g6nQralis6e. I1 nous reste maintenant B examiner le coefficient de * A o : (7.43)

En tenant compte des relations

3*]l? dui duk d2t 3k ds ds ds2 I

d4t ds4 -

(7.44)

121

Kentaro Yano.

52

D'autre part, les (7.42) peuvent ktre hcrites comme il suit:

donc en contractant Gihnh et en tenant compte de

G.ah

ds

on a de (7.42)

(7.45) Donc, (7.44) devient

et on voit bien que cette equation est identiquement satisfaite le long de la courbe. O n en conclut que les fonctions ui(s) qui dijinissent une circonfiwnce gbniralisie satisfont aux bquations diffhentielles

122

Sur la thdorie des espaces L connexion conforme.

63

Pour trouver les Qquations de circonfkrences gknkraliskes dans un espace de Riemann, prenons les composantes de la connexion de M. 0. Veblen, U flJ-"j i .- -i /I& = = 11; = = 0 , nq3k = - Rjlc gjj.,R n-2 2(72-1)(n-2) ' (7.48) Il$ = {&} , 1

n.

= g .3k

316

,

alors, on aura les kquations diffkrentielles pour les courbes

et l'kquation schwarzienne pour le paramktre projectif (7.50)

{t,s} = 1 - g . ~

2

ds

cEs

ds

ds

oc s est l'arc de cette courbe mesurk par le tenseur fondamental gjk,

(7.51)

Si Yon niultiplie gjlc par un facteur pZ, s se transforme en S. et Yon a (7.52)

I1 n'est pas difficile de vkrifier que les kquations (7.49) dkfinissent des courbes inva.riantes pour le changement de la mktrique (7.52) et le parametre projectif t est aussi invariant & une transformation homographique p r h

123

64

Kentam Yano.

En effet, remarquons d'abord que d'apr8s

Oil

En Dosant

124

@k

se transforment en

fl:k

Sur la thhorie des espaces ii connexion conforme.

66

par consbquent, on obtient

et

Les Bquations (7.55) et (7.56) nous assurent que (7.49) et (7.50) sont invariantes par rapport h la transformation conforme du tenseur fondamental. Cela htant, rappelons-nous les formules de Frenet pour l'espace de Riemann a-1 a a = l , 2, ..., n, AEi= -k a-1~i +ka+l p (7.57) kO = kn = o , ba d'oij

oii

a dBsigne la d6rivBe covariante le long de la courbe. 8s

125

66

Kentaro Yano.

En remarquant que . __ dui p=

ds’

1

on tire de (7.49)

Rii

RiEk

]=O. +[-* + 2(n- 1)(n- 2) 1

Grice ii la relation

g j k E ’ i t k = gjiGEiik= 1, 11 2 2

cela devient

(7.58)

Donc, on a : Le transformi! de la tangente Ei par le tenseur de Ricci Ri se 1

trouve dans l’espace linbaire formi! par la tangente, la premiere normale et la deuxikme normale, les composantes sur ces trois vecteurs ortho1

12

gonaux ktant R ~ ~ E( nG- 2~) , dk et ( n - 2 ) kk respectivement. ds 11 Si l’espace est euclidien, c’est-&-dire Rjkh=O, on tire de (7.58) que 1

2

k=constante et k=O : donc on a en vertu de (7.49) (7.59)

si l’on prend un systeme de coordonnCes dans lequel gJk=constantes

et

{1}=0 ,

on peut facilement intbgrer (7.59) : (7.60)

1

1

du~=Aicosks+Bsinks, ds

oti nous supposons que

(7.61)

gijA”A’Z1 ,

gijB’B3=1 .

giJAiB’=O,

On a encore de (7.60) (7.62)

.

uE=-i-sm k

1

1

~cs- B c o s k s + ~ .

i

126

Sur la thbrie des espaces A connexion conforme.

67

Ce sont les Qquations de la courbe. Mais on a . . 1, (7.63) g, (ui-ci) (u' -8 )= ~

ik

c'est-&-dire la longueur du vecteur ui-ci est constante et se trouve toujours dans le plan A deux dimensions form6 par Ai et Ri, donc cette courbe coincide avec la circonfkrence ordinaire. Dans ce cas, (7.50) se r6duit & 1 k2, ' {t, s} =-

(7.64)

2

1

k &ant une constante. En r6solvant 1'6quation diff6rentille

d3t ____

ds on trouve (7.65)

t=,tan-(s+u)+c. 2b

2 k Pour chercher la significance gbm6trique de t, posons

c=O,

b=l,

u=O,

r 6tant le rayon de cette circonf6rence. On aura alors, (7.66)

t=2r tan

s ~. 2r

DQsignons par 0 et A les points correspondant aux deux valeurs o et rr de s, respectivement. Alors, en d6signant par B le point correspondant & la valeur s, on a s

L BAO=-.

2r

On a donc 0 ~ = 2 r t a n- S , et par suite t = O P , 2r

P 6tant le point d'intersection de A B et la tangente en 0. Tokio, le I" Janvier 1939.

127

68

Kentaro Yano.

Bibliographie. E. Cartan. (1) La dbformation des hypersurfaces dans l'espace conforme reel & TZ 2 6 dimensions. Bull. Soc. Math. France, 46 (1917) 67-121. (2) Sur les espaces conformes g6n6ralisBs et 1'Univers optique. C.R. 174 (1922), 857860. Les espaces A connexion conforme. Annales de la SOC. Polonaise de Math. 2 (19231, 171-221. Rapport sur le MBmoire de J.A. Schouten intitulb ,,Erlanger Programm und ifbertragungslehre. Neue Gesichtspunkte zur Grundlegung der Geometrie ". Bull. SOC.Phys.-Math. Kazan 2 (1927), 71-76.

D. van Dantzig. On conformal differential geometry, I. The conformal gradient. Amsterdam. 37 (1934), 216-221. L. P. Eisenhart. Riemannian Geometry. Princeton University Press. 1926.

Proc. Akad.

A. Haimovici. Directions concourantes le long d'une courbe sur une surface d'un espace conforme. C.R. Roumanie. 1 (1937), 296-301. Directions concourantes et directions parallbles sur une varibt6 d'un espace conforme. Thhe, Jassy. 1938. V. Hlavatjr. Zur Konformgeometrie I. Eichinvarianten Konnexion. Proc. Akad. Amsterdam. 38 (1936). 281-286. Zur Konformgeometrie 11. Anwendungen, insbesondere auf das Problem des Affinnormale. ibid. 706-708. Zur Konformgeometrie 111. Anwendungen auf die Kurventheorie. ibid. 1006-1011. Systsme de connexions de M. Weyl. Acad. Tch&queSci. Bull. int. 37 (1936), 181-184.

J. Levine. Conformal-affine connections. Proc. Nat. Acad. Sci. 21 (1936), 165-167. New identities in conformal geometry. Duke Math. Journal 1 (1936), 173-184. Conformal scalars. Bull. Amer. Math. Soc. 42 (1936), 116-124. Groups of motions in conformally flat spaces. ibid. 418-422. J. A. Schouten. Uber die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit guadratischer Massbestimmung auf eine Mannigfaltigkeit mit euklidischer Massbestimmung. Math. Zeitschr. I1 (1921), 58-88. On the place of conformal and projective geometry in the theory of linear displacements. Proc. Akad. Amsterdam. 27 (1924), 407-424. Der Ricci-Kalkiil. Berlin. Springer. 1924. Sur les connexions conformes et projectives de M. Cartan et la connexion linbaire gbnbrale de M. Konig. C. R. 178 (1924). 2044-2046. Erlanger Programm und Ubertragungslehre. Neue Gesichtspunkte zur Grundlegung der Geornetrie. Rend. Circolo Mat. Palermo 60 (1926)' 142-169.

128

Sur la thQrie des espaces B connexion conforme.

69

(6) Uber die Projektivkriimmung und Konformkriimmung halbsymmetrischer Ubertragungen. Bull. SOC. Phys.-Math. Kazan 2 (1926), 90-98. (7) Projective and conformal invariants of half symmetrical connections. Proc. Akad. Amsterdam. 29 (1926), 334-336.

J.A. Schouten e t J. Haantjes. (1) Uber allgemeine konforme Geometrie in projektiven Behandlung. I. 11. Proc. Akad. Amsterdam. 38 (1935), 706-708, 39 (1936), 27. (2) Beitrage zur allgemeinen (gekrummten) konformen Differentialgeometrie. I. 11. Math. Annalen. 112 (1936), 594-629, 113 (1936), 568-583.

J. A. Schouten et D. J. Struik. (1) Un theoreme sur la transformation conforme dans la g6ombtrie diffkrentielle A ndimensions. C. R. 176 (1923), 1697-1600. (2) Einfiihrung in der neueren Methoden der Differentialgeometrie I. Noordhoff. Groningen. 1935.

D. J. Struik. (1) The theory of linear connections. Berlin. Springer. 1934.

J. M. Thomas. (1) Conformal correspondence of Riemann spaces. Proc. Nat. Acad. Sci. I1 (1926), 257-259. (2) Conformal invariants. ibid. 12 (1926), 389-393. T.Y. Thomas. (1) Invariants of relative quadratic differential forms. ibid. I1 (1925), 722-725. (2) On conformal geometry. ibid. 12 (1926), 352-359, (3) Conformal tensors (First Note), ibid. 18 (1932), 103-112. (4) Conformal tensors (Second Note), ibid. 18 (1932), 188-193. (5) The differential invariants of generalized spaces. Cambridge University Press. 1935. J. Vanderslice. (1) Conformal tensor invariants. Proc. Nat. Acad. Sci. 20 (1934), 672-676.

0. Veblen. (1) Conformal tensors and connections. ibid. 14 (1928), 735-745. (2) Differential invariants and geometry. Atti del congresso internazionale dei matematici. 6 (1928), 181-189. (3) Formalism for conformal geometry. Proc. Nat. Acad. Sci. 21 (1935), 168-173. H. Weyl. (1) Reine Infinitesimalgeometrie. Math. Zeitschr. 2 (1918), 384-411. (2) Zur Infinitesimalgeometrie. Einordnung der projektiven und konformen Auffassung. Gottinger Nachrichten. 1921, 99-112.

K. Yano. (1) Remarques relatives 3 la t h b r i e des espaces B connexion conforme. C.R. 206 (1938), 560-562. (2) Sur les circonf6rences g6n6ralisbs dans les espaces B connexion conforme. Proc. Imp. A d . Tokyo. 14 (1938), 329-332,

129

MATHEIIATICS Vol. 55, No. 1, January, 1952 Printed in U . S . A . A N N A LS OF

ON HARMONIC AND KILLING VECTOR FIELDS BY KENTARO YANO (Received January 25, 1951)

$0. Introduction

S. Bochner [2, 3, 41 has shown on various occasions a remarkable contrast between harmonic wctors and Killing vectors. For example, he proved, among others, the following thpee Theorems A, 13, C: THEOREM A. I n a compact Iiiemanniaji space with positive deJnite nietric, there exists no harmonic vector jield, other than the zero vector, which satisfies the relatioil

Izl,tJEk 2

0,

(i,j , k ,

*

*

=

I , 2, * .

*

,n)

unless we have E 3 ! k = 0. I f the space has positive Ricci curvature throughold, then the exceptional case cannot arise. THEOREM B. I n a compact Riemannian space with positive definite metric, there exists no Killing veclor jield, other than the zero vector, which satisfies the relation

Rl~s'€k6 0 , unless we have f l , k = 0. I f the space has negative Ricci curvature throughout, then the exceptional case cannot arise. In these statements, RILdenotes the Ricci curvature tensor defined by Rlk

where the curvaturc tensor

RlIkl

= Iialk.

,

is given by

{h} denoting thc Christoffcl symbols formed with the fundamental metric tensor g j k of the space, and the semi-colon denotes the covuriant derivative with respect to the Christoffel symbols. THEOREM C. I f , i n a compact Riemannian space with positive dejnife metric, there exist a harmonic vector field 4, and a Killing vector Jeld q', then we have i,q'

= constant.

In the first section of the present paper, we shall prove a general formula. Jn the second section, wc shall show that this formula gives us immediately the proofs of Theorems A and €3 for an orientable space and enables us to see clearly how the contrast between harmonic and Killing vector fields arises. From this general formula, we can also deduce a theorem which states that, in a compact orientable Riemannian space with positive definite metric, if the 38

130

O N HdRMONIC AND KILLING VECTOR FIELDS

39

space has negative Ricci curvature throughout, then there exists no continuous group of conformal transformations other than the identity. In the third section, we shall prove that if the space admits a one-parameter group of motions, then the Lie derivative of any harmonic tensor with respect to this motion vanishes. Theorem C stated above is a corollary of this theorem. On the other hand, G. de Rham [5] has proved that the necessary and suflcient condition that the vector Jield ti be a harmonic vector field is that it satisfy

gakti;j;. - R i l l 1

=

0,

where

nil = gjkRijkr= gi'Rjl

,

In the last section, we shall obtain a theorem corresponding to the above for Killing vectors, from which we can deduce the theorem which states that, in a compact orientable Riemannian space with positive definite metric, an affine collineation is neckssarily a motion of the space into itself. $1. A fundamental formula

In a compact orientable Riemannian space, we have a general formula [l]

/ X';idv

0

=

for an arbitrary vector field Xi, where

dv

=

l/S

dx'dx'

dx",

* *

and g denotes the determinant formed with the fundamental metric tensor the integral being taken over the whole space. Now, take an arbitrary vector field ti and calculate ( [ ' ; k E k ) ; j . We have =

( t j ; k t ) ;j

t j ; k ;j t k

+

tj;kt;j

On the other hand, if, in the well known formula of Ricci: ti;k;

- ti;1;k

=

R'jjkdj,

we contract over i and I, we obtain t j; ~ = k R ' t I 3k i

f j; k ; j -

and therefore f j ; k ;j

=

fi;j;k

+

R j k tj *

Substituting this into the above equation, we find (1.1)

( t J ; k t k ) ;j = t j ; j ; k t k + R j k t j t k

Next, we shall calculat,e (6'; (1.2)

j&;k

(t';jtk);k

+

t i ; k t k ;j

and we have = fi;i;ktk

131

+

ti:jt;k

-

.

gjk

,

40

KENTARO YANO

Thus, subtracting (1.2) from (1.1), we obtain

- (f’;jfk);k

(fi;ktk);j

=

RjkEiSk

-I-

tj;kP;j

-

(j;jtk;k

1

and, taking the integral over the whole space, we find ( 1.3)

1

(Rjkiifk

f

{j”t/c;j

- f ’ ; j f k ; k ) dv

= 0,

where we have put

tAk=

tiiagak

and t!+= tag&.

52. Harmonic and Killing vector fields

Formula (1.3) enables us to deduce very easily Theorems A and B referred to in the Introduction. In fact, if we suppose that the vector f j is a harmonic vector field, then we have ,$ik

=

and

fk;j

fj;j

= 0.

Thus formula (1.3) gives

1

(Rjklilk

+

.$i;k.$j;k)

dv = 0 ,

from which we can see that, if the harmonic vector field then, l j ; k t j ; k being also positive or zero, we must have

fi

satisfies h ! j k t j t k 2 0,

= 0,

6j;k

and also Rjk&!+ = 0 in consequence of the last equation, and if the space has positive Ricci curvature throughout, then the vector ti must be identically zero. This proves Bochner’s Theorem A for an orientable space. Next, if we suppose that the vector ti is a Killing vector field, then, the Lie derivative Xg, of the metric tensor g j k must vanish, where X denotes the operator of the Lie derivation with respect to ti[6]. Thus we have Xgjk

,$j;k

-b l k ; j

= 0,

and automatically (’;j

= 0.

Thus, formula (1.3) gives

1

(Rjktitk

-

[’;‘tj:k)

dv = 0,

from which we can see that, if the Killing vector ti satisfies -ti;k,$j;. being also negative or zero, we must have tj;k

= 0,

132

Rj&jtk

5 0, then,

ON HARMONIC AND KILLING VECTOR FIELDS

41

and also R j & j t k = 0 as a consequence of the last equation, and if the space has negative Ricci curvature throughout, then the vector timust be identically zero. This proves Bochner’s Theorem B for an orientable space. Finally, if we suppose that the vector ti defines a one-parameter continuous group of conformal transformations, then the Lie derivative X g j k of the metric tensor g j k must be proportional to the metric tensor [6]. Thus we have x$jk

Ej;k

+

=

tk:j

%gjk,

the factor of proportionality being determined by

On substituting tk:j

=

- ti;k

%gjk

and

t’;j =

into the fundamental formula (1.3), we obtain

/

[ R j k t J t

- n(n - 2)d,2 - t ” k ( j ; k ]

du

=

0.

Thus, we can see that, if the vector ti defining a group of conformal trans0, then, -n(n - 2)d,’ and -fi‘‘tj$ being also formations satisfies R j k f i t k negative or zero, we must have tj;k

0,

=

and also R&tk = 0 and d, = 0 in consequence of the last equation, and if the space has negative Ricci curvature throughout, then the vector ti must be identically zero. Thus we have THEOREM 1. In a compact orientable Riemannian space with positive definite metric, there exists no vector ti which defines a one-parameter group of conformal transformations and satisfies the relation Rjkt’c

2 0,

unless we have [ j : k = 0. If the space has negative Ricci curvature throughout, then the exceptional case cannot arise. This theorem has been already obtained by S. Bochner [2] for n = 2 .

83. Lie derivatives of a harmonic tensor A harmonic tensor tiliz...ip in our Riemannian space is defined by the following three conditions: (i) (ii)

is anti-symmetric in all lower indices,

tili2,..ip tiliz

...i p ; k

=

tkiz

...i p ; i l

+

tilkit

...i P ; i z

+

133

‘

.. + t i l i z . . . i p - l k ; i p

42

KENTARO YANO

Now, suppose that our space admits a one-parameter group of motions generated by an infinitesimal transformation 2' = xi

+ qi(z)dt,

where dt is an infinitesimal, we then have 7j;k

xgjk

+

qk;j

= 0,

where the X represents the operator of the Lie derivation with respect to q', and, from this, we can easily prove that

X (f k }

+ Rijkrq'

qi;j;k

= 0.

The last relation is equivalent to the fact that the covariant derivation and the Lie derivation are commutative [6]. Thus, from three conditions for the harmonic tensor, we have (i) (Xtili2...ip)is anti-symmetric in all lower indices,

+ + (Xtiliz...ip-Ik);ip

+

(ii) (XtitiZ...i p ) ; k = (XtkiZ**.ip);il (Xtilkia.*.ip);iz

* * *

3

(iii) (Xfilil...ip-lj);kg~k = 0, which show that X f i l i2...ip is also a harmonic tensor. On the other hand, we have

Xtiliz ...ip =

= ( f a i t ...ip;il

=

+ + +

tili**..ip;aqa

(tai2...ipqa);il

tai2-..ipqa;il

tilaia

+

.a.

ip;i2

+ ... + + filaia

* * *

tail...ipqa;il

+

ipqa;i2

+

.

a * +

tiliz...ip-latlD;ip

tili2...ip-1a;ip)~a

+. +

tilaig*..iptla;i2

+.+

(tilaia.-.ipqa);i2

*

*

* *

tili2...ip-la~a;ip

(Eili2...ip-laqa);ip

,

--

A dxip is which shows that the harmonic form ( X tili2...ip)dxil A dxi2 A the exterior derivative of the form n([ai2...ipqa) dxi2 A . A dxip, from which, we conclude that +

Xtili2 ...i,, = 0. Thus we have THEOREM 2. If a compact orientable Riemannian space with positive definite metric admits a one-parameter group of motions, the Lie derivative of any harmonic tensor with respect to this group vanishes. This theorem was also found by Y. Muto. For the Lie derivative of a harmonic vector t i , we have

Xti

= ti;aTa = fa;iT"

+ +

= (taqa);i

134

taqa;i

ta7";i

43

ON HARMONIC AND KILLING VECTOR FIELDS

and, from the vanishing of this, we obtain constant.

fhq" =

This proves Bochner's Theorem C for an orientable space.

54. Group of a f h e collineations

If we take an arbitrary vector

tiand calculate g j k ( t $ ) , j ; k , then we have =

gik(fjfi);i;k

2[igPti;j;k

+

2tj''tj;k

.

Integrating over the whole space, we find

1

(4.1)

(tigjkt';j;k

+

[';'tj;k)

dv

= 0.

If we subtract (1.3) from (4.1), we find

1

[([igikfi;j;k

- Rjk-$jtk) f (tiik[j;k

- [jiktk;j)

+

[j;jtk;k]

dv = 0,

which may also be written as (4.2)

1

[fi(gjkfi;j;k

+

- R'L~')

3(tjik

-

tk;')([j;k

- t k ; j ) + [ j ; j t k ; k ] dV

=

0.

Now, if the vector field ti satisfies gikti;j;k

- RilE1 = 0,

then the other two terms in the integrand being both positive or zero, we must have 5j;k

-

&;j

=0

and

[';j

= 0,

that is to say, the vector must be a harmonic vector field. Conversely, if the is a harmonic vector field, that is to say, if it satisfies the above equalities, then, from the identity tj;k;i

-

[i;i;k

=

-R".i k a.to ,

or

5..] , k ; a. - t t.; .j ; k -R".i k a.ta -

9

we have, multiplying by g" and contracting over j and k, jk

g ti;kk

- RilS1

= 0.

Thus, the necessary and sufficient condition that the vector field ti be a harmonic vector field is that ti satisfy the conditions (4.3)

gjkti;j;k

- Ri l l1 = 0.

This is de Rham's theorem referred to in the Introduction.

135

44

KENTARO YANO

Next, if we add the equations (4.1) and (1.3), we find

1

[(tigjkti:j:k

f

+

(E’;k[j;k

RjkE’t’)

E’;k&;j)

-

+

Ek:j)

[j;jtk;k]

dv = 0,

which may also be written as (4.4)

1

[€i(g’kEi;j;h

+ Rilt’) +

+

$(tjik

Ek;’)(Ej;k

- E’;jtk;kl

dv

=

0.

Now, if the vector field t i satisfies gjkfi:j;k

+

= 0

R i l t l

and

[jij

=

0,

then me must have Ej;k

+

6k;j

=

0,

that is to say, the vector t’ must he a Killing vector field. Conversely, if the 6 , is a Killing vector field, then, from the equations

we find

and

XI:,}

= Ei;j;k

+

=

R’jklE‘

X denoting the Lie derivation with respect to have

0,

ti. From the last equation, we

+ Ri1t1 = 0.

gikti;j;k

Thus we have THEOREM 3. I n a compact orientable Riemannian spaccwith positive definitemetric, the necessary and suficient condition that the vector ti be a Killing vector field i s that we have (4.5)

gikEi;j;k

+

Rillf‘

t’;i

= 0 and

= 0.

Now, suppose that the vector ti defines an infinitesimal affine collineation in our Riemannian space with positive definite metric, we then have (4.6)

x { i k )

Ei;j;k

+

Rijkil

1

= 0,

from which, on multiplying by 8% and contracting, we have gjkti;j;k

+ Ri&‘ = 0.

Next, contracting over i and j in (4.6), we find (Ea;a):k

136

=

0,

O N HARMOKIC A N D KILLING VECTOR FIELDS

by virtue of the identity

Raokt = [a:a

45

0. Thus we must have = c (=constant.)

But, as we have dv = 0, we must have c = 0 and consequently we have =

0.

Thus, by Theorem 3, we have THEOREM 4. I n a compact orientablr Riemannian space with positive dcjinile metric, one-parawreter group of a$ne colliiiealioiis mwst be that of motions. The author wishes to thank Professor S. Rochner for many stimiilating discussions. INSTITDTEFOR ADVANCED STI~DY l%lBLIOGRAPHY 1. 2. 3. 4.

DOCHSER,S . Remurk: o n the Iheorctn of Green. Duke hlath. J., 3, 334-338, (1937). BOCHNER, S . Veclorfields and R i c c i curvature. null. Amer. Math. Soc., 52,776-797, (1916). BOCHNER, S . Curvature a n d B d t i nunibers. Ann. of Math., 49, 379-390, (1948). ROCHNER,S . Vector fields o n complex a n d real manifolds. Ann. o f M a t h . , 52, 642-649, (1950).

5. U E RHAM,6. and I<. KODAIRA. Harmonic integrals. Lertiires delivered a t the Institute

for Advanced Study, (1950). D. YANO,Ti. Groups of transformations in generalized spaces. Tokyo, (1949).

137

Reprinted from Transactionsof the AMS, volume 74, pages 260-279, by permissionof the American Mathematical Society. @ 1953 by the American Mathematical Society.

ON n-DIMENSIONAL RIEMANNIAN SPACES ADMITTING A GROUP OF MOTIONS OF ORDER n ( ~ 1)/2+ 1 BY

KE m . 4 ~o

1. As is well known [2 ; 4 : 113 ( I ) ,

\$

vriNO

e have the following theorem.

THEOREM A. If a n n-dzmensional Rieinannian space adtnzts a group of inotions of the m a x i m u m order n ( n + l ) / 2 , then, the spuce is of constant curvature. Thus, it might be interesting t o ask whether an n-dimensional Riemannian space can admit complete groups of motions of order n(n+1)/2-1, n(n+1)/2-2, . * or not, and if so, then what the structure of the corresponding space is. In this connection, we have a very suggestive theorem due t o G. Fubini [4,p. 229; 51:

THEOREM B. A n n-diinensioiaul Rieinanninri .spore for cotrsplete group o j motions of order n(n+ 1)/2 - 1.

11

> 2 tiinnot

(idinit (L

0 1 1 tlic otlicr l i , ~ n t l ,it \\asa11o l ~ c nprohlriii to clrtcriiiiiic I Iic rr-diiiic*iision..il Finsleri,in sp;icc it hich adinits a group of motions of tlic n~axiiiiutiiordci n(n+1)/2. Recently, H. C. Wang [8] p v c thc n i i s w ~ rto this prohlcrn by proving the following beautiful theorem.

THEOREM C. I f an n-dimenszonal Finslerilcn :haw jor 11. > 2 , n 24, admzts a group of motions of order greater than n ( n - l ) / L + 1 , then the space is Riem a n n i a n and of constant curvature( 2). T o prove this theorem, Wang used, among others, the first of the following theorems due to D. Montgomery and H. Samelson [ 6 ] .

THEOREM D. In a n n-dimensional Euclidean space for n f 4 , there xzsts no proper subgroup of the rotation group of order greater than ( n- 1) (n - 2)/2. THEOREM E. In a n n-dzmensional Euclidean space for n 2 4 , n 2 8 ,a n y subgroup of the rotation group of order (n- 1)(n- 2)/2 fixes one and only one direction. Wang's Theorem C not only generalizes Theorem A, but also gives the following interesting Received by the editors December 28, 1951 and, in revised forni, March 24, 1952. (l) The numbers between brackets refer to the bibliography at the end of the paper. ($1 Professor H. C . Wang pointed out t o the author that Theorem D below, and consequently this Theorem c, are not true fsr R =4.

2 60

138

W-DIMENSIONALRIEMANNIAN SPACES

261

THEOREM F. A n n-dimensional Riemannian space for n > 2, n 2 4 , which is not of constant curvature cannot admit a group of motions of order greater than n(n- 1)/2 1 .

+

On the other hand, by studying the integrability conditions of the socalled Killing equations, I. P. Egorov [3] has proved recently the following two theorems.

THEOREM G. The maximum order of the complete groups of motions in n-dimensional Riemannian spaces which are not Einstein spaces I S n(n- 1)/2 +1.

THEOREM H. The order of complete groups of motions of those n-dimensional Riemannian spaces which are diferenf from spaces of constant curvature i s not larger than n(n- 1)/2 +2. According t o Theorem C, if a n n-dimensional Riemannian space for n > 2, n # 4 , admits a group of motions G, of order r >n(n - 1)/2+1, then the space is of constant curvature. T h e largest group of motions in a space of constant curvature being of order n(n+1)/2, if we denote it by G, then G, must be a subgroup of G. But, as will be seen in the next section, by exactly the same method as that used by Wang to prove Theorem C, we can prove t h a t the group G, coincides with the largest group G, t h a t is to say:

THEOREM 1. I n a n n-dimensional Riemannian space for n 2 4 , there exists no group of motions of order r such that

Thus, i t might not be useless t o study the n-dimensional Riemannian spaces which admit a group of motions of order n ( n - 1 ) / 2 + 1 . This is the main purpose of the present paper, in which Theorem E plays an important rble. T h e main result appears in the last Theorem 9. 2. We begin with a sketch of the proof of Theorem I. We consider an ndimensional Riemannian space V,, with positive definite fundamental metric form ds2=glk(x)dxjdxk (i,j , k, I, m = 1, 2, . . , n) and assume t h a t the space V,, admits a continuous group G, of motions of order r > n ( n - 1 ) / 2 + 1 . We take a n arbitrary point Po(xi)in the space V , and consider all the motions of G, leaving this point Po fixed. These motions constitute a subgroup G(P0) of G, consisting of the motions:

(1)

2' = f ' ( x ; a)

T,:

with the property i

i

xo = f ( x o ;

139

4,

KENTARO YANO

262

[March

being r o z r - n essential parameters [4, pp. 64-65]. The subgroup G(P0) is called a subgroup of stability of G, a t the point PO. T o each transformation T, of G(P0) corresponds a linear transformation S, defined by

CY

I t is easily seen that if T,+S, and T,t-fS,t, then T,T,r+S,S,r, where TUTU# is a product of two transformations while S,S,f is a product of two matrices. Thus, all the S’s forming a linear group L(Po),it is readily proved [8] that the correspondence T,-& is an isomorphism between G(P0) and L(P0) in the sense of topological groups, and consequently that G(P0) and L(P0) are of the same order. The group G(P0)being a group of motions fixing the point Po in a n n-dimensional Riemannian space, the group L(P0) is a rotation group in an n-dimensional Euclidean space. On the other hand, we know that the order ro of G(Po)or of L(P0)satisfies the inequality ro 2 r - n.

But, we are assuming that r

1

> -n(n - 1) + 1, 2

and consequently we have

Thus, from Theorem D, we must have, for n Z 4 , 1

ro = -n(n 2

- l),

and consequently the group L(P0)coincides with rotation group O(n). Thus, G(P0) contains a motion which carries any given direction a t Po into any given direction at Po. We note here that, in the above discussion, the point Po was an arbitrary point. Now, we take two arbitrary points PI and Pz in V , such that they are sufficiently near to each other and consequently they can be joined by a geodesic. We consider a midpoint M of this geodesic segment and a direction a t M tangent to this geodesic. Then, in the group of stability G ( M ) , there exists a motion which changes the direction of this tangent into the op-

140

19.531

263

n-DIMENSIONAL RIEMANNIAN SPACES

posite direction. Since a motion does not change the length of a curve and carries a geodesic into a geodesic, this motion carries P1 into Pz. If there are given two arbitrary points A and B in V,,, then we join these points by a curve, and choose a series of points on it P1,P2, * , PN in such a way that A and PI,Pt and Pz,P2 and P1, . . , P N - and ~ P N ,PN and B can be joined by geodesics. If we denote the midpoints of the geodesic segments , MN respectively, APi, P 2 2 , P2P3, * * * , PN-lPN, PNB by Mo,MI, then, applying suitable motions belonging t o G(Mo),G ( M l ) , * , G ( M N ) successively, we can carry the point A into the point B. T h e points A and B being any points in V,,, this means th at our group G, is transitive and consequently t ha t

-

--

--

r = ro

+ n = n(rt + 1 ) / 2 .

Thus, Theorem I is proved. 3. Next, we assume th at the space Vn admits a continuous group G, of motions of order r=n(n-1)/2+1. If we denote r infinitesimal operators of the group G, by

and the rank of the matrix (&(xo)) a t a point PO(& by q o , then n z g o , and the subgroup G(P0) of stability a t POis of order 14, p. 651 r - qo

1

=

-n(n - 1) 2

+ 1 - 40.

Now, suppose th at n>qo, then we have

Thus, the subgroup G(P0) of stability at Po, and consequently the corresponding rotation group L(Po), is of order greater than (n- l ) ( n- 2)/2. Thus, from Theorem D, we conclude that, for n Z 4 , the rotation group L(P0) coincides with O ( n ) . Thus, if we denote the generic rank of the matrix ( & ( x ) ) by q and assume t ha t n > q, then our Riemannian space admits free mobility around a n y point of the space, and, consequently, our group becoming transitive (see, for example, [2]), we have n =q, which contradicts our assumption. Thus, we must have n=p and consequently:

THEOREM 1. If an n-dimensional Riemannian space for n Z 4 admits a group of motions of order n ( n - 1 ) / 2 + 1, then the group i s transitive. In the following, since we need always Theorem E, we assume hereafter

141

2 64

"SNTARO YANO

[March

that n 2 4 , n f 8 . If we suppose that our V , admits a group G, of motions of order r = n(n- 1)/2 1, and if we fix a point P Oin Vn,then the above mentioned rotation group L(P0)is of order ( n- l ) ( n- 2)/2, and consequently, by Theorem E, it consists of all rotations around a fixed direction. Thus, with every point P of the space ITn, there is attached one and only one direction which is left invariant under the subgroup G ( P ) of stability a t the point P. We shall denote this direction by [(P). Now, we take two arbitrary points P and Q in our Riemannian space. Since the group G, of motions is transitive, there exists a motion T carrying the point P into the point Q. If we denote an arbitrary motion fixing the point Q by T Q,lhen the motion T-'TQT fixes the point P. Thus, applying T-~TQT to the direction [ ( P ) , we obtain

+

T-'TQTl(P) = ( ( p ) ,

[(P)being invariant under T-lTQT.From the above equation, we have TQTl(1')= r ( ( P ) , which shows that the direction T [ ( P ) a t Q is invariant under any TQ.Thus, we must have TE(P) =

t(Q),

and consequently :

THEOREM 2. If a n n-dimensional Riemannian space V,, for n # 4 , n # 8 admits a group of motions of order n(n - 1 ) / 2 + 1 , then there exists a jield of directions such that the direction [(P)at P i s transformed into the direction ( ( Q ) at Q by a n y motion of the group carrying the point P into the point Q. Now, we consider the geodesic which is tangent t o the above mentioned direction [ ( P )a t P. Since the group G (P ) of stability a t P is a group of motions and fixes the point P and the direction [(P), it fixes also t h i s geodesic pointwise. Thus, if we take an arbitrary point Q on this geodesic, G ( P ) fixes the point Q. Now, we consider an orthogonal frame a t P whose first axis is in the direction of E(P) and transport it parallelly along the geodesic to the point Q. Then we have, at Q, an orthogonal frame whose first axis is tangent to the geodesic. The parallelism of vectors along a curve being preserved by a motion, a motion belonging t o G ( P ) gives the same effect on the orthogonal frame a t Q as on that at P. This shows that the group G ( P ) behaves, a t Q, as a group of motions fixing the point Q and of order ( n - l ) ( n - 2 ) / 2 , and consequently that G ( P ) = G ( Q ) . The group G ( Q ) fixing the tangent to the geodesic and [ ( Q ) , the tangent must coincide with [ ( Q ) , and consequently the geodesic is a trajectory of the direction 5. Since there is one and only one trajectory passing through an arbitrary

142

19531

n-DIMENSIONAL RIEMANNIAN SPACES

265

point of the space, these geodesics depend on n - 1 parameters, and are transformed into each other by a motion belonging to G,. Thus we have [2, p. 2941:

THEOREM 3. I f a n n-dimensional Riemannian space V , f o r n Z 4 , n#8 admits a group of motions of order n(n- 1 ) / 2 + l , then there exists a f a m i l y of geodesics such that, passing through a point of the space, there i s one a n d only one geodesic of the f a m i l y and the geodesic passing through P i s transformed into the geodesic passing through Q by a motion of the group carrying the point P into the point Q. 4. Now we take a point 111 in V,,; then there is associated a direction t ( M ) with this point. We attach to this point a n orthogonal frame of reference [e,] i n such a way that the first axis el is in the direction of t ( M ) and we consider all the frames of reference which are obtainable from this original one by all the motions of the transitive group G,. Such a family of orthogonal frames of reference is said to be adapted to the group of motions under consideration. The frames of reference thus attached to different points of the space depend on n(n-1)/2+1 parameters, the first n of which are coordinates x1, x2, . . . , x” of the origin M and the last (n-l)(n-2)/2 of which are parameters v l , v2, . . . ,z ~ ( ~ - - l ) ( ~ - fixing ~)/~ the directions of the axes e2, . . . , e,,. Now, with respect to these moving orthogonal frames of reference, we write down the formulas (3)

dM

=

ole,,

de,

= u,,el

defining the Euclidean connexion without torsion of the Riemannian space under consideration. Here the w i and w,, are Pfaffian forms with respect to x and v . The frames of reference being orthogonal ones, we must have (4)

U S )

+

uJ% =

O.

Moreover, the frames of reference being adapted ones, the Pfaffian forms w , and w,, are invariant under the group [2, p. 2741.

We can see th at the forms w , are linear homogeneous in axi, because they must vanish when the point ( x i ) is fixed, tha t is to say, when the dxi vanish, and moreover th at the forms wlJ are also linear homogeneous in dxi because the vector el must be invariant, say, del=wl,e,=O, when the point (xi) is fixed. Thus, w I being n linearly independent forms, we must have relations of the form

(5) cjk

Wlj

= CjkWk,

being functions of x and v .

143

KENTARO YANO

266

[March

But, all the motions belonging to G, leave w l j and wg invariant, and consequently they leave c j g invariant. Thus, the group being transitive on all the frames, the C j k are all constants. T o find the values of these constants c j k , we shall follow a method given by E. Cartan [2, pp. 293-2951. At two infinitely nearby points M and M’ of the space, we consider the orthogonal frames of reference ( R M )and (Rw)both adapted to the group G,. Next, we effect, on both of them, an infinitesimal rotation around the first axis, these rotations being defined respectively with respect to ( R M ) and ( R M ~by) a bivector having the same components [ii.By the assumption, [ i j must satisfy (6)

Elj

=

= 0.

Ejl

We denote the orthogonal frames of reference thus obtained from ( R M ) and by (a,) and ( E M * respectively. ) Then the figure consisting of ( R M )and (%I) is congruent to the figure consisting of (RAP)and ( E M ’ ) , that is to say, there exists a displacement which carries at the same time ( R M into ) (RM’) and ( E M ) into (EM,). This displacement is analytically represented with respect to ( R M )by the set of vector w i and bivector wij. But, under the transformation of the orthogonal frames of reference which carries ( R M into ) (&), these components w i and w i j will receive the variations (&I)

(7)

6Wi

=

tikWk,

6Wij

=

fikokj

+

EjkWik.

But, from (5), we have 6Wlj

=

Cjkhk.

Substituting (5) and (7) into this equation, we find ElkWkj

+

tjkCklW1

=

CjkEklWlr

or (8)

fjkckl

- Cjktkl

=

0

by virtue of [ 1 k = 0 and the linear independence of First p u t t i n g j = 1 in (5), we find (9)

Clk

= 0.

Next, putting 1 = 1 in (8), and taking account of fjkckl

which must be satisfied by any conclude (10)

Ckl=O,

we get

= 01

(= -&) ckl

01.

= 0.

144

satisfying (6), from which we

19531

Finally, since

267

n-DIMENSIONAL RIEMANNIAN SPACES

we may consider that the summation index

&j=kjl=O,

k in (8) takes the values 2, 3, . * , n only. Then, in (8), putting j = r , (1, s, t , u , v = 2 , 3 , . . , n ) , we obtain 1

-

Erscst

Z=t

= 0, *

Cl85S'

which may be also written in the form (11)

Euv(6urcvt

-

= 0.

cru6ut)

Equation (11) must be satisfied for any which we get (durcue

-

-

cr u8 ut)

fuv

satisfying

( S u r ~ u t- ~ r u 6 u t )=

&,+fvu

= O , from

0-

Contracting, in this equation, with respect to r and v , we find

(12)

+ ctu

(n - 2)cut

=

cUuL

If n = 3 , then we have (13)

Gut

+

ctu

=

Cvu6u1,

and consequently, we can conclude from ( 9 ) , (lo), and (13), that the matrix ( c j k ) has the form

(: :). 0

(14)

(cjk)

=

c

-a

0

c

Since this case was throughly studied by E. Cartan [ 2 , pp. 300-3061, we assume hereafter n > 3 , n f 4 , n Z 8 , that is to say, n > 4 , n Z 8 . Then, taking the anti-symmetric part of both members of (12), we find ( n - 3 ) ( c U t - c e , ) = 0 , from which =

6.1

CfU,

and consequently 1

cut

=

Thus, in this case, the matrix

-c.v&bt. n-1 (cik)

has the form

0 c

o*..o o***o

0 0

c * . - 0

0 0

. . . . . . 0 0 0 . e . c

145

268

KENTARO YANO

[March

Consequently, we have, from (S), (16)

Wl*

=

CWI.

Thus, from the equations of structure dwi =

[ ~ i j ~ j ] ,

we find do1 =

Thus, the Pfaffian form (17)

w1

[w1*osJ

=

0.

is an exact differential: w1

= dg(x),

and, consequently, we can see that, in our space, there exists an m 1 family of hypersurfaces g ( x ) =constant along which we have w1 = 0, or

dM = me2

-

+ - - + wnen.

, enare always tangent to one of these hypersurfaces, Since vectors e2, we can see that these hypersurfaces, regarded as ( n- 1)-dimensional Riemannian spaces, admit the free mobility. Thus, these hypersurfaces regarded as ( n- 1)-dimensional Riemannian spaces are all of constant curvature. I t is clear that the orthogonal trajectories of these hypersurfaces are geodesics referred to in Theorem 3. Since any of these geodesics which are orthogonal trajectories of these hypersurfaces is transformed into any of these geodesics by a motion of G,, we can see that any of these hypersurfaces is also transformed into any of these hypersurfaces by a motion of G,. Thus, these hypersurfaces, regarded as ( n- 1)-dimensional Riemannian spaces, must be of the same constant curvature. Now, we must distinguish here two cases: (I) c = O and (11) c # 0 . We shall first assume that c = O in (16). Then we have (18)

Wlr

= 0,

del

=

and consequently (19)

0,

which shows that the el is a parallel vector field. Thus, the normal to the hypersurfaces referred t o above being always parallel, the hypersurfaces must be totally geodesic, their orthogonal trajectories being geodesics. We next assume that c Z 0 in (16). Then we have 1

ws =

018, t

and consequently

146

19531

n-DIMENSIONAL RIEMANNIAN SPACES

269

from which

(20) which shows that

along the hypersurfaces w1=0, that is to say, the vector el is a concurrent vector field [ l o ] along the hypersurfaces referred to above. The normals to these hypersurfaces being concurrent along them, these hypersurfaces are totally umbilical hypersurfaces with constant mean curvature and their orthogonal trajectories are geodesic Ricci curves. Thus we have :

THEOREM 4. If a n n-dimensional Riemannian space V , for n > 4 , n # 8 admits a group of motions of order n(n- 1 ) / 2 1 ; then (I) there exists a n 00 1 f a m i l y of totally geodesic hypersurfaces whose orthogonal trajectories are geodesics, these hypersurfaces regarded a s (n- 1)-dimensional Riemannian spaces being of the same constant curvature, or (11) there exists a n 00 f a m i l y of totally umbilical hypersurfaces with constant mean curvature whose orthogonal trajectories are geodesic Ricci curves, these hypersurfaces regarded a s ( n- 1)-dimensional Riemannian spaces being of the same constant curvature. In both cases, the group leaves the f a m i l y o f geodesics and that of hypersurfaces invariant.

+

5 . We shall first study case (I). If case (I) in Theorem 4 occurs, then, the normals t o these hypersurfaces being a parallel vector field, by a well known theorem [ l o ] ,there exists a coordinate slstem in which the fundamental metric form of the space takes the form

the form gst(xr)dxadxLbeing the fundamental metric form of an ( n - 1 ) dimensional Riemannian space V,-1 of constant curvature. Conversely, if there exists a coordinate system in which the fundamental metric form of the space V , takes the form ( 2 1 ) ,g,,(xr)dxadxLbeing the fundamental metric form of an (n-1)-dimensional Riemannian space V,-1 of constant curvature, then it is evident that case (I) in Theorem 4 occurs and the space admits a group of motions G, of order n ( n - 1 ) / 2 + 1 :

147

2 70

KENTARO YANO 3' = x1

+ t,

37

[March

= f ' ( x ; a),

where %r = f ( x ; a ) represent the group of motions of order n(n- 1)/2 in the (n- 1)-dimensional Riemannian space lTn-l of constant curvature. T h u s we have :

THEOREM 5 . A necessary and su&cient condition that case (I) in Theorem 4 occur i s that there exist a coordinate system in which the fundamental metric form of the space takes the f o r m (21), glt(xr)dxsdxtbeing the fulzdamental metric f o r m of a n (n- 1)-dimensional Riemannian space of constant curvature. In this coordinate system, the fundamental tensors being of the form

if we calculate the Christoffel symbols of V,:

we then find (22) the other

being zero, where

denotes the Christoffel symbols of Vn-l:

Next, calculating the Riemann-Christoffel curvature tensor of Vn:

we find (23)

148

19.531

271

n-DIMENSIONAL RIEMANNIAN SPACES

the other R i j k l being zero, where R*StU denotes the Riemann-Christoffel curvature tensor of V,,-I:

But, we know that

R* (24)

R*rartL

=

(gars:

- 2)

(n - l)(n

-

g8Us:),

R* being a n absolute constant, and consequently we have, for the Ricci tensor

Rjk=Rijki

the other =gikRjk

Rjk

Of

Of

v,,

being zero. From (25), we obtain, for the scalar curvature R

v,,

R

(26)

=

R*.

Thus if we p u t

we then find Tll

=

791

=

R* 2(n - l ) ( n - 2)

(28)

1

A,t

= -

7

Art

=

R* 2(n - l ) ( n - 2)

-

R*g,t 2(n - l ) ( n - 2)

R*6: 2(n - l ) ( n - 2)

1

the other t ' s being zero. Thus, for the Weyl conformal curvature tensor: i

(29)

c jkl

=

R

i jkl

-k

i njksl

i

- Ajl8k

+

i gjkr 1

-

i g j l a k,

we find (30)

C"j k l

= 0.

Thus, since we are assuming n > 4 , our space must be conformally flat. Concersely, if we assume that our space is conformally flat and admits a parallel vector field, then there exists a coordinate system in which ds2 = ( d x ' ) 2

+ g,t(xr)dx'dd

149

KENTARO YANO

2 72

[March

and

the other

and

Rij&l

Rjk

Tll

=

TI1

=

being zero. From these we have 2(n -

R* l ) ( n- 2 )

'

R*

2(n - l ) ( n

- 2)

!

+

A,:

=

R*a: R*ga: -n-2 2(n - l>(n- 2 ) '

=rt

=

R*': -n-2

+ 2(n -R*6: l > ( n- 2 ) '

the other T ' S being zero. First, from clatl

+

= rat

g * t d = 0,

we find

R* R*,:= -g a t , n-1

and consequently a,t =

-

R*gat

2(n - l ) ( n - 2 )

Trl

J

= -

R*6:

2(n - l ) ( n - 2 )

Next, from

we find R*',:, =

R*

( n - l ) ( n- 2 )

(gat&

- gaud),

which shows that the hypersurfaces x1=const. regarded as (n- 1)-dimensional Riemannian spaces are of the same constant curvature. Thus we have:

THEOREM 6. A necessary and suj'icient condition that case (I) in Theorem 4 occur i s that the space be conformally $at and admit a parallel vector field. T. Adati and the present author [ 1 2 ] proved that a necessary and suffi-

150

19531

2 73

n-DIMENSIONAL RIEMANNIAN SPACES

cient condition that a space be Kagan's subprojective space is that the space be conformally flat and admit a concircular vector field. Thus Theorem 6 shows that the space under consideration is Kagan's subprojective space. Next, we shall try to get a characterization by curvature tensor of the space referred to above. First of all, there exists, in our space, a parallel vector field E': (31)

tj;k

=

0,

semi-colon denoting the covariant differentiation. We assume that t i is a unit vector field and, f i being a gradient field, we Put

First, from (31), we find (33)

Rijklti

= 0,

(Rijkl

=

girnRmjkl).

The sectional curvature a t a point of the space determined by a 2-plane containing the unit vector f i and an arbitrary unit vector vi orthogonal to ti is given by -Rijk&;qj[kql. But, the space admitting a transitive group of motions which carry the field 5' into itself and any vector orthogonal to ti into any vector orthogonal to t i ,this sectional curvature must be an absolute constant. But, from (33), we have

-

(34)

RijkltiQitkg1

=0

for any vi, which shows that this sectional curvature is always zero. On the other hand, we know that the hypersurfaces given by g(x)

(35)

=

constant

are totally geodesic and are of the same constant curvature. T h u s , representing one of (35) by parametric equations: xi

= xi("),

and putting Vri

=

axi -,

aur

we have, from the equation of Gauss, (36)

R*rsttr

= Rii/cttl,'tlaitltk7u1,

where R*ratu are components of curvature tensor of the hypersurface, and consequently have the form

151

KENTARO YANO

2 74 (37)

~ * r e t u=

**

[March

* *

K(g,tgru - g,ugrt),

g: being the fundamental tensor of the hypersurface: g:

=gjk%.jqlk.

*

gat =

gjkqajqtk*

The K in (37) represents the sectional curvature determined by a 2-plane orthogonal t o [i. The space admitting a transitive group of motions fixing Ei invariant, K must be an absolute constant. Now, putting qaj=gijg*r,'qri,we have i r

(38)

Vr

q

j

=

i 6j

i

- t ij,

*

6

gatq

t

jq k

=

gjk

- [jib

.

Multiplying both members of (36) by r]raq'b?fc?pd ( a , b, c, d = 1, 2, and contracting, we have, by virtue of (37) and (38),

**

K(g8tgru

*

*

r

a

t

, n)

u

- gaugrt)q a 7 b?l c7

d

=

Riikl(6;

- fi[a)(S'a - [ ' [ b ) ( S c

k

k

- [ [,)(ad

l

1

- 4 Ed),

or, by virtue of (33) and (38), K[(gbc

- tbgc)(gad

- [aid) -

(gbd

- tb[d)(gac

-

ia[c)]

=

Rabcd,

from which (39)

Rijkl

=

K[(gjkgil

- gilgik) - (figik

- ijgik)fl + (tigjl - tigil)lk]*

Conversely, suppose that the curvature tensor o f the space has the form (39) where K is a constant and f i is a unit parallel vector field. T h e vector [i being a gradient, if we put 4i=dg/dxi, then the hypersurfaces g(x) =constant are totally geodesic and their orthogonal trajectories are geodesics. Representing one of these hypersurfaces by xi = xi(ur), we have, from (39) and the equation of Gauss,

where R*rstuis curvature tensor of the hypersurface. This equation shows that the hypersurfaces regarded as (n - 1)-dimensional Riemannian spaces are of the same constant curvature, Thus we have:

THEOREM 7. A necessary and suficient condition that case ( I ) in Theorem 4 occur i s that the curvature tensor of the space have the f o r m (39) where K i s a constant and f i i s a unit parallel vector field. From (39), we have

(40)

Riikl;m

= 0.

Thus, we can see that our space is symmetric in the sense of E. Cartan [2]. 6. We shall next study case (11). If case (11) in Theorem 4 occurs, then

152

19531

%-DIMENSIONAL RIEMANNIAN SPACES

2 75

the normals to the hypersurfaces being Ricci directions, by a well known theorem [9], the space admits a so-called concircular transformation [9] and consequently there exists a coordinate system in which the fundamental metric form of the space takes the form [9]: (41)

+

ds2 = ( d ~ ' ) ~f(x')f,;(xr)dxaddxt,

the form g.,dx"dd =f(x1)frt(xr)dx8'dxtbeing that of ( n- 1)-dimensional Riemannian spaces V,+I of the same constant curvature. Here, if the functionf(xl) reduces t o a constant, then our case reduces t o case (I). Consequently, in this case ( I I ) , we assume thatf(xl) is not a constant. Calculating the Christoffel symbols of V,,, we find

the other

being zero, where f' = d f l d x l and the

or, what amounts to denote Christoffel symbols formed from g, =f(xl)f&') the same thing, fromfir(xT). Next, calculating the Riemann-Christoffel curvature tensor Rijklof V,,, we find

(43)

the other Rijkl being zero, where R*r,tudenotes Riemann-Christoffel curva ture tensor of V,+l. From (43), we get

153

-

KENTARO YANO

276

[March

the other R i j k l not related to these being zero. From the first equation of (44), we see that the sectional curvature determined by two unit orthogonal vectors (1, 0, 0,

*

a

*

I

O),

(0, T 2 , q 3 ,

*

'

9

V")

is

(45) and does not depend on (0, 72, . . , 7"). But, the space admitting a transitive group of motions which carry the field (1, 0, 0, * * , 0) into itself and any vector orthogonal to it into any vector orthogonal to it, this sectional curvature must be an absolute constant. From the second equation of (44), we see that the sectional curvature determined by two mutually orthogonal unit vectors

-

(0, TIa,

9,' '

9

(0, S2,

T">,

Sat *

' *

I

S")

is

This having to be independent of the choice of 7' and f',we must have (46)

R*r,tu

= K*(gatgru - gaugrt),

and consequently (47)

The group being transitive, this scalar must be also an absolute constant. Equation (46) shows that the hypersurfaces x1 =const., regarded as (n- 1)-dimensional Riemannian spaces, are of constant curvature. But we know that these must be of the same constant curvature. Thus, K* is also an absolute constant. On the other hand, we have gat = j(xl)fat(xr>,

and consequently R*ratu

= Fratup

where Patuis the Riemann-Christoffel curvature tensor formed with fet(xr).

154

%-DIMENSIONALRIEMANNIAN SPACES

19531

277

or

where

F = fK*

(50)

is an absolute constant. Now, we know that F and K * are both absolute constants. But, we are assuming that the function f ( x ' ) is not constant. Thus, we must have

K* = 0,

F = 0,

from which (51)

Frstu

=

R*,,t, = 0.

0,

Moreover, the right-hand side of (47) being a constant, we put 1 f'2 _ _ -- k2,

4f2

k being a constant different from zero, from which we get

f = a2e2kr'l

(52)

a2 being an arbitrary positive constant.

Thus, the fundamental metric form (41) takes the form

+

ds2 = ( d ~ l ) ~0 2 e 2 k z ~ 8 t ( x r ) d x a d x 1 ,

(53)

where the form fat(xT)dx8dxtis, as equation (51) shows, the fundamental iiietric form of an ( n- 1)-dimensional Euclidean space. Moreover, substituting (52) into (44), we get Rlelu

=

+

K2gsu,

RrstiL

=

- k2(gstgrtt - gaugTt)t

which may be also written as (54)

R z.j.k l - - k2(g.j t g i t - g j l g i k ) .

Thus, the space is of negative constant curvature. Conversely, if an nLdimensiona1 Riemannian space is of negative constant curvature-K2, then it is well known [ l ] that its metric can be written in the form (53), or

155

,-,

278

[March

KENTARO YANO

or, on putting

in the form

ds2 =

(56)

du2

+ (dx2)* +

* * *

+ (dx")'

k2u2

Thus, the space admits a group of motions of order n(n- 1)/2

+ 1 given

by ii = au,

(57)

3'=

r

a(a,x

e

+ a ), r

where a is a parameter and *r

x

= a:x'+

d

represents a general motion in an ( n- 1)-dimensional Euclidean space. Thus we have:

THEOREM 8. A necessary and suficient condition that case (I I) in Theorem 4 occur is that the space be of negative constant curvature.

7. Gathering all the results, we can state the following:

THEOREM 9. A necessary and suficient condition that a n n-dimenszonal Riemannian space V,, for n > 4 , n Z 8 admit a group G, of motions of order r =n(n- 1)/2 1 is that the space be the product space of a straight line and a n ( n- 1)-dimensional Riemannian space of constant curvature (this i s equivalent to the fact that the space i s conformally $at and admits a parallel vector jield) or that the space be of negative constant curvature.

+

The author wishes to express here his gratitude t o Professor D. Montgomery and t o his colleagues, Professors K. Iwasawa, H. E. Rauch, and H. C. Wang, discussions with whom were very valuable during this research. BIBLIOGRAPHY 1 . L. Bianchi, Lezioni d i geometria diferenziale, 3d ed., vol. 11.

2. E. Cartan, LeCons sur la gLomLtrie des esbaces de Riemann, 2d ed., Paris, GauthierVillars, 1946. 3. I. P. Egorov, On a strengthening of Fubini's theorem on the order of the group of motions of a Riemannian space, Doklady Akad. Nauk SSSR. N.S. vol. 66 (1947) pp. 793-796. 4. L. P. Eisenhart, Continuous groups of transformations, Princeton University Press, 1933. 5. G. Fubini, Sugli spaaii che ammettono un gruppp continuo di movimenti, Annali di Matematica (3) vol. 8 (1903) pp. 39-81.

156

ri-J~IRII~NSIONAI,IIIEMANNIAN SI’ACES

195.1J

2 79

6. D. hlontgoinery and H. Samelson, Transformation groups of spheres, Ann. of Rlath. (2) VOI. 44 (1943) pp. 454-470. 7. P. Rachevsky, Caractires tensorirls de l’espace sous-projeclq, Abhandlungen des Seminars fur Vektor- und Tensoranalysis. hloskou vol. 1 (1933) pp. 126140. 8. H. C. \Vniig, On Finsler spaces z d h completely integrable equations of Killing,J . London hlath. SOC. vol. 22 (1947) pp. 5-9. 9. K. Ynno, Concircular geottidry. 11. Integrability conditions of c,,” =+gUr, Proc. Imp. Acad. Tokyo VOI. 16 (1940) PI). 354-360. 10. , Sur le paralltlisine et la cortcourance duns l’espace de Riemanrt, Proc. Imp. Acad. Tokyo vol. 19 (1913) pp. 189-197 11. __ , Groups of transforvtations in generalized spaces, Tokyo, 1949. 12. I<. Yano and T. Adati, On certain spaces admitting concircular transformations, Proc Jap. Acad. vol. 25 (1949) pp. 188-195. ~

INSTITUTE FOE ADVAXCED Sruw, PRINCETON, N. J.

157

This Page Intentionally Left Blank

KONINKL. NEDERL. AKADEMIE \’AN WETENSCHAPPEN - AMSTENDAM Reprinted from Proceedings, Series A, 58, No. 4 and Indag. Math., 17, No. 4, 1954

MATHEMATICS

ON GEOMETRIC OBJECTS AND LIE GROUPS O F TRANSFORMATIONS BY

NICOLAAS H. KUIPER

AND

KENTARO YANO

(Communicated by Prof. J. A. SCHOUTEN at the meeting of June 25, 1956)

Introduction EHRESMANN defined prolongations (prolongements [ 1, 21) of a Cs-manifold. A prolongation is a principal fibre bundle which is for a great deal determined by the base space a Ca-manifold and a natural number, the object class t 5 s. I n differential geometry one often is led to fibre bundles which are not principal, but of which the associated principal fibre bundle is of the kind defined by EHRESMANN. Following HAANTJES and LAMAN[3] these fibre bundles will be called geometric object bundles. Geometric objects or geometric object bundles have been defined by GOLAB, HAANTJES, LAMAN, NIJENHUIS,SCHOUTEN, WAGNERand others. I n Q 1 we give a definition of a geometric object bundle. I n 5 2 we consider transformations in the base space of a geometric object bundle, and their prolongations in the bundle. I n Q 3 we obtain a main theorem on geometric objects on a Lie-group space with applications. 5 4 deals with Lie groups of transformations of a manifold and the existence of invariant geometric objects. I n 5 5 we define Lie-derivatives of geometric object fields and give some applications. More applications will be given in a second paper. I.

Q 1 . Geometric object bundles I n this paper A will denote a fixed pseudo-group of homeomorphic mappings of differentiability class Ca, s > 0, in n-dimensional number space R”, which contains the group of translations : TCA. Two homeomorphic mappings yk : Uk + v k k- i , j’ of neighbourhoods U k of an n-manifold X into Rn are called A-compatible l) if 1.. 12 = y 1. K 1I Fi(Vi n Ui)E A . (The mapping 1, is only defined in the neighbourhood mentioned on the right hand side of the vertical bar.) -4manifold with a local A-structure or a A-rna?iiiold is a manifold covered by a complete Set of mappings v k : u, --f v k of the above kind, 1)

Compare

\-EBLEN

and WHITEHEAD [4].

412

any two of which are A-compatible. The homeomorphic mappings are called A-coordinate systems, A-reference systems or just reference systems. I n the sequel X will be a A-manifold. Points of X will be indicated by x ; points of R” will be indicated by z ; in particular the point (0, 0, ... , 0) by 0. Two reference-systems qi and q ~ ,both covering x C X are called jetequivalent a t x if their restrictions to some neighbourhood of x are identical. The jet-equivalence class of {x,vi} is called a jet of kind A or A-jet and it will be denoted by j(vi(x),x ;vi)=j(vj(x),x ; cp,); z is called the 8ource of this j e t , z= vi(x) is the butt of this jet 2). If f C A , z E R” is covered by f , then the jet determined by f with source x is denoted by j(z’, z ; f ) = j(z’, z ) , where

z’ = f ( z ) .

Jets of this kind will sometimes be called auto-jets. If the butt of a first jet coincides with the source of a second jet, then the product can be formed : j(%,z l ; f a f l ) = i ( z 3 ,z 2 ; f z ) . j ( z z , 21; f l ) j@2, x;fcp)=j(z,, 21; f).i(z,, x ; v). The jet with source z1 E R” and butt z2 E R” obtained from a (unique) translation t(z2, zl) is itself denoted by t(z2,3). P r o p o s i t i o n 1. The jets of the kind j(0, 0 ; f ) form a group d. We introduce a non-Hausdorf topology in d with respect to a Ca-A-manifold by the definition: a neighbourhood in A consists of all jets that can be represented by functions whose systems of derivatives up to the sth, a t the source of the jet, forni a neighbourhood in the suitable number space, P r o p o s i t i o n 2 . Any jet of the kind j(z’, z ; f ) admits a unique factorization as follows

f(z’,O).j[O, 0 ; t ( 0 ,:’).,/.t(z, O ) ] ’ t ( O , z ) short : j(z’, x ) = t ( z ‘ , O ) . j ( O . ( ) ) . I ( ( ) ,

2).

The mapping Oj : j ( z ’ , z ; f ) + j[0, 0:t ( 0 , c ‘ ) . f . l ( z , O)] is a homoniorphism of the pseudo-group of auto-jets onto the group of jets with source = = butt = 0. P r o p o s i t i o n 3 . Two jets j ( z , x ; ql) a i d j(i’, x ; p2) with the source .7: E X determine a unique auto-jet j(z‘, z ) by division:

stxilie

x ,;pJ. j(z‘, x ; v2)=j(z’,z ; y, q c l ) . j ( ~ P r o p o s i t i o n 4. If ql,pz, y 3 determine three jets vitli the same source x and with butts zi= v,(x), then the quotient-auto-jets obey i ( z 3 ,zl;v3 ~

l

= j ()c 3 , x z ;

e3q p l ) . j ( z z .z l ; p2 pi-])

160

413

and this product rule also holds for the images of these jets under Oj, which we denote as follows: ')

j31('3

or

1'

j32(',

L1=j32.iZl

*j21(0,

iii E

O)

A.

T h e o r e m 1 . T h e entities X , Y , Q?h defined below determine a unique fibre bundle B with base space X , fibre of the kind Y , group G , and homomorphism h : A + G , which, i s called the geometric object bundle over X of the kind ( Y ,G , h).

X is a C-A-manifold of dimension n. Y is an analytic manifold. G is a Lie group of analytic transformations of Y . h is a continuous homomorphism of d onto G. The bundle is defined as follows. Let pi : U i.+ Vi(l J i C X , Vf C Rn) be A-reference-systems covering X 3 ) . In t,he set of triples ( i ,x E Ui, y E Y ) we introduce the equivalence, called identification :

( i ,x.y) gii

=

N

(j,x , qji?y) for x E U i U i

hj..: 71 j.. 11 = t ( O , Z j ) *j(zi.zi:qliqli') -t(z,,O)EA zi = ( P < ( X ) zj = (Pj(2).

An equivalence class is by definition a point of the fibre bundle. The equivalence classes with a fixed .r form the fibre of the bundle a t x. The bundle projection n is the mapping of the fibre a t x, onto x. The fibre z-'(x)= Y , is homeomorphic with Y . The mapping: class of (i,2, y) + (x,y)

(1.1)

is a homeomorphic mapping of n-l(7Ii) onto U ix Y and will be denoted by i = (nx pi*). We t,hen have for b E n-'( U i )

n-'(U,) i lj -+

If b

E Z-1

-5

rch x

( U in Ui), then $4

Ui x Y 2-t

qlab =

vix Y

% cpinb x pfb.

gii Tab, and if b

E

n-l ( U in C, n tJk),

then j(z,,zi:p&y1)

= j(z,,Zj:qlkql;')

jki =

*

j(zj,Zi,qljqq')

. . 9... il 7%

hence, because h is a continuous homomorphism, glii = g k j

Also gii is a. continuous function of

'

Qji.

2.

3) Jf we require moreover t h a t the sot of A-reference systems is complete, that is not contained in a (A-c~ompatlhle)bigger set, then the definition4 are independent of the particdar 3et of snhsets (Vi}of X . Compare STEENROD[Y].

161

414

The mappings (1.1) which fulfill all the conditions just mentioned [8]. define the structure of fibre bundle in the point set n-'(X). STEENROD The fibre bundle B so obtained is the object-bundle required in theorem 1. If Y' C Y is invariant under G , then X, Y ' , G , h determine a unique object-bundle B', which can be considered as imbedded in B . We call B' a subbundle of B. A cross-section of B is called a geometric object field or geometric object of the kind ( Y , G ,h). One point of B : b E Y , is called a geometric object at x. Example : Let A be the pseudo-group of all C8 reversible homeomorphisms in R". r be the invariant subgroup of A consisting of those jets that can be obtained from homeomorphisms in A , that are expressed by functions ,

I

2%)

(21,

zk=zk

<

which have the same L-th derivatives for k = 0, I , ..., 1 5 t s a t the point 0, as the functions that express the identity homeomorphism : Zk = Zk

h is the homorphism which maps a jet in A onto its image in G'= A/r. The object-bundle B so defined is a manifold of differentiability class s - t . A C'-cross-section in B ( r 5 s - t ) is called object (field) of object class t and (of course) of differentiability class r . HAANTJES and LAMAN[31 determined all transitive geometric objectbundles of object-class t = 1 and dimension n 1 (dimension Y = 1). Tensor-bundles are bundles of class t = 1. iiffine connections (parallel displacement) can be defined in the bundle of tangent vectors of a C2-manifold. The affine connections are themselves cross-sections in a bundle of object-class t = 2. Every affine connection belongs to a class of projectively equivalent affine connections, which class determines a unique normal projective connection. E. CARTAN [5]. Such a normal projective connection is a cross-section in a bundle of objectclass t = 2 . It is not easy to determine whether a given connection which is defined in a general way in a fibre bundle can be considered as a geometric object field. As an example we mention a projective (conformal) connection in a fibre-bundle with fibre the projective n-space (n-sphere), with group the projective (Moebius) group and without a fixed oblique cross-section. EHRESMANN [B], KUIPER["I. Every product bundle is a geometricobject bundle however (G = d / A = 1 ) . Another example of a geometric object bundle is obtained from the tensor-bundle of covariant tensors of kind f X A under the identification :

+

La

N

e

~ X A

e > 0-

If t,, is symmetric and positive definite then the geometric object is called a conformal metric. It is of class t = 1. The normal conformal con162

413

neotion determined by a conformal metric is a geometric object of class t= 2.

Other examples of geometric object-bundles are obtained by taking for A a subgroup of all reversible C8(s= 1 , I ,... 00 or O J ) homeomorphisms, for example consisting of all homeomorphisms that leave invariant a fibred structure or a complex structure in R". \T'e conclude this paragraph with : T h e o r e m 2 . If X i s a Ca--product space of k circles and a euclidean n-k-space, then a n y geometric object bundle over X is a product bundle. The same is true for a n y open sub-spuce of such a space X . P r o o f : The universal covering 3 of' X admits the space R" of rows of 71 numbers (zl,..., 2,) as one coordinate system x that covers 2 and such that the fundamental group of X is generated by the k transformations 2; = 2, b:

+

for j = 1 , ... and I; respectively (6: is the unit matrix). \Ye chose coordinate systems in X that are the product of the natural induced homeomorphism of a sufficiently small neighbourhood in X onto one in 2 and this coordinate-system 1c: 3 + R". The only element in A that is obtained from pairs of such coordinate systems for X , is the identity. Then the only element in G' that occurs in the description of the object bundle is also the identity, and therefore the bundle is a product bundle. ,4s the circle is base-space of non-trivial fibre bundles, this implies Theorem 2 ' .

R o t ever!/ fibre bundle is

u

georndric object bundle over

its base space. An interesting problem is the characterisation of all object-bundles among the fibre bundles.

8

I . Prolongations of A-transfownnfions in X

Let b be a cross-section or a geometric object-field in the object-bundle ( B ,X, Z, Y , C, h, A ) . b(x)= Y , n 6. A homeomorphism 71 : U' + U in X is called a A-point-transformation if in case y : li -+ 1' is a A-referencesystem, the same is true for yq : IJ' + TI.

t

f V*

1'

163

416

The A-point transformation q has a unique prolongation { EHRESMANN [ I ] ; cp* and (cpq)* are defined in ( I . l ) } :

defined by, if b' (4.1)

E

n-l(u')o n t o +

n-l

(U)

n-l ( U ' ) , :

q*b'

= z-l

q n b' n (cp*)-I (cpq)*b'.

Prolongation commutes with projection : nq* = qn. (4.1) can be understood as follows: If we use the reference systems cp and cpq for U and U' then the prolongation q* of q is expressed by the pair {q : U' -+ U , and identity in Y } . An expression in terms of two arbitrary reference systems cp : U --f V cpU and v : U' -+ V'=yL7', instead of cp and cpq respectively, is as follows (2'= nb'). (4.2)

. j(plp', WX'; cpqy-l) -

~j*b'= n-lqb'n n (cp*)-'h [t(o,cp&) * t(vx', 0) ] v*b'.

Substituting y = q q in (4.2) we obtain (4.1) again. (4.2) is independent of the particular reference system cp for U . This can be seen from straightforward computation. From (4.1) we have: (q-1)* = (q*)-1.

Therefore (4.2) is also independent of the reference system y for U' hence independent of reference systems used. Now suppose we have two A-point transformations. q : U' + U and U" -+ U ' , hence the product qv : U" + U . Using the reference systems q : U -+ V , cpq : U' + V and cpqv : U" --f V , we observe that the prolongations q*, Y*, ( q v ) * are respectively expressed by :

v :

( q , identity) ( v , identity)

(qv, identity). Hence (qv)*=r]*v*, and we have the T h e o r e m 3 . Every A-transformation in X has a unique prolongation in B. T h e mapping which assigns to every A-transformation its prolongation is a group-isomorphism. From the definitions we also have: Any subbundle B' C B is invariant (not point wise) under the prolongation of any A-transformation in X.

8 3. Geometric objects o n Lie g r o u p T h e o r e m 4 . Let X he a Lir group of transformations operating on the left o n the group space H of X . Any object bundle B with base space H and fibre space Y i s a n analytic product bundle H x Y with left invariant analytic cross-secfions H x y (?J E Y ) . 164

417

P r o o f : Choose a fibre Y, C B, and a point b E Y,. Let ? i ( t )E A? be a transformation of H , q*(t) E H *its prolongation in B , and t a point of the abstract analytic group of %'.The set .@*b cokists of one point in each fibre and is an analytic cross-section because q(t) is an analytic transformation, which depends also analytically on t . A point b' C X * b is characterised by b E Y , and n(b')=x'. The analytic correspondence 6' -+ x' x b of B = n - l ( H ) onto H x Y,, so obtained, is 1- 1, and q* E X * the prolongation of q E S is represented under this representation by q*(x' x y)=qx' x y.

Application : T h e o r e m 4 ' : An n-dimensional Lie group has the following left invariant geometric object fields : a n absolute parallelism ; m a n y ufine connections among which symmetric connections ;Riemannian metrics of a n y signature ; for n even m a n y almost-complex structures and almost-hermitian metrics ; Finder metrics. In all these cases we define suitably the geometric object a t one point of the group and the required object-field consists of the images of this geometric object under the prolongations in the fibre-bundle of the tr&nsformations of the group.

Geometric objects and transitive groups of transformations I n this 9 H is a transitive group of C"-A-point transformations of a C"-A-manifold X. base space of an object-bundle B. I, is the subgroup of all transformations in H that leave x E X fixed. The prolongations in B of I , and H are I,* and H * . I , and I,* are called group of isotropy of H ( H * ) a t x. fj 4.

T h e o r e m 3 . T h e object bundle B over X admits a cross-section b invariant u n d e r all prolongations in H * , if and only if the isotropy group I t at x has u fixed point in Y , (for some z E X , and then for a n y x E X ) . P r o o f : The necessity is obvious. To prove the sufficiency we consider a point b, E Y , invariant under I:. For any two transformations f l and in H , wliich map x onto the same point 5 ' . the prolongations f:, f: obey:

(fWf r E I,* (I;)-' f : b?==b,. f; ox=/; bL. Therefore the point set (/* 6L.. f * E H * . contains cssctly oiie point in the fibre Y,, for any .r' E X. The group properties imply that this crosssecfion {/* 6,) is invariant under H * . In the applications it often occurs that the homomorphism-onto f: --f I,* 1 Y,, defined by restriction of the transformations of I, to the fibre Y,. is an isomorphisin. This is the case when the restriction of an element q* of I,* to Y , uniquely determines q*. In the proof of many

theorems on groups of transformations leaving invariant some geometric object we therefore may restrict ourselves to considerations concerning one fibre Y,. For example: T h e o r e m 6 . {Z} Let X" be a space with a) a Rierrmmian metric, b ) a n afine connection, c ) a Kahleriun metric, d ) a n afine connection with a n invariant almost complex structure, with a N-dimensional group of structure preserving transformalions. ( I I ) Let NO be the dimension of the group of a ) motions in a space of constant curvature, 6 ) affinities in the affine space, c) motions in a F u b i n i - s y w , d ) complex-anal?Jtic cr ffini tiee in complex afine space. Then N 5 NO, and equality N =

hT0

implies th at X" is of the kind mentioned under (ZZ).

N Z = n ( n + 1)/2. N ! = n ( n t 1) and putting n=2m N: = m2 + Zvn, Xfl= '7rta2-C 2 m . P r o o f : I n a space with an affine coiinection, of which cases abcd are examples, a n affine point transformation with fixed point x is determiiied by its prolongation restricted to the tangent space a t .c. This ensures the faithfulness of the representation of I: in the tangentspace. The dimension I) of the isotropy-group obeys *V-n j I1 5 ,IrO- 7 1 , hence N 5 XO. Next suppose Ly=i+'O. I n all cases abcd, there is an affine connection. Let Q be the curvature tensor of this connection and s' the (anti-symmetric) torsion-tensor ( S = 0 for the cases CL and c), the vanishing of which characterises the cases mentioned under { I I } . In cases 6 and d we find, among the prolongations of the point transformations in 9 with invariaiit point x , restricted to the tangent space Y,, those which are geometrical multiplications of the tangent space ITz. The curvature-tensor and the torsion tensor must be invariant under the representation of these multiplications ill the related teiisor-spaces. I hese representations are also non-trivial geometrical multiplications. Hence Q= 0, S = 0. This proves b a i d d . I n cases a ( c ) the (holomorphic) sectional curvature is invariant, under all orthogonal (unitary) transformations in the tangent space a t x. As H is transitive the (holomorphic) sectional curvature is the same for all (holomorphic) sections at all points and the space is " of constant (holomorphic) curvature ". This proves cases a and c. 17

$ 5 . Lie-derivatiws Let 6 be a geometric object field iii the bundle B over X . Let N b~ a neighbourhood of the identity of a Lie group H . which operates as a group of A-transformatioils q ( 1 ) : I' --f r:(r) in X. t

E

*I7 is a point in the grou1wpace. ,/(t) is the corresponding trans166

419

formation in X. Suppose .K 'z-'(x) = Y , is defined by :

E

C'(f)for all

L :2

(3.1)

--f

f E X.

Then a mapping of AT into

Y, n [ q * ( t )b. ] .

In case AV= H . H acts as a Lie group of transforniations on the image point set L ( H ) in Y,. Under the mapping ( 5 . 1 ) the tangent space at the unit-element of H is mapped into the tangent space of the point h ( x ) = b n Y , with respect to the fibre. This mapping is called the Lie-diflerential of the geometric object 6 , a t x, with respect to the given Lie group. For any parametrised differentiable curve t ( s ) in H , with t ( 0 ) is the identity, the iniage of the vector dtlds under the Lie differential is called the Lie-derivative of b , a t x, with respect to the parameter s. It is a tangent to a parametrised curve in Y,. For x variable we get a field of such tangents also called the Lie-derivative. T h e o r e m 7 . The Lip-derivative of a geomefric object (field) of diferentiability-class 2 1 i s a geometric object (field).

P r o o f : If X, Y , G, h are the entities which determine the fibre-bundle B, in which b is a P-cross-section r 2 1 , then the Lie-derivative 9 b is a C-?-' cross-section in the fibre-bundle determined in a unique way by : X , Yl, G, hl where Yl is the space of all tangent vectors a t all points of Y and h1 is obtained from h by replacing any analytic transformation in Y by its prolongation in Y 1 . This replacement is an isomorphism according to theorem 3. T h e o r e m 8. A P-geometric object field b in the bundle B ouer X , r 2 1 i s invariant under the prolongations of a connected Lie-group H of A-transforinations of X , if and only if the Lie-digerential of b at a n y point x E X with respect to H vanishes. The necessity is obvious. The suficiency is not equally obvious however (!). Suppose the Lie-differential of b a t every point x E X with respect to H vanishes. Suppose, for a fixed .r, that the set of points

Y, nq*(f).h

t

EH

is not one point. Then a curve t ( s ) with a point that the tangent vector

f ( 1) = t,

exists in H

siicli

does not vanish for s - 1. aiid this Let .T , / ( f , ) . . r ' . The prolongation ,,*(tl) maps I.,,, onto innpping is under reference systems represented by an element of G operating in Y. Hence it carries no11 vanishing tangent vectors of FJ,onto such vectors of I', and vice versa. Therefore the curve with parameter s : ~

167

420

has a non-vanishing derivative for s = 1 , that is a t the point Y,, n b. The Lie-differential of b a t x’ with respect to H is then not zero in contradiction with the assumpt,ions. REFERENCES 1. EHRESMANN, CHARLES,Les prolongementv d’une varikt6 differentiable. Attidel

IV Congress0 dell’ Unione Maternatico Italiana Taormina, 9 pages, 25-31 Ott. (1951). , Introduction 8. la theorie des structures infinitesimales et des pseudo2. groupes de Lie. Colloque international du C.N.R.S. G6om6trie differentielle Strasbourg, 97-1 17 (1953). 3. HAAwrJEs, J. and G . LAMAN,On the definition of geometric objects. Prof. Akad. Amsterdain, 56, Series A = Indagationes Math. 15, 208-222 (1953). 4. VEBLEN,0. and J. H. C. WHITEHEAD, The foundations of differential geomettry. Cambr. Tract., Ch. I1 § 8 and Ch. 111, 29 (1932). 5. CARTAN, E., Sur les vari6t6s 8. connection projective. Bull. SOC.Math. France, 52, 205-241 (1924).

6. EHRESMANN, CHARLES, Les connections infmitesimales dans un espace fibre differentiable. Colloque de topologie 1950 du C.B.R.M., 29-55. 7. KUIPER, N. H,, Einstein spaces and connections, Proc. Akad. Amsterdam 53, = Indagationes Math. 12, 604-521 (1950). 8. STEENROD, N., The topology of fibre-bundles (Princeton University Press, 1951).

168

KONINKL. NEDERL. AKADEMIE VAN WETENSCHAPPEN - AMSTERDAM Reprinted from Proceedings, Series A, 58, No. 3 and Indag. Math., 17, No. 3, 1955

MATHEMATICS

ON INVARIANT SUBSPACES I N THE ALMOST COMPLEX

X,,

BY

J. A. SCHOUTEN

AND

K. YANO

(Communicated a t the meeting of March 26, 1955)

Introduction

I n a former paper l ) we considered real Em-fields in the almost complex XZn2) whose 2.m-direction a t each point was invariant for the transformation F . The connecting quantities of the field being Bt,Ct, the n.8.s. conditions for this invariance are

Fih B i C t

(0.1)

=

0.

We also studied the possibility of the construction of X2,’s having a t each point an invariant tangent E, and we proved that the integrability conditions of the system of differential equations for these X2,’s are identically satisfied if the X,, is pseudo-complex. I n this paper we consider the case that a normal system of m 2 R - 2 m real Xz,’s with invariant tangent EZm’sexists and we call these X2m’s invariant. $ 1. Quantities i n X,, and X,, From the invariance of the tangent E , we get immediately3) (1.1)

a)

Fib BZb = ‘F:

Bt;

b)

Fib Cz = ”p,”cy

where ‘F;Pis a tensor in the X,, and ‘IF: a tensor in the EM,, m ’ = n - m , arising from the local E , by reduction with respect to the tangent E,. These two tensors are uniquely determined. From (1.1) we get (1.2)

a) ‘ F ; = - BE; Bt d% b ) “F;””F,’x= - CX. Cx 2 6% Y)

u-

Y

hence SCHOUTEN and YANO,On the geometrical meaning of the vanishing of the 1) NIJENHUIStensor in an X , with an almost complex structure, Proc. Kon. Ned. Akad. Amsterdam A58 ( =Indagationes 17) 133-138, (1955) hereafter referred to as s - Y 11. 2, We call an almost complex X , almost Hermitian if there is a fundamental tensor such that F i r Fik glk: = gii and almost Klihlerian if moreover P,, = 0. If Nj:ih= 0 we change “almost” into “pseudo”. Cf. the list on p. 25 of YANO,On three remarkable affine connexions in almost Hermitian space, Proc. Kon. Ned. Akad. Amsterdam 58 ( = Ind. Math. 17) 2 P 3 4 , (1955) hereafter referred t o as Y3c. Cf. for instance S - P I 1 (2.8), p. 134.

262 T h e o r e m I. I n a n almost complex X,, a n almost complex structure i s induced at every point in every invariant E , in the local E,,, hence a n invariant X,, in this X,, i s itself a n almost complex X,,. From the definition of N;ih l ) and (1.1s) it can be deduced easily that

N;ihB$ Bt

(1.3)

B:

=

where INEZ is the NIJENHUIStensor of the X,.

Hence

T h e o r e m 11. An invariant X,, in a pseudo-complex X,, i s itself a peudo-complex X,,,,. The X,, is almost Hermitian if a fundamental tensor gih of rank 2n is introduced such that

Pik F?

(1.4)

=

9th *

Then the fundamental tensor induced in X,, (1.5)

lgbo =

is

B: B: 9,

and from (1.1) and (1.4) we get immediately

' F f IF;"lg&

( 1 -6) hence

= Igba

T h e o r e m 111. An invariant X,, in a n almost (pseudo) Hermitian X,,,i s itself almost (pseudo) Hermitian.

From (1.1) we get (1.7)

'FbaE'F:

'gca =

IF: Bt Bi ghj

2

Pih Bi Bi, ghi

=

& BZ F a ,

Now it is well known that the section of the rotation of a bivector in X,, with the X,, equals the rotation of the section.2) Hence (1.8)

b,,

'Fbal=

B L B%Bt bG Fa].

and

.

T h e o r e m I V An invariant X,, in a n almost (pseudo) Kahlerian X,, i s itself almost (pseudo) Kahlerian.

3 2. Connexions in the almost complex

XZn3)

We prove f i s t that it is always possible to find a linear connexion be the parameters of an arbitrary symsuch that Vj Fib= 0. Let metric connexion and let us write

f'$

(2.1) 1) 3) 8)

T;ih

rh.- $h ?a

21'

Cf. for instance S-Y I1 (2.8), p. 134. Cf. Schouten, Ricci Calculus 1954, hereafter referred t o as R.C., p. 89 Cf. Y 3 c.

170

263

Then we wish to find values of T;ih such that (3.2) or (2.3)

0

*

Vi F:h Irn F;m-T;;Z F;h % = V . Fib + T:'h

=

* *

07;Timz= -

/2

(

Gi Fiz)Fib; 62 = l/, (A? A: + Fim Fib).

Here 0 is the idempotent but non invertible operator already used in a former paper l ) , hence there exist mow solutions. One of them is for instance 2) (2.4)

T::h 31 =

/,

(

-1

cjFit)Pih

and every other one can be found by adding to the right hand side of * (2.4) a term that is pure in (that means made zero by applying 0 to t) for instance the term (2.5)

l/,

(

FciF;;) Fib +

l/,

(

c'Ij F;;) 1";'.

This leads to the connexion with

..

crj

Tiih= - 11, ( b(jF;;) Fib + ( Ff;) F;'. 3 ) For every linear connexion F; the following identity is true.

(2.6)

(2.7)

1

N : : h= 2 F;: ( VI11FiF - Vil Fib) + +"2 (Si'ih- F'? F;k Sikh + F;l F;h #;;k -

F;h sit?)

hence in a pseudo-complex X,, the torsion tensor Siih of every connexion for which V j Fib= 0 satisfies the identity (2.8)

S,:;h - Fit

F;k S;l,h + Fg:lF;h S;;k - F;l F;h S;;k

=

0.

In a pseudo-complex X,, it is always possible to make F$ symmetric. In fact, for the connexion (2.6) we have on account of (2.6) and (2.7) (2.9)

S::h 71 = - 1/4 (

tLi F;,?)F;; +

*

1/4

(

v1Pi;)Fill =

118

N;ih *)

hence 6, Theorem V. I n a n almost complex X,, there exists a symmetric linear connexion with Vj Pih=O if and only if the X,, i s pseudo-complex. SCHOUTEN and YANO,On an intrinsic connexion in an X, with an almost 1) Hermitian structure, Proc. Kon. Ned. Akad. Amsterdam 58 ( = Ind. Math. 17) 1-9 (1955), hereafter referred to as S-Y I.

*

0

If we take for V the operator V of the Riemannian connexion in an almost Hermitian X , this is the first connexion dealt with in Y3c. Sur les structures comJ ) This gives a connexion that occurred in ECKMANN, plexes et presque complexes, GBom. Diff. Coll. Inter. de C.N.R.S., Strasbourg %)

151-159 (1953). ') Cf. S-Y I, (1.26); Y 3 ~ . 5) ECKMANN, 1.c. ; cf. FROLICHER, Zur Differentialgeometrie der komplexen Strukturen, Math. Ann. 129, 50-95 (1955).

171

264

From (2.9) we see that 0:; SIih=Sl:ihand this means that the connexion (2.6) satisfies the condition of infinitesimal parallelograms in every in-

N1:ih variant E,'). But neither this condition nor the condition S2:;h=1/8 is sufficient to determine the linear connexion completely. We now return to the general case of an almost complex X,, and suppose only that ViFih=O. Then (2.7) can be written in the form (2.10)

N7:ih= - 2 6:: S;Jhi4 6;4kS;,;;.

If there is a fundamental tensor we can lower the index h and get (note Y the change of 0 into 0 in the second term (2.11)

' I 2 N "9th = - 2 6g 8 l l ; h + 4 Olfih S l l i ] k .

If the connexion satisfies the condition of infinitesimal parallelograms we have

*

0;; Sikh = 0 .

(2.12)

Hence for such a connexion the identity (2.11) splits up into (2.12) and (2.13)

Niih =

lk Orilhl

s

llilk

and this is in accordance with what was proved in a previous paper for the metric connexion leaving Pih invariant and satisfying the condition of parallelograms. ,) It is remarkable that (2.11) and (2.12) hold also if the connexion is not metric, that is if Of g,# 0. I n the paper just mentioned we proved that there is one and only one oonnexion in almost Hermitian space that is metric, gives Vj Flh= 0 and eatisfies the condition of parallelograms. For this connexion that we call hereafter the intrinsic connexion, Tj'lhwas found to be3) 0

(2.14)

0

Tl:Z:"= ' / a ( V h Flj) Pi1- ( V ( iFllla) Fil - '/4

Nhii

and the first term of the right hand side is obviously alternating in ji.

0 3.

The connexion induced in a n invariant X,, If an X,, is imbedded in an X2,&with a linear connexion, a linear connexion is induced in the X,, provided that the X,, is rigged. But starting from an almost complex X,, we have only F;h and we can deduce 'I"?, " F P and 'N::, but there is no rigging. I n the X,, we have invariant E2's and these coincide with those invariant E2's of X,, at points of X,,, which lie wholly in X2m.But ' F 2 , 'NEb(l and these E2's are not sufficient to fix a linear connexion in X,,. If the X,, is almost Hermitian, there is a fundamental tensor gih and S-Y I, p. 8. S-Y I (2.23). The right hand side of this formula (2.23) is automatically alternating in ij. a) S-Y I (2.25), p. 8. ')

a)

172

265

from this the fundamental t'ensor 'g, in X2m, Hence there is now an intrinsic connexion fixed in X,, and also one in X2m. On the other hand the X,, is now automatically riggid (by orthogonality) and consequently there is also a connexion induced in the X,, by the connexion jn, the X,, and the rigging. We prove the T h e o r e m V I . If an invariant X,, is imbedded in a n almost Hermitian X,,, there is one and only one intrinsic connexion in each of these spaces. The intrinsic connexion in X,, is identical with the connexion induced by the intrinsic connexion in X,, by means of the orthogonal rigging. After the introduction of gih and the rigging there exist next to Bt, Bg, Cr, C; also the quantities Ci, Ct, BP, BZ and the m'drection of the rigging is spanned by Ch, and also by B7.l) For these quantities we have )3J)

a ) Fi'b B:

= 'Fba

B: ;

b)

Fih

Ci = 'IF'," C:

that can be easily deduced from (1.1) by raising and lowering of indices. (Note that Bh,='g, Bq g g b ; etc.) Using any metric connexion in X,, with vj Fib= 0 we get by differentiation of (1.la) ),

D, Fih Bb

=

D,'F: B;

where D , is an operator of VAN DER WAERDEN-BORTOLOTTI, or (3.3) I n the same way we get by applying the operator D, to ( l . l b )

2;;.

2mr

I n these equations and HI; are the curvature tensors of valence 3 of the X;; and the X;;' respectively for the connexion V . The equations hold also if the X f is anholonomic. But in our case D , in (3.3a) can be substituted by 'VCbecause the X i ; is holonomic. From (3.3) i t follows immediately that (3.4)

a ) 'Po'F:

=

b) D, "F',"

0;

=

0

but 1) Cf. R.C. p. 253 ff. For convenience we do not deal with one single X, with an Xgr and the rigging Xi:' with the condition that the Xi: is X,,-building. Of course the Rzm,'s are invariant if the RZm'shave this property because the transformation 25, is a rotation. a) Cf. R.C. p. 254 ff.

173

266 Now we have

giih

bemuse lies with the index h in the rigging Xi;'. With the indices j i this quantity lies in the tangent R,, and accordingly the equations (3.5) can also be written

or

Because 'V,'F,"=O and ' V c ' g b , = O we have now only to prove that the induced connexion satisfies the parallelogram condition. This is geometrically evident because an invariant E, in X,, is also an invariant E , in X,,, and the parallel displacement of a contravariant vector in an invariant X,, is just the projection of the displacement of this vector looked upon as a vector of X2%.But it can also be proved algebraically. We * know that O ~ ~ X = , , ,0 (cf. 2.12) and we have to prove that 'Scda = 0. Now we have 'Scba= B', BL B: Sjih and accordingly

'&

(3.9) ( @ Bt+ 'FAe'Fbd)'Seda= Bi Bk ( A ;A:

+ Fil pik)Xlkh Bk = 0 .

From (1.1) other equations could be deduced applying D, to ( 1 . h ) and D, to (1.lb). But because the connexion is supposed to be metric, this leads only to equations equivalent to (3.3a, b). Of course differentiation of (3.la, b) leads for the same reason also to equivalent equations. So the only equations that have to be discussed from the point of view from the imbedded X2,'s are (3.5a). I n order to discuss (3.5a) we first only suppose that the connexion G' satisfies ViFih= 0 and V i = 0 and we use the well known formula l) (3.10)

Hcbx 2m..

~

ern - 2"" + cb

Bib Cxh & "i hi a

9

2, z"x cb

B:: Cxh Qhii

2m..

where Q;i is the anholonomity object and Zcbxthe tensor whose vanishing is n.a.5. for the X,,-forming of the Xi:. So in our case we have 2m..

(3.11)

2m.. = HW,?

+ B'cbiCxh S"h. ii

Now we know that 0

(3.12) l)

*)

r; = q;- s g : ; h + s;;~ + ,yi 2)

R.C. (7.22), p. 257. R.C. (3.5), p. 132.

174

267 0

hence writing Hi;;" for the curvature quantity of Xi: of valence 3 with 0

respect to V

By transvection of (3.5a) with 'gcb we get on account of (3.11, 13) (3.14)

\-

0

0

lgcb

Hi: +. 2 fgiiS:h, Cx 3 . t h - 9

f

eb

IF'"

b

f'F: H;;u +

C; S;;h . - 2 tgcb IF: "FZ &: Ci &(;hi)

+. 'gob 'F: "F; or (3.15)

0

'gcb

HI.2 = 2 'giiS3:hic; - 'Fii siih "FZ = 2 'gii Sihi"ghZ - 'Fii Siih"Fh".

This equation is valid for every metric connexion satisfying Vi Fih=O. Now we may suppose that the intrinsic connexion is chosen. Then we know that Siih is pure in the indices j i l ) and the consequence of this is that 'FiiSi,= 0 because 'Fii is hybrid. Further we know that 2, 0

(3.16)

Sjih = ' 1 2 ( v h F L j )

Fi'l - ' 1 4

Nh[ij]

and that Nhziis pure in all indices. Hence (3.15) takes now the simple form, valid in the almost Hermitian X2,3)

'M" is the mean curvature vector of the X,,, whose vanishing is n.a.s. for the X,, to be a minimal subspace in X2,. 0

I n an almost Kahlerian X , we have Vij Fihl= 0 , hence in such a space 'Mxtakes the form 0

(3.18)

2m'M"

=

'1, ( V h Fii) 'Fii "ghX =

0

( O h "Fgi)'Fii "gh

=

0

because 'IFii can be written as a sum of products of vectors in the rigging each of which gives a zero transvection with 'Fii. That proves (cf. theorem I V ) T h e o r e m V I I . Every invariant X2, in a n almost Kiihlerian X,, is minimal.

It is remarkable that the X,,, need not be pseudo-Kiihlerian. This is due to the fact that N7r;i=0.4) 1)

s - Y I (2.21).

*)

S-Y I (2.24). Cf. R.C. (9, 16a.),p. 271. Cf. S-Y I1 (2.16), p. 136.

s, 4,

175

268

I n an almost Hermitian X,, we call the divergence of Fih (3.19)

Zh

+i

Fih = g-'h bi

g'ls F i h

the divergence vector of the space. We prove T h e o r e m V I I I . Every almost Kahlerian X,, i s divergence free and accordingly in such a space the bivector Fih i s harmonic ,).

I)

I n fact, in every almost Hermitian X,, (3.20)

The converse is only true for 2 n = 4 because for that case Fih=&FkiIkiih. We consider now the case that in an almost Hermitian X,, both the Xz: and the X;;' are X2m-(X2m,-) building. Then we have the two mean curvature vectors

end next to Z h we have the divergence vectors (3.22)

a)

'zh= 'pi0 ' p h ;

b)

"Zh =

"vt"pa 0

in the XZm's and the X2m,'srespectively. From these equations we get (3.23)

2, Fih

=

'Mh + "Mh + 'zi 'Fib + "Zi 'IF*

or (3.24)

1) - 'Zh- ''MtF'h* 1 ' a) ZzBh

Fib. b) ZzCh1 -- "Zh- 'Mz

Hence T h e o r e m I X . Let there be in a Hermitian X,, a normal system of invariant XZm's and a normal system of XZm,'s orthogonal to each other. If the X,, is divergencefree then the X2m's are minimal if the X2,,'s are divergencefree and vice versa. If both the XZm's and the X2m,'s are minimal and divergencefree the X,, i s divergencefree, but this condition i s not a necessary one. Comparing this theorem with theorem VII we see once more that the condition of being Kahlerian is much stronger than the condition of being divergencefree. If there is only one invariant X,, in the almost complex X,, we can choose a normal system of invariant XZm'scontaining it, Then the rigging For the Kahlerian X , , this was proved by Apte, S u r certaines varibt6s hermitiques, C.R.Paris 238, 1091-1093 (1964). But here it is obvious because even 0

the stronger condition Vi Fhl = 0 holds. %) This fact can obviously be used to find global properties in compact almost Kiihlerian spaces.

176

269

Xgr' is invariant but in general aaholonomic. Then we get from (3.22b,24b) (3.25)

2m 'M"

=

zfcF: - "Z'F;

or 0

(3.26)

- " g Z V (0, "Fux) "Fxh 2m 'Mh = zi "Frh t

or (3.27)

0

ZzCf= ''fu (0, "Fwx) "9"- 2m 'MI Pih.

I n the special case that the Xi:' can be taken holonomic the first term of the right hand aide of (3.27) reduces to "Zh. It is remarkable that this term only depends on Zhaand 'Mh and not on the way how the normal system of Xzm'sis chosen. So in the case where a holonomic Xt' is possible, the divergence vector "Zh of the X2m,'sat points of the X,, is fixed if the X,, is given. If the X,, is minimal "Zh is the component of Zh in the R,, of the rigging.

177

ANNALSOF MATHEMATICS Val. 61, No. 1, January, 1965 Printed in U.S.A.

ON REAL REPRESENTATIONS OF KAEHLERIAN MANIFOLDS BY KENTARO YANO

AND IsAMU

MOQI

(Received September 21, 1953)

The main purpose of present paper is to study the real representations of Kaehlerian manifolds, especially those of Kaehleriaii maiiifoltls of cwstant holomorphic sectional curvature. In the preliminary $1, we state the characteristic properties, found recently by B. Eckmaiin and A. Frolicher [5],' of real analytic manifolds which cati represent complex analytic manifolds. Such manifolds h a w t o contain a mixed tensor 4'j satisfying [5, 6 , 8, 181

r#l')r#l1=

-6;

and

(r#l'J

I

- 4'1

j h ' l

-

(4'J1 -

4J1L,jMJ/.

=

0.

In t2, we discuss affine coiinections which do not change the rolliiieatioii defined by 4iJin every tangent space of the manifold [9, 161. In $3, we come to the discussions of Hermitian and Kaehleriaii metrics i i i such manifolds. In $4, we discuss the curvature in a Kaehlerian mailifold and we examine i i i detail the case in which the so-called holomorphic sectional curvature is coilstant. The real representation (4.1 1) of the curvature tensor of such a matiifold seems to be new. In $5, me shall show that if we assume that the so-called axiom of planeb holds not for all the planes hut only for the holomorphir planes, then the matiifoltl will be of coilstant, holomorphic etirvatiire, Similarly in $6, we shall show that if we assume the so-caalled holomorphic free mobility, that is, if we assume that the manifold admits a groiip of motions which carries any two vectors t i i a i d +'l(.r)uJ a t a point P ( s ) into ally t no \.cctoi.s ii" and 4'J(d)u'J at aiiy point P', theti Ihe m:tiiifold will he of' cwistanf holomorphic curvature. 111 $7, we study the distsnc~eI)etwccw t\\o cotisecwti\rcJ roiljugate points oil R geodesic in a Kaehleriaii manifold of positi\re constaiit holomorphic curvature. If we denote the constant holomorphir taiin-atiire hy It* ( >O), then this c1istaiic.c. is constant and is given by 2n/&. In the last 58, we disrrtss analytic v o r licltls i i i L: g(~iiwi1Iitichleriaii maiiifold. The authors wish to express here their sitic*crcgi.atitiidc to I'rofessor S.Bocliiici. who suggested this iiitcrestitig pri)l)lcm to o t i ~of thc a i i t h o i ' q (Yaiio) whcii hc tvas stayiiig at l'riiicetou. __

. See the Bibliography at the end of the paper. 170

178

171

KAEHLERIAN MANIFOLDS

81. Complex analytic structure We consider a Zn dimensional real analytic manifold L7?,, covered entirely by a system of neighborhoods endowed with real coordinates (.r'), where atid in the following the Latin indices _ - a,- 0, c, . . . , i, j , li, . . . are supposed to run over t h e r a n g e l , 2 , 3 , * * ' , n , l , 2 , 3 ,. . . , n , E . Now, if we put

and if the following Greek indices a , p, y , . . . take the 1-alues 1 , 2-, 3, . . . n and - _ consequently ? p i, , 7, . . . run over the range of symbols 1, 2, 3, . . . , ii, t h e ~ ~ , we can consider (zp,2") as complex coordinates of a point in our 2n dimensional real analytic manifold Vzn. If we m i choose a system of complex coordinate neighborhoods in such a way that, U1 and IT? being any two complex coordinate , the complex coneighborhoods both coiitaining a point 1' of the manifold ordinates zlU of the point P in one of these complex coordinate neighhorhoods are complex analytic functions with non-vanishing Jacobiaii of the complex coordinates zu of the same point in the other cwmples coordinate neighborhood: I

(1.2)

2'"

=

4 " ( z ) and consequently 2'"

=

$"(E),

where &"(.i!) denotes comples conjugate of the function +"(L), theti \re say that the manifold has a complex analytic structure, and call it a comples analytic manifold of real dimensioii 2 1 ~and of comples dimensioti 7 1 . If we put z' = za, then equations (1.2) may he written in the form z" = f ( z " ) , and it is easily seen that the Jacobiati of this transfoi~mationis real and pofiitive and consequently that the complex analytic manifold is always orientable. h i a complex analytic manifold, the differential geometric objects such a vectors, tensors, affiiie connections are defined with respect to z" = f'(z"), having the special form (1.2), in just the same way as in real case. For example, a mixed tensor Z', has the law of transformation

But, by virtue of the special form (1.2), the law (1.3) separates into four blocks:

This fact shows that,, if T i jare components of a mixed tensor, then (1.5)

6i6;Txp,

6i6:Tx3,

179

6i6;Ti,,

6i$Tip A 1

172

KENTARO T A N 0 AND IR.iMU U O G I

aiid

are respertivelyalso coinponciitsof a miscd teiisor of the same kind as the original one, wherc 6;. are Krotiec-ker symhols aiid H = ii! if i = a and B = a if i = C, thc har on the m i t r a l letter tlenoting c~omplcxroiljugate. 'l'hc tensor defiiied by (1.6) is called the adjoint of the original tensor aiid a tensor which is equal to its adjoiut is said to be self-adjoint. Now, equatioiis (1.3) show that there exists a mixed self-adjoint tensor which has the compoiients

+',

all complex analytic- coordinate systems ( x " ) . Conversely, if, in a 2n dimensional real analytic mailifold V Z n , there exists a mixed tensor field +', aiid moreover there exists a system of complex coordinates z 1 = (z", 2") in which the tensor has the cornpoileiits (1.7), then, as we can see from ill

the trailsformatioils of coordinates in the overlappiiig domain of two coordinate neighborhoods have always the form ( I .2), and consequently the manifold has a complex analyt,ic structure. The tensor which has components (1.7) in a complex analytic coordinate system x p = (z", 2") has c~ompoileiits

+

in a real analytic coordiiiat,e system (xi)[9, 161 which is related to (xi) by

(1.10)

We iiot,ice that this tensor field satisfies the equality (1.11)

cbi,+'n.

=

-st!+

.

Thus, in order that an 2n dimeiisioiial real analytic manifold have a complex analytir structure, it is necessary atid sufficient that the maiiifold contain a mixed tciisor field +*, sat isfyiiig ( I .11) and there exist a coordinate system ill n4iich the ficltl +', has numericaal cwmporients ( I .9).

180

liA 1sH LEltI A S MA I\‘1FOLDS

173

Sow, ifif aa 9rk 2 t b dimciisioiial tlimciisioiial real real analytic. aiialytiv manifold maiiifolcl coiitaiiis roiitaiiis aa teiisor teiisor field field 4Ij Sow, satisfying (1.1 (1. I I1)),, we wc say say that that the thc maiiifoltl manifold has has maiii almost almost complex complex analytic ailalyt ic satisfying stritctmrc [ 16, 8, 151. 15). stiuctiire t i , 8, hssiimiiig that t.hat2thc thc matiifold maiiifold has hns ail ail almost almost. vomplcx t.omplcs aiialytic uiialyt ic. sst.ructure 4*,,, .Issiimiiig t r ~ r t u l c4', order ~ that the thc teiisor tensor ficld ficld + I$’, give aa compleu complex aiialytic aiialytic structrire st.rric.titi*eto to the the maiiimaiiiiii ii order that IJ give fold, itit isis iiecessary iicc*essaryaiitl aiicl suffic.irlit suffiviciit that that thrix t.hcrc cxist esist. aa coordioatc coordiiiate traiisformatioii traiisformatioii fold, x” = = .r"(.r) d i ( . r )s11c1i stich that that. .r"

0

ax1 4' = axfa ~axx ax'@

(1.12)

axt' axJ

-6" "

6; = __ -#J, ax* ax'p

3

zt

axfa ax' a T p 4'J Y

a d G ax'

0 = __ az1B -4%

I'

or what what amounts amoiiiitsto to the the same, samc, or

-

(1.13)

We iiotSic*cthat, if we ttikc t i v t w i i t of ( l . l l ) , the two cqiiatioiis of (1.13) are equivalent. Now, R. Erkmaiiii aiitl Friilivhcr j ) r o v ~ [3J ~ l that : fJ’ atr nhnosl comp1r.r ntrn1,tltic striirlnrc +iJ(.r) of class ( ‘ I girics n, contplrx ntial!jlic slriictim~,lhcn wr haw whrt IJw cu

cM))plry QI(

slricclwrc lo thc tmviijolil.

2. Mine connections ia complex analytic manifolds I'll 111 a complex aiialytic. muiiifolcl C,, , wc define an rrffiiic coiwicc.t.ioti hy tbc comlwiieiitw I.jik(x, t ) \diic*h haw thc tratisformatioti law

for thc t.lic! traiisform:itioii tritiisformatioii ((II 2 2) or complex c.omples analytic. tiiialytic c*oordiiiatcs coordiiiatcs ((zz,, 25)) . ) of From thc spcc*ial form of t hc traiisformathi (I 2 ) we can can see, see, ffor (wmplc, From tlic sprc.ial of the trarisformatioii (1.2) o r f~xaniple,

\vliit.h slio\vs tlint

arc wmpoiieiits of it mixctl teiisor. Similarly,

181 181

174

KENTARO YANO AND ISAMU MOGI

are all components of mixed tensors. Also if we assume that the affine connection is self-adjoint, theti the self-adjoiiitness of a teiisor is preserved hy the coyariant differentiation with respect to this afFine connection. Now, we know that in our manifold there exists a mixed tensor h a ~ ~ i nthe g components (1.7) and the covariant derivative of 6') : +I,

if written out fully, is = -2irpa/, ,

+"alk

+"!w

=

0,

+'blk

=

'Lirs'L., +'Blk

=

0.

These equations show that if we consider an affine connection

rjik

such t.hat

rg"k = rs"k = 0,

(2.2)

ramr being arbitrary, then

we h a w

(2.3)

=

0.

Since the tensor + ' j defines a collitieatioii in the affiiie space tangent a t a point of the manifold, equation (2.3) shows that it is possible to define an affine connection in such a way that the group of holonomy of this affinely connected manifold fixes the collineation defined by the tensor + ' i j [O, 161. Moreover, since the components of such an affine connection are still arbitrary, we can see that there is a large class of such affine connections. Now, let us consider a 271 dimensional real analytic maiiifold V27rwhich has a complex analytic structure, that is, which contairis a mixed tensor +ij(z) such that (2.4) &+ik = -6; and ( + i j , k - +ik,j)@'l - ( + ; j , l - 4 i l , j ) + 3 k = 0, (2.5) and introduce an affine coniiectJioti rjikwhich does not chatige the c.ollitieation defined hy 4 7 j:

= -

+'.

(2.6)

+ii,k

+ +(Ljrail. - +iar;lk 0. =

From ( 1 . G ) and the equation ohtained from (2.6) Ijy interchangiiig .j and k , we find 4L. 311; - +* ' k , j = +ij,k - +ik,i +"jiiaik - +(lkraii - &ia,~jk ,

+

from which (+ijlk

-

+iklj)+il

-

(+ijll

-

- +illj)+jL = - +il,j)+jl;

(+ij,l

(4lj.k

+

-

2Sikl

182

+l.j)+jl

- 2Sih f#A+cl

+

24ia+bkA9"bl

-

24'a+'I*~"bk

,

IiAEHLEFlIAN iMANlFOLDS

1i 5

'I'hiis we have

I s a 2n diinensioriat! rcnl nntr[!lfic niatzijold has ail n/ttiost comp/c.r analytic slrrictrire dcjined h!j oj class P, then the recccwary and siificicnt cotidition that + a 3 induce a complex analytic strrictriro ~ f thc ' matiifold i s that il be possiblc to ititroduce, iii the manifold, an qffinc, contiertiorb whzch does riot cliairye the rollineation defined by the tcrisor field utid sdisjics (2.8). If the coniiection is symmetric: ' ~ I I E O I W M2.1.

+I,

+IJ

S'jh

=0

or more generally, if the connection is semi-symmetric :

S',n

=

s;x/. - slx,,

theii (2.8) is always satisfied. Thus \vo ha\^ THEOREM 2.2. If a 2n tlimcnsional r e d anal!jtic ninnij'old has u n nlmo.st comp1c.r analytic sfructurc defined b!j + I J of class Cu, then the necrssary a n d sriflciont condition that indiicc a comp1c.e analijtic strirctiirc oj' the nintiifold is that it bc possible fo introdrice i n thc mnn~foltla symmetric (or t w w generall!j a serni-s!jrnt~ietrac)nflpic connection which docs not chungc th(Jcoilincation rlcjitictl b ! ~the tmsor .field +',. +I,

$3. Hermitian and Kaehlerian metrics in complex analytic manifolds 11, 2, 3, 4, 171 We now assume that, in our comples aiialytic matiifold, there is gi\-eii a positive definite quadratic dif'ferential form (3.1)


"2'

dk,

where gJk is t: symmetric aiid self-adjoint tetisor satisfying

(3.2)

gafl = g q i = 0.

'I'hc metric' (3.1) has thc form

(3.3)

d N L=

zgmId d z a (1155

by 1-irtue of' (3.2) such a metric8 is called IIermitiaii rnetrir. The Christoffel symhols

(3.4)

183

176

ICENTAAO YANO AND ISAMU YOGI

have the values

the other symt)ols givcii hy symmct~~y niid self-adjoitit~less. Sow, if the affine connectioii tlcfiiied by the Christoffel symbols does riot change the rdlineation defined by d', , then we r d l siich a metric I
from which

(3.5) and consequently (3.6)

For a Kaehlerian metric, the Christoffel symbols have the values (3.7) the other heiiig zero. For the curvature tensor

we can prove that

the other (3.10)

Rijkz

being zero and if we put R$jki

= BraROlki

184

,

177

Map =

R.8

=

0.

S o n , \vc cwnsider a 2ri dimensional real aiialytic manifold V?, I\ hivh has a complru analytic structure, that is, which coiltailis a mixed tewor field satisfying (2.4) and ( 2 . 5 ) ,and assume that there IS given a positive definite quadratic8 diReretitinl form + I l

ds'!

(3.11)

=

{Jjk

(!zJdzh.

First of all, we notice that cwiditioti (3.2) may he written, iii a real coordiiiate system, as (3.12)

Itideed,

gbd"J$"A.

ill

=

.

!Ilk

a complex cmrdinnte syhtem, (3.12) g i l w -g,&!6? +Biiy@:

= {la", =

+gai@?

=

P i -

-gg8rSa6, -

g a y ,

CJpc

,

gpi

,

whic.11 show that (3.2) atid (3.12) are equivale!it. Thus we have 'I'HEOREM 3. I . I n a 2 n dimrnsiorial rcal analytic manifbld which has a comp1P.r nnal!ytic striicture, that is, whzch contairis a tctisor j e l t l satisjyzng (2.4) arid ( 2 . 5 ) , thc ticwssar!j arid siLJ9cicnt condition thnf a p o s i t i w d(.Jinifcmetric +IJ

(1s'

=

I/jh

(12'

dz

h

be H ~ ~ r m i i i ais t i that the icrisor +', a d g J b be r d a t d by (3.12). 111 a 2n dimensional real analytic manifold which has a romplex analytic structure and a Hermitiail metric (3.11), we introduce a metric coniiectiou rJtkwhich is not necessarily without torsion. Then we have

(3.13) from which

(3.14) where

s'.Ik

= gjagcigkc.

Now, we moreover assume that the affiiie coiiiiectioii not change the colliiieation defined by + i j ; (3.15)

4'jjlk

=

+ij,k

+

+"jraiS

185

-

+iar(i(lk

rjik

= 0.

thus iiitroduced does

178

KENTARO YANO A N D ISAMU MOGI

From (3.13) and (3.15), we obtain (3.18)

4ijlk

where we have put +ij The condition 4i)$3k

=

gia4"j

. +.$I

if we compare this with gajtpaz$ik =

- -9. t k

k

2)

, or

gik

4ii4jk

I

+'j

I

=

,

= gik

# 0, we get

4.. 21

(3.17)

Now, writing down f p i j l k

0,

being written as

= --6ik

and take account of the fact

=

- 4 .11.

=

0 fully, we have

- & j , k - 4,,jriok 4.. =

- 4iariek = 0,

from which (3.18)

4ijk

=

+

2(4iaS"jk

4jarki

+

&aSaij),

where (3.19)

4iik = 4ii.k

+

4ik,i

+

4ki.j

are coefficients of the exterior derivative of the exterior differential form

34ij dxi d d . On the other hand, we know that (2.5) is equivalent to (2.8) and (2.8) can be written as (3.20)

Sijk

= - ( s i b ~ $ ~ j $ ' ~ kf Sjb&bA4ci

+

kykb&$*i$cj)a

From (3.18) and (3.20), we obtain (3.21)

Sijk

=

$#JaD<#J"i4bj$ck

.

I t is evident that equations (2.8) and (3.21) are equivalent, and consequently we have THEOREM 3.2. In a 2n dimensional real analytic manifold which has a positive definite mctric tensor g j k and contains a n anti-symmetric analytic tensor 4ij satisfying gbc4b,$cL = g j k , the necessary and suficirnt condition that the tensor + i j = ginrpaj give a complex analytic struct,ure lo the manifold i s that the torsion tensor Sijk of the metric afine connection satisyfing 4 ; j l k = 0 be g i v m b ! ~(3.21) and consequen,tly be a.nli-symmetric in all the indices. Now, 4 i j l k = 0 is written also i l l tJheform (3.22)

4ijlh.

=

4jj;k

-

&jl!?;k

186

-

4inlyjk

=

0,

179

KAEHLERIAN MANIFOLDS

where the semi-colon denotes the covariant derivative with respect to t,he Christoffel symhols { . i l k } . From (3.18) and (3.22), we find (3.23)

+ij,k

=

4ij;k

-

$fJijk

-

Sn;j+ak

=

0.

From (3.23), we have TUEOREM 3.3. I n a 2n dimensional real analytic man
When the manifold has a complex analytic st,ructure induced by + i j , we defined the Kaehlerian metric as one for which C#j;k

=

0

&j;k = 0.

or

Thus, for a Kaehlerian metric, we have, from (3.24), [ l l , 12, 13, 14, 151 (3.25)

0.

=

+ijk

$4. Curvature in a Kaehlerian manifold [2] In this section, we consider a 2n dimensional real analytic manifold which can represent a Kaehlerian manifold, that is, which is a real Riemannian space with a positive definite metric ds2 = g j k dx' dx' and contains an anti-symmetric tensor field satisfying (4.1)

= gtj

ghc+br+rj

and 4 s.j .; h -

(4.2)

1

Z4ijk =

0.

From the IZicci identity 4'j;k;l

- 4'j;i;k

=

+aj8'akl

- 4'nR"jn.l

and 4'j;e = 0, we find (4.3)

42aRo,kl= + " j R L n k l

or (4.4)

+niRnj~l

=

+"jRcz,xl.

Equation (4.4) shows that +niRoj~l is symmetric in i and j . Similarly +'kRijal is symmetric in k and 1. From (4.41, we find (4.5)

Rijki

=

+",#Jbj&b/ii

187

.

180

KENTARO YANO AND ISAMU MOGI

Multiplying (4.3) by g J k and summing up, we obtain +'or1

=

= $ # ~ ' ~ ~ ( R ' a b/ Rabat),

+abRaabl

from which

.

-

(4.6)

~#~~~ =f i ?&ba'JR'lab ~l

This equation shows that + , a R a l is anti-symmetric in i and 1 and consequently, we obtain

+

+zaKal

0,

=

+,aRnl

from which

R',

(4.7)

= -+'a+bJRab

and (4.8)

+ntRnl

+

=

+'&at

0.

Now, if we assume that the manifold is of constant curvature, then the tensor should have the form

IZtJnl

=

Rtjkl

K(gjkgr1

- gjlgak).

Substituting this into (4.4),we find K(gJd2L -

gll@kl)

= K ( g d + l ~-

gzl+k,)t

from which, by contraction with g'k,

K(2n -

=

1)+1t

K+1a

or

K=O for n

>

1.

Thus, if a Kaehlerian manifold is of mistant curvature, it is of zero curvature

PI.

We next assume that the mailifold is conformally flat, so that the curvature tensor has the form Rtlii

=

1 2n - 2

(Rjhgti

-

Rj1gzL

+

gjbRi1

-

Yi1Rak)

R + (2n

- 1)(2n - 2)

(glkg11

-

gllglk).

-

~ L / ~ I J ) ,

Substituting this into (4.4), we find R j ~ + i t

f gJh+aaRal

-

Rji+rt

=

R214iJ- R,i+rl

+

-

R

gJi+'1Rah

gtk+ajRal

(glkd'lt - g J l + h * ) f 2___ n - 1

- gti+ajRar

188

+

R

r l (gth+li

181

KAEHLERIAN MANIFOLDS

g", from which, by contraction with g",

2R4li

+ 2(n - l)4'iRal

= -+alRai

+

+ 2n R- 1

+aiRat

~

dli

or

4n - 3 2(n - 2)4,"iRa1= 2n - 1 Rdil

,

from which

I2

=

0 and R l k = 0,

corisequently and corisequeiitly R , j ~ i= 0. Thus we have THEOREM 4.1. Zj a linchlcrian manifold is conformallyjlaf, then it is of zero c iirvat w e . Now, we consider a sectional curvature determined by vectors u' and 4',u' and call it holomorphic sectional curvature with respect to the vector u' [2]. +elzi' is also a vector orthogonal to u', If we assume that u2 is a vector, then c$~,zL' and coiisequeritly the holomorphic sectional curvature with respect to the vector t i a is given by

(4.9)

If the holomorphic sectional curvature is always caonstant with respect to any vector at every point of the manifold, then me call the mailifold that of constant holomorphic curvsturc. [a,71. Now, if this is the rase, then (4.9) or

shoiild he satisfied for any u ' , from which we have

by virtue of the symmetry of R,t,,dC$b,&,d, in i, q and equatioii by ~ $ q ? + ' i and contracting, we find

T,

s. Multiplying the above

by virtue of' (4.5). On the other hand, we have

Thus, iiiterchhaiigiilg k and 1 in (4.10) and siibtracting the resulting equation from (5

in)

WP

ohtnin

189

182

KENTARO YANO A N D ISAMU MOGI

or (4.11)

Rtlkl

=

k

4 [(gjkgzl

-

gilgik)

+ ( 4 i k 6 1 - 4il4rJ

- 24*i&lI*

It is easily seen from the Bianchi identity that if the curvature tensor has the form (4.11) then the scalar h is an absolute constant. Thus, we have THEOREM 4.2. If a 2n dimensional real representation of a Kaehlerian manifold has constant holomorphic curvature, then the curvature tpnsor has the form (4.1I ) , where li i s an absolute constant. If the curvature tensor has the form (4.11), then we obtain (4.12)

and (4.13)

=

Rzlki+'l

-(n

+ 1)k+tj.

In a Kaehlerian manifold of coiistaiit holomorphic curvature, we consider a general sectioiial curvature Ii; determined by two orthogonal unit vcctors u' and v'. It is given by

K

a

i

k

= - R E J k 1 2 1 u 11 u

l

.

Substituting (4.11) into this equation, we find (4.14)

K =

k 4

- (1 +3Xz),

where (4.15)

x

= (PiJU'V'.

But, since v' is a unit vector, the vector 4'pj is also a unit vector, and subsequently, if we denote by 0 the angle between u' and +',v', we get (4.16)

x

= COS

8.

Thus, from (4.14), we have THEOREM 4.3. I n a Kaehlerian man
z -I K s k k

k 5 K 54

when

k

> 0,

when

k

<0

and the upper (1owc.r) limat i n the ,first (the second) i n r q i d i t y a s attained whrn the section i s holomorphic and the lowcr (iipper) limit i n thc first (thr scconri) inequality is attained when u2 and (P',vJ are orthogonal.

190

183

KAEHLERIAN MANIFOLDS

86. Axiom of holomorphic planes

We assume in this section that, when there is given a holomorphic plane element, that is, a plane element determined by vectors 7~' and +t,zil a t a point of the manifold, we can always draw a 2 dimcnsiotial totally geodesic surface passing through this point a i d being tangent to the given holomorphic plane element. If this is the case, we say that the manifold satisfies the axiom of holomorphic planes. If we represent such a surface by parametric equations 2' = X * ( P " ) ,

where we understand that indices a, b, c, d take the value 1 and 2 in this section, then the fact that the surface is totally geodesic is represented by equations

where { b(lc) are Christoffel symbols formed with the fundamental tensor

axt axk Y j k b

av a v

of the surface. The integrability conditions of these differential equations are

where

arid R''b,d is the curvature tensor of the surface. If we put R; = 111, BZ = +altdl euqatioii (3.1) should tie satisfied by any unit vector u ' . Thus me should have

(5.2) From the first equation of ( 5 . 2 ) ,we obtain

191

184

KENTARO YANO AND ISAMU MOGI

Subtracting from this the equation obtaiiied by ii~terchang~iig of It and 1, we firid -4li'jhl

=

(Y(26)#'Al

+ +

-

g~A+'i

-

8jl'fJ'h

-

P(2'fJtj$Al

qJh6;

f

+jl6i)

+ 9116; -

+

+jA+'Z

from which, by coiitractioii with respect to z and 3 , we get a quently we get

B R',AI= 3

8; -

( ~ J A

YJz~;

+

-

+ik+'l

+114'k

=

d'jl'fJ'h),

0, and conse-

- 24'3+k~),

which showa that the manifold is of constant holomorphic curvature. Thus, we have THEOREM 5.1. If a real represenlation of a Kaehlerian manafold satisfies the axiom of holomorphic planes, then the manifold i s of constant holomorphic curvature. 56. Holomorphic free mobility

If a real representation of a Iiaehleriati manifold admits a group of motions which carry any two vectors uz and + 1 3 ua~t a point P to any two vectors u'' and +'2ju'' at any point P', we say that the manifold admits a holomorphic free mobility. If we denote hy .r"

(6.1)

+ f'(.x)6t

x'

=

an infinitesimal transformation of the group, thcn the fact that this is a motioii is represented hy [lo] (6.2)

-Ygjh

+

i

fA

J

and the fart that this cwiies a pair of vevtors 71'' and +"ju" is represented 1)y

X'fJ']= F"

(0.3)

- f'

]+IO

0,

=z

I('

and +' ju' into a pair of vectors

&J"/

=

0

=

0,

From (0.2), we get [ I 9 1 (6.4)

A ~ { J * A ]

=

E'

j

h

+

R'jIZF'

and the integrability ronditioiis of these differential cqustioiis are given hy [19] (6.5)

X'RsJ~i

R ' j h l nEn

-

6'n l Z n j / Z

f

4"

jR'ahZ

+ F"

hfl)'jnl

+ F"

/Jf'jio

=

0.

Soiv, at a fixed poiiit P of the manifold, we cmsider tn o arbitrary holomorphic~ plane elements, then hy hypothesis, there evists always :L motloti which fixes this point and carries one of these holomorphic plane elements t o thc other. The point P being arbitrary, the manifold must he of cwnstaiit holomorphic crinrature, and suhscqucntly the curvature tensor has the form R'jk1

=

1; 4

- [gjL6;

-

gjl6;

4Jh4'l - 4 ~ ~ 4-' i24aj 4111

192

KAEHLERIAN MANIFOLDS

185

Conversely, if the crirvaturc tensor has the above form, then it is easily seeti that the integrability condition SRZJal= 0 is always satisfied for any t',, satisfying X g , , = 0 and .YC#I'~ = 0, and differential equations (6.4) have solutions. But eqitatioii (6.4) is equii~aleiitto [ 1 Y] (xgJh),l

=

and coiisequently if the equation Xg,n = 0 is satisfied t)y initial conditions, then it is satisfied hy any solutions. 0 1 1 the other hand, if t Lsatisfies (6.4), then we haire

(LYC#I'J) 1 =

- $'l,a.hJ$a,

E",j.h6'a

= ( -R",hi$'a

-k RZoii+aj)lr = 0,

aiicl coiisequeiitly, if the equation X+IJ = 0 is satisfied by initial conclitiolis, then

it is satisfied hy any solutions. Thus, the manifold admits the holomorphic free mobility and we have THEOREM 6.1. Thc ncccssnry and sii.flicicnt conditioii that a real representation of a Iiarklr.rmit manifold admits a Iiolomorpliic ,free mobilzty, is that thP manifold br of co iistarr t h o/omorphic c i d riat 11 rc. $7. Distance between consecutive conjugate points on a geodesic in a Kaehlerian

manifold of constant holomorphic curvature To discus the rlistaiires hetwecn two consecutive conjugate points on a ge.otlrh,sic*in a Riemannian manifold, we consider the equations of .Jacohi

(7.1) along a geodesic. .r'(s), where 6 ds denotes covariant differentiatioil along the curve. It' the solution of this equatioii vanishes a t a point Po(.~'(so)) and at aiiothcr point P,(.r'(sl)) and if' it does r i o t vanish t)etn-eeti P, atid P I theii the point P,, and PI :we said to tw coiljugate on this gt:odesic. Wr ai'r goitig to ronsidei. the eqitations of .Javohi i n a manifold of positive constatit holomorphic cun-ature. Sul~stitiiting(4.1 1) into (7.1), \IT find

(7.2) where we have put

(7.3)

and supposed that 7' is o~-thogotialto the geodesic. Dift'creiitiating (7.3) twice \\-ith respect to .r and taking account of (7.2), wr get (7.4)

d'A-- -LA. dS?

193

186

K E N T A R O Y A N O A N D ISAMU MOGI

Thus, equation (7.2) may be written in the form

(7.5) and consequently, if me choose a system of Fermi coordinates along the geodesic, then the above equation gives

where iliand Bi are constants. Now we assume that q i = 0 and consequently X have

x

(7.7)

=

C sin

=

0 when s

=

0. Then we

diis

from (7.4), where C is a constant and

(7.8)

qi

+ X4ia dx" ds

-

=

d is A i sin __ 2

from (7.6). From (7.7) and (7.8), we can see that, if q' vanishes at a point, then X vanishes a t this point and consequently sin $&s vanishes also a t this point. Conversely, if sin $&s vanishes a t a point, then X vanishes a t this point and consequently q' vanishes also a t this point. Thus the point a t which 7' vanishes immediately after Pa coincides with the point a t which sin $&s vanishes immediately after s = 0, that is, the point given by s = 2 n / z / x . Thus we have 7.1. I n a Kaehlerian manifold of positive constant holomorphic curvaTHEOREM ture k (>0), the distance between two consecutave con.jugate points i s constant and i s given by 27r/&. From the fact that in this case the general sectional curvature K satisfies the inequality

k O<-IKSlc 4and Ronnet's theorem, it is known a priori that the distance between two consecutive coiijugate points is less than 2n/&. But the above theorem assures that this is always exactly equal to this value. $8. Analytic vector fields [20]

In a complex Kaehlerian manifold the fact that the components of a self-adjoint covariant vector field ( g a , l ; ) are analytic functions of the complex coordinates is expressed by the formulae (8.1)

tm,6= 0 and i;,B

194

=

0

187

KAEHLERIAN MANIFOLDS

The same fact is expressed by (8.2)

+

+"i[a;j

+"jti;a

= O

the real representation. We call such a vector a covariant analytic vector. Differentiating (8.2) coi-ariaritly with respect to xk,arid contracting the resulting equation by c)", we find

ill

+aigjk[a;j;k + " i g J k [ a :j ; p

+

-

(pob[i;a;b =

= 0,

&ab[lR'iab

+"i[gjk[a: j ; k

-

0,

[lR'a]

=

0

and consequently (8.3)

s j k t a ;j ; k

- E ~ R '=, 0,

which is a iiecessary arid sufficient condition t,hat a vector field ti i n a compact orientable Riemannian space be harmonic. Thus in a compact Kaehlerian manifold, a covariaiit analytic vector field is necessarily harmonic. Coii~wsely,if [i is harmonic., then we have (8.3) from which we can deduce yJ'((~a+''z); j ; r

which shows that

ta+"i

-

I

t a + o l ~i

=

0,

is also harmonic. Thus we have ((a+"<); j

=

(+a j E a ) ; i

which is equivaleiit to (8.2) by i7irt)ue of +"i;j = 0 and E.;, = $jra , Thus 4 ; is covariant analytic. This proves a famous theorem stating that in a compact Iiaehlerian manifold, the necessary and sufficient condition that a covariant vector be analytic is that it be harmonic. 111a complex Iiaehlerian manifold, the fact that the components of a selfadjoiiit contravariant, vector field (.$", .$') are analytic functions of the complex coordinates is expressed by the formulae (8.4)

i " : =~ 0 aiid t ' ; b

=

0.

The same fact is expressed by (8.5)

.y+zj =

+

-[":+aj

=

0

in it,s real representation. We call such a vector a contravariant analytic vector. From (8.5),by exactly the same method as above, we can deduce (8.6)

gi"Ei;

j;k

+ Ri&' = 0.

On the other hand, we kilo\\- that in a compact orientable Riemannian space, the necessary and sufficient coliclition that, a \-&or field E' be a Killing L-ector is that t isatisfy (8.6) and ti;/= 0.

19.5

188

KENTARO YANO AND ISAMU MOGI

Thus we have THEOREM8.1. I n a compact Kaehlerian manifold, a contravariant analytic vector field ti satisjying t % ,=, 0 i s necessarily a Killing vector. Equation (8.2) can be written as (8.7)

t',a+"3

+

=

tfl.j+'a

0.

C o m p a h g (8.5) aiid (8.7), we have THEOREM 8.2. I j a vector as at the same time, covariant and contravariant analytic, then it is a parallel vector field. Now, we coiisidcr a contravariant analytic vector field satisfying (8.8)

=

0.

Then, equation (8.6) and (8.8) give and which show that the vector +'&" is a Killing vector. Thus, we have THEOREM 8.3. I f a contravariarit analytic vector l asatisfies +'a[n i s a Killing vector field. For such a vector, we have

tE;,+"= 0 , then

or (8.9)

Comparing (8.5) with (8.9), me find Ei;a

=

Fa:t

.

Thus \\re have THEOREM 8.4. Ij*a contravariant analytic [ a satisfies tClgi' = O and ti;j+i' = 0. then it is a. harmonic vector. Comhiiiiiig Theorem 8.4 and 8.2, we obtain THEOREM 8.5. I j a contravariant analytic vector ti satisfies [ i ; j g i ' = 0 and = 0 , then it i s a parallel vector field. THEOREM 8.6. If a contravariant. analytic vector ti satisjies t;;jgi' = 0 and = 0, then it i s a Killing vector. UNIVERSITY OF TOKYO TOKYO UNIVERSITY OF EDUCATION BIBLIOGRAPHY [l] BOCHNER, S. Vector Jields and Ricci curvnlure. Bull. Amer. Math. Soc., 52 (1946) 776797.

196

KAEHLERIAN MANIFOLDS

189

BOCHNER, S. Curvature i n Hermitian metric. Bull. Amer. Math. Soc., 53 (1947) 179-195. BOCHNER, S. Curvature and Betti numbers. Ann. of Math., 49 (1948) 379-390. BOCHNER, S. Curvature and Betti numbers. ZI. Ann. of Math., 50 (1949) 77-93. ECKMANN, B. and FROLICHER, A. S u r l'intdgrabilitd des structures presque complexes. C. R. Acad. Sci. Paris, 232 (1951) 2284-2286. C. S u r les varidtds presque complexes. Se'minaire Bourbaki. 1950. Proc. Int. [6] EHRESMANN, Cong. Math. 1950. 171 FUBINI, G . Sulle metriche dejini da una forma Hermitiana. Istituto. Veneto., 63 (1904) 502-513. 181 HOPF,H. Z u r Topologie der komplexen Mannigfaltigkeiten. Studies and Essays. Presented to R. Courant on his 60th Birthday J. 8 (1948) 167-185. 101 IWAMOTO, H. O n the structure of Riemannian spaces whose holonomy groups jix a null system. Tbhoku hlath. J., 2nd series, 1 (1950) 109-135. 1101 KAEHLER, E. Uber eine bemerkens,werte Hermitesche Metrik. Abh. Math. Sem. Hamburg. Univ., 9 (1933) 173-186. [Ill LICHN~ROWICZ, A. S u r les varidtds rienianniennes admettant une forme quadratique extdrieure d ddrivde covariante nulle. C. R. Acad. Sci. Paris, 231 (1950) 1413-1415. [12] LICHN~ROWICZ, A. Formes ct ddrivde covariante nulle sur line variitt riemannienne. C . R. Acad. Sci. Paris, 232 (1951) 146-147. 1131 LICHN~ROWICZ, A. S u r les varidtds riemanniennes admetlant une forme d dkrivie covariante nulle. C. R. Acad. Sci. Paris, 232 (1951) 677-679. 1141 LICHN~ROWICZ, A. Sur les variitds symplectiqiies. C. R. Acnd. Sci. Paris, 233 (1951) 723725. j15j L I C H N ~ R O W IA. C ZGdndralisations , de la ge'otne'trie KiihlCrienne globale. Colloque de gbometrie diffbrentielle, Louvain, (1951) 99-122. 1161 SASAKI,S. O n the real representation of spaces with, Hermitian connection. Sci. Rep. TGhoku Univ., 1st series. 33 (1949) 52-61. [I71 SCBOUTEN, J. A. and V A N D A N T Z I G , D. Uber ztnitdre Geometrie. Math. Ann., 103 (1930) 319-346. [lS] WEIL, A. S u r la the'orie des formes difle'rentielles attachds d ime varidte' analytique compleze. Comment. Math. Helv., 19 (1946) 110-116. 1191 Y . 4 ~ 0 I<. , Groups of transformations in generalized spaces. Akademein Press, Tokyo, 1940. 1201 YANO, K. On hnrmonic and Killing vectorJields. Ann. of Math., 5.5 (1952) 3 8 4 5 . 121 [3] [4] 151

197

Reprinted from Transactions of the AMS, volume 80,pages 72-92, by permission of the American Mathematical Society. @ 1955 by the American Mathematical Society.

A CLASS OF AFFINELY CONNECTED SPACES BY

HSIEN-CHUNG WANG AND KENTARO YANO

I t is known that the existence of a group G of motions of sufficiently high dimension in an n-dimensional Riemannian space imposes strong restrictions on the space [ l o ; 15; 161. This restriction is related to the fact that the isotropic subgroup, being a “high” dimensional subgroup of the orthogonal group O(n),is very restricted [ 1 2 ] . Some results of this nature for affine connections have been known [4;5 ; 6 ; 7 ; 8 ; 11; 131;in particular, if an affinely connected space 8, of dimension n admits a group Gof collineations with dim G 2 n 2 - n + 5 , then 8, is projectively flat. We know however that, unlike the Riemannian case, there are numerous non-equivalent, simply-connected, homogeneous( l ) affinely connected spaces with the same local structure. For example, over the Euclidean space (in the topological sense), many flat and homogeneous affine connections can be defined such that any two of them are different in the global sense(2). This tells us that the above class of spaces U, needs some further clarifications. In this paper, we give a more detailed study of the symmetric affinely connected spaces 8, having the above property. All the possible curvature tensors of U, as well as the isotropic subgroup of G are determined (Theorem 2, $9). In case G is transitive, we exhibit all the simply-connected 8,’s (Theorem 3, $10). They fall into three individual cases and two classes depending on a non-negative constant. Among them, there are three affinely flat spaces, two of which are homeomorphic with the euclidean space while the other one homeomorphic with the product of a line and an ( n - 1)-sphere, n > 2. As for the nonflat cases, they are all homeomorphic with the euclidean space. For the transitive case, all the isotropic subgroups given in Theorem 2 are realized; some of them can moreover be realized over nonequivalent affinely connected spaces. 1. Reducibility of certain linear groups. Let P, denote the special linear group of n real variables xl, * * , xn. In this section and the next, we shall determine all the closed subgroups of P, with dimension not less than n z - 2 n f 4 . This information plays an important r61e in our further discussions. Let us first establish the following

-

Received by the editors December 28, 1953 and, in revised form, October 16, 1954. (1) An affinely connected space is called homogeneous if it admits a transitive group of collineations. (s)Let 8,be the entire n-space with coordinates d , xs, * , x n and 8.(m) the subset of a, consisting of all the points whose first m coordinates are positive. The affinely connected spaces (B,(m); V,&)=O\ (m-0, 1, 2, , n) are homogeneous, flat, and homeomorphic with one another. But no two of them are equivalent.

-

-

72

198

73

.4 CLASS OF AFFINELY CONNECTED SPACES

LEMMA1. Let G be a proper subgroup of P,.I f dim G 2 n 2- 2n +4, then G i s reducible. Proof. M'e shall prove it by using the method of contradiction. Suppose G to be irreducible. Let denote the same group G when the range of the variables X I , . . . , x n is extended to the field of complex numbers. T h u s q is a transformation group of the complex vector space of n-dimensions. Of course, dim

=

dimG 2

11.~

- 2n

+ 4.

is reducible, then as a direct consequence of Cartan's arguments [ l , p. If 1.551we would have dim G S ( ~ 2 / 2 )which ~ is impossibIe. Thus q i s irreducible, and thus the complex form I' of qis also irreducible. We know that [ I , p. 1.511 a complex irreducible linear group is either semi-simple or the direct product of a semi-simple group and the homothetic group defined by yi =piwhere p is a n arbitrary nonvanishing complex number. From our assumption GCP,, no homothetic transformation can appear in I'. Hence I? is semi-simple. Here we find it convenient t o divide our discussion into the following two cases: CASE 1. I' is simple. Since the complex dimension of F which is the same as dim G is not less than n2-2n+4, we know that I' cannot be isomorphic with a complex linear group in less than n variables. Such simple linear groups have been determined by Cartan. Taking account of the fact dim I' 2 n 2 - 2 n + 4 , a survey of Cartan's table [3, pp. 147-1481 tells us at once that dim I' = n 2 - 1 where dim 'I means the complex dimension. Hence dim G = n 2 - 1. This contradicts our assumption t h a t G is a proper subgroup of P,. CASE2. I' is not simple. From representation theory, 'I can be written as a Kronecker product I'lXI'z of two nontrivial irreducible linear groups Fl and I'z. Let ri and ni denote, respectively, the complex dimension and degree(3) of I'i ( i = 1, 2). Then n1n2

= n,

rl

+ r2 = dim 'I 2 n2 - 2n + 4,

nl

2 2,

n2 L 2.

Without loss of generality, we can assume r 1 2 (n2-22n+4)/2. Since n l S n / 2 , it follows that rlz n: - 1. A contradiction is thus obtained. The contradiction in both alternatives proves our lemma. 2. Subgroups of the a 5 e groups. Let H , be the general linear group in n real variables. Then each element of H , can be regarded as a nonsingular real matrix ( a i j ) , and the special linear group P, becomes the totalit). of matrices (aij) whose determinants det (aii) are equal t o one. For simplicity, we shall use the following notations throughout: H i + = {(aij):det ( a i j ) > 01,

K = { ( a i j ) : a i j = A&), X = positive number] (4), (a) By the degree of a linear group, we mean the dimension of the vector space on which it acts. (4) 6i,, 6; denote the Kronecker deltas.

199

74

HSIEN-CHUNG WANG AND KENTARO YANO

I, =

rdi =

{ (az,):all = 1, anl = 0 , det (ail) = 1; a { (a t j ) : a l l = 1 , al, = 0 , det (ai,) = 1; a

= 2 , 3,

[September

. . . , n),

.. ,n), M = {(ai3):oll > 0 , a , ~= 0, det (ail) = 1 ; a = 2, 3 , . . . , n ] , M ' = { (a,,):all > 0, al, = 0, det (ai,) = 1; a = 2 , 3, . , 1 2 1 , where, and i n the following, the indices a , b, c, . . , i, j , K , . . take the values in the range 1 , 2 , 3 , . . . , n. We see at once that they are closed and connected subgroups of H Z , and = 2, 3 ,

9

dim K = 1,

dim H $ = n2, dim L = dim L'

= 1t2

-

ii

dim M

- 1,

=

dim M i = n2 -

ti.

LEMMA2. Let G be a closed and connected subgroup of M w'th dim G l n 2 -2n+4. T h e n either G=L, or G=M.

Proof. Let P,-1

=

{ (at,):all

=

1, ale

a,l

=

0,det ( a z 7 )= 1; a = 2, 3 ,

1

. . , PZ).

a11 subgroups of M , we have

Since G, L , and P , (2.1) (2.2)

=

+ dim P,-l- dim M 2 n2 - 3n + 4, dim ( L n Pn-l) = dim L + dim P,-l- dim M 2 n2 - 3%+ 3. dim (G

Pn-J

=

dim G

I t is a well known result (due t o S . Lie) that the projective group Pk has no proper subgroup with dimension higher than k2- k. Thus Pn-l cannot have proper subgroup with dimension higher than n2-3n+2. ( 2 . 1 ) then implies that GnP,-I = Pn-1, or what is the same, Pn-ICG. Thus P n - l C G A L C L . By using matrix multiplication we can easily verify that P,-l is a maximal subgroup of L. I t follows then that G n L is either Pnplor L. On account of (2.2), the first alternative cannot happen, and therefore, G n L = L , i.e., LCG. But we know that the difference between the dimensions of L and M is equal to one. I t follows then that G is either L or M . Lemma 2 is proved.

THEOREM 1. Let G be a closed and connected subgrouP of P,. If dim G 2 n 2 - 2n-I-4, then G is conjugate to one of the groups P,, L , M , L', M'. Proof. If G=Pn, our theorem evidently holds. Now, assume GZP,. Lemma 1 then tells us that G is reducible. I n other words, G leaves invariant a linear subspace of m dimensions with O<m
200

1955)

.4 CLASS OF AFFINELY CONNECTED SPACES

'e(1fb)t

0 0

0

0

ebt 0 0

0.

75

..0

0 . e . O

eb( O . . . O

. . . . . . . . . . .

.o

0

0

O...eb'

where t runs through all real nutnbers. Then each closed and connected subgroup G of H , w'th dim G z n 2 - 2 n + 5 i s conjugate to one of the groups: H$, P,, K X M , K X M ' , K X L , K X L ' , I ( b ) X L ,I ( b ) X L ' , L , L'.

Proof. From the connectedness of G, we know GCII,+. Let G * = G n P , . Since H$ = K X P, and dim K = 1, it follows th a t G* is a normal subgroup of G and dim G *Sd im G s d i m G * + l . This tells us th a t dim G * 2 n z - 2 n + 4 . Then on account of Theorem 1, we can assume G* t o be one of the groups P,,L, M , L', M'. When G* = P,, G is evidently either P, or H,+. We shall only discuss the cases G* = L and G* = A{; the remaining two cases can be reduced t o these two by a dualit,.. CASE 1. G* = M . Since G* is a normal subgroup of G, G is contained in the normalizer KXiM of M . From the fact th a t M C G C K A M , dim K = l , we know t ha t G can only be M or K X M . But M = I ( - l / n ) X L . Thus our corollary holds in this case. CASE2. G* = L. The normalizer of L in H,? is K X 111.Hence L C G C K X M . Passing from these groups to their Lie algebras, we find by a short calculation tha t G is either L or K X L or I ( b ) X L for a certain b. This completes the proof of the corollary. 3. Croups of affie collineations. Let et, be an n-dimensional spice with symmetric affine connection I'jk(x) covered b y a system of coordinate neighborhoods ( x l ) . Then the paths of this space are defined to be integral curves of the differential equations (3.1) s being the so-called affine parameter on each path. A point transformation

(3 * 2 )

*i

= f"(x)

is called an affine collineation when it carries a n y path into a path of the space and preserves the affine character of the parameter s. A necessary and suffi-

20 1

76

HSIEN-CHUNG WANG .AND KENTARO YANO

[September

cient condition that (3.2) be an affine collineation is that (3.3) If (3.2) is an infinitesimal transformation 5? =

(3.4)

xi

+ ('(x)dt,

4'i(rilieing a vector field and dt an infinitesimal, then (3.3) gives

where X is the operator of Lie derivation with respect t o (3.4). Equation (3.5) may be written also in tensor form: (3 * 6)

X r i j k E (';j;k

$-

=

Rijkltl

0,

where the semi-colon followed by an index denotes the covariant differentiation with respect t o r i j k and R i j A l the curvature tensor formed with The integrability conditions of system (3.6) of partial differential equations, or those of (3 * 7)

(ij;k

=

- Riikl(I

are given b y a sequence of equations (3.8)1 (3.8)~

X R i j k l E taRijkl;a XRijkl;rn

- t i ; a R a j k l + (';iR'akl

EaRijkl;m;a

- ti;aRajkl;m

+

+

ta;kRijal

ta;jR'ahl;m

+

+

(";iRijkQ

=

0,

-I- E a ; 1 ; R i j a l ; m

(a;lRijka;m

Sa;rnRijkl;a

= 0,

. . . . . . . . . . . . . . . . . . . . . . . . . Thus, in order that system (3.7) of partial differential equations be integrable, it is necessary and sufficient that there exist a positive integer N such that equations (3.8)1, (3.8)z, . , (3.8)N+1are automatically satisfied b y tiand satisfying equations (3.8)1, (3.8)2, * . . , ( 3 . 8 ) ~ . In this case, if N equations (3.8)1, (3.8)2, . , ( 3 . 8 ) ~give just r linearly independent equations with respect t o t i and t i ; j l then the space admits a group of affine collineations with dimension n 2 + n - r . Conversely, if the space admits a group of affine collineations with dimension n2+n - s, then there exist s linearly independent relations between 5' and si:j and the integrability conditions 3.&, * . should be automatically satisfied by ti and ti;jsatisfying these s relations [9;151. Now, when the space admits a group G of affine collineations with dimension r , if we take a point P in the space and consider all the transformations of the group which fix this point P,then such transformations form a sub1

202

group Gp, called the isotropic subgroup a t P. This subgroup Gp consists oi transformations T:3‘ = p ( x ;a)

such that .I-; =

j’(x0; a)

where xh are the coordinates of the points P,and a denotes the parameters. To each transformation T , in Gp, there corresponds a linear transformntion

of the tangent space [p a t the point P.I t can be easily proved t h a t this linear representation 7 of G p is an isomorphism in the sense of topological groups D61. Now consider the matris (l:) (a = 1, 2 , . . . , r ) of the components of a basis of the infinitesimal group of G, and denote by p the generic rank of this matris. .A point is called an ordinary point if, a t this point, the matrix assumes the maximum rank q, and is called a singular point if otherwise. be an n-dimensional space with a symmetric affine connection adLet mitting a group G of affine collineations of dimension greater than or equal t o n.Z--n+5. We confine ourselves to an open domain containing only ordinary points. Let G p denote t h e isotropic subgroup a t P. Then evidently dim Gp =dim 7(GI’)2nn?-2n+5. Thus, by corollary t o Theorem I , the identity component A P of r(Gp) should be conjugate t o one of the groups H 2 , P , K X M , K X M ’ , K X L , K X L ‘ , I ( b ) X L , I(b)XL’,L , L’. 4. The case by which A p is conjugate to H i or P,.In these two cases, the group G is transitive. Because if G is not transitive, there would be an invariant subvariety passing through P,and consequently, A p would leave invariant a proper linear subspace of the tangent space [ p a t the point P which is impossible. 1. CASEA p = H , t . In this case, G is of dimension n2+lz. T h u s t h e integrability conditions (3.8)1should be satisfied identically by a n y t8 and ti;+Thus by writing (3.8)1in the form i

(4.1)

.$aR’jka;a

- tb;a(6bRajkl - 6qRibkl

-

6;Rijbl

- 6;R’jkb)

we obtain (4.2)

=O

Rijkl;a

and (4.3)

6:Rajk~ -

a

6qRibkl

.

- 6kR’jbl - 6fRijkb

203

= 0.

=

o(‘),

78

HSIEN-CHLJNG WANG A N D KENTARO YANO

[September

From (4.3), ive get, by contraction with respect t o a and b, R iJ.k l = 0. (4.4) I t is evident t h a t if R i j k l = o , then conditions (3.8)1, (3.8)2, . * * are automatically satisfied. Thus, in order t h a t A p = H,+, i t is necessary and sufficient that the space is flat. 2. CASEA p = P,. I n this case, the group A p being of dimension n2- 1 and the group G being transitive, we know t h a t dim G=n2+n- 1. Since A P =P,, ti;jshoulcl satisfy

p ; n= 0

(4.5)

and the integrability conditions (3.8) should be satisfied identically by any and E ‘ ; j satisfying (4.5). Thus comparing (3.8)1and (4.5), we see t h a t there should exist functions Fijjn.l such t h a t EaRijtl;n

- Ei;aRUj/cl

become identities in

+

ta;jXiakl

+

{“;t.Rijal

+

la;lRij/.a =

- ~‘;$‘,II

ti and 4;;;i. Thus we must have Ri. jkt;a

(4.6)

=

0

a I1 d a

(4.7)

6;Rajkl

.

- 6 g R a b k l - 6 k R Z j b l - 6;Rijsb

=

6>ijkl.

By Contraction with respect t o a and b , we find from (4.7) F i j k l = - (2/.)Ri

jtl,

and by contraction with respect to i and b, we get

(4.8)

iiRajkl

-

6;Rjk

+ 6iR,1 + 6J(Rk1- Ri/J = -

(2/n)Rajkl,

where R,A= R a l k ( L . Contracting again with respect t o n and 1, we find R j k = o for f z > 2 . Thus we have, from (4.8), X i j j n L = ( ) . 5 . The case in which A r is conjugate to K X M , K X M’, K X L or K X L’. 111 these cases, the group G is transitive. \VC shall prove this by method of contradiction. We first suppose t h a t Ap=KXII.I or K X L and t h a t the group G is intransitive. Then the invariant subvariety passing through P should be onedimensional, because the linear manifold tangent to this subvariety at P is left invariant by K X M or K X L which fixes one and only one direction. Thus the rank of the matrix (4;) is equal to 1 a t P and, consequently, is equal to 1 a t every point of the domain under consideration. I t follows t h a t through every point of this domain there passes one and only one invariant curve. Now take an invariant curve passing through a.point Q which is not on the invariant curve passing through P and which is in the domain under

204

19551

A CLASS OF AFFINELY CONNECTED SPACES

79

consideration, .ind consider all the paths joining P to the points on the invariant curve passing through Q. These paths constitute a t u o-dimensional surface. This surface is left invariant by the isotropic subgroup G p . Consequently, the corresponding linear group A p must fix the two-dimensional plane tangent t o this surface a t P Lvhich contradicts our assumption. We next suppose t h a t A p = K X A I ' or K X L ' and t h a t the group G is intransitive. T h e invariant subvariet!. passing through P should be ( 1 1 - 1)dimensional, because the linear manifold tangent t o this subvariety a t P is left invariant by K X M' or K X L' which fixes one and only one hyperplane. Thus the rank of the matrix (ti) is equal to n - 1 a t P and, consequently, is equal to n-1 a t every point of the domain under consideration. I t follows t h a t through every point of the domain there passes one and only one invariant hypersurface. Now, consider a path through P which intersects these invariant hypersurfaces; then the points of intersections can be transformed by K into one another (except the point P, of course), Lvhich is a contradiction. Thus, in these cases, the group G is transitive, and consequently two isotropic groups a t a n y two ordinary points in the domain under consideration are conjugate t o each other. T h e groups K X M , K X M ' , K X L , K X L ' being respectivelj. with dimension n 2 - n + l , n 2 - n + l , n 2 - n , n 2 - n , and the group G being transitive, the group G is respectively with dimension n2+1, n 2 + 1 , n2, n2. Now, a t the point P of the domain, we choose the normal coordinates x z whose origin is P, then the space admits a one-parameter group of affine collineations

(5.1)

3%=

elxi.

In this coordinate system, the vector ladefining the infinitesimal transformation of this one-parameter group is given by

(5.2)

ti

= xi.

Thus, the integrability condition (3.8)1 becomes

which shows t h a t Ra,Ll are homogeneous functions of dc.gl-t*e- 2 of xi. But we know t h a t the components R i l k l of the curvature tensor are \Tell defined a t the origin of the normal coordinates system. T h u s the components Rijkr must vanish a t P and consequently a t a n y point of the domain. Thus, i n these cases, the space is affinely flat. 6. The case in which A p is conjugate to I(b)X L or L. In these cases, the group G is transitive. This c a n be proved b\- the same argument as that used a t the beginning of $ 5 .

205

80

HSIEN-CHLTNG M'ANG A N D I<EN'l';lRO YANO

[September

The group G being transitive, the isotropic groups at any two points of the domain under consideration are conjugate to each other. On the other hand, the isotropic group GQ a t an ordinary point Q fixes one and only one direction which we denote by uQ. Thus, a t every point Q of the domain under consideration, there is associated a direction UQ. Consider a path which passes through a point Q and tangent to u Q ;then the isotropic group GQ,being an affine collineation, fixes this path. We take a point R different from Q on this path and consider the transformations of GQ which fix this point R. These transformations form the group L. Now, we consider an affine frame a t Q whose first axis is in the direction U Q and transport it parallelly along the path to the point R. Then we have a t R a n affine frame whose first axis is tangent t o the path. The parallelism of vectors along a curve being preserved by an affine collineation, the transformation of GQ fixing the point R gives the same effect on the affine frame at R as on t hat a t Q.This shows th at the subgroup of GQ fixing R coincides with the subgroup of GE fixing Q. Th e subgroup of Gn fixing Q fixes the tangent to the path and U R , and consequently the tangent must coincide with U R , which shows that the path is the trajectory of the field of directions u. Now, the isotropic groups I(b)X L and L being respectively with dimension n 2 - n and nz-n-1, and the group G being transitive, the group G is respectively with dimension n2 and n 2 - 1. Now, the group G of affine collineations being transitive, we denote by T a transformation of G which carries a point Q into a point R. Then, by the same method as in [l6], we can prove that Tug

= UR

and that U Q is a parallel vector field. If we denote this vector field by u l ( x ) , then we have

where cr is a certain scalar and XI,a certain covariant vector field. 1;roin ( 6 . 2 ) , we find (6.3)

t 6 ' R T , ~= z

z~'XA~

where

- hl;h. p = I ( b )XL. Then equations (3.8)1 should be

(6.4)

hkl

=

Xk;l

We first suppose that A satisfied by any t band 41;1 satisfying (6.5)

(1

+ F2b)XUL = (1 + b),y,"I".

We see that conditions

206

.A CLASS OF .-\FFINELY CONNECTED SPACES

19551

81

and = 0

(6.7)

put together are stronger than (6.5). Hence a n y ti and ta;jsatisfying (6.6) and (6.7) must satisfy (6.5) and hence satisfy (3.8)1. T h e group being t h a t of affine collineations, the covariant differentiation and the Lie derivation are commutative and consequently, from (6.2) and (6.6), vie find xXk= 0. But the group G p does not fix a hyperplane and consequently we should have X p =O. Consequently we have (6.8)

uiik =

0

and

uiRijkl =

0.

Thus the integrability conditions (3.8)1 should be satisfied by a n y ti and ti;j satisfying

ii;a~a = 0 and l a ; ,= 0

(6.9)

and consequently there must exist functions (6.10)

Rijkl;a

=

Fijkl

and

G',jklb

such t h a t

0

and (6.11)

a

6bRajkj

- 6;RRibr;l -

6iR'jbi

- 6eRijkb

= 62';kl

+

UaGijklb.

From (6.11), after some calculation, we can deduce R i , k l = O . The case A p = L is characterized by (6.6) and (6.7) and consequently the above discussion shows t h a t when A p = L the space is also affinely flat. 7. The case in which A p is conjugate to I @ )X L ' or L' and G is transitive. T h e group G being transitive, two isotropic groups a t a n y two ordinary points in the domain under consideration are conjugate t o one another. On the other hand, the isotropic group GQ a t an ordinary point Q fixes one and only one hyperplane which we denote by V Q . Thus with every point Q of the domain under consideration, there is associated a hyperplane VQ. The isotropic groups I(b)x L' and L' being respectively with dimension ns-n and n 2 - n - 1 and the group G being transitive, the group G is respectively with dimension n2 and n 2 - 1. By exactly t h e same method as in [16], we can prove t h a t TVQ=

OR,

where T is an arbitrary transformation carrying a point Q into a point R. Furthermore, if we represent this hyperplane by a covariant vector 'uj(x), then we can prove t h a t

207

82

HSIEN-CHUNG WANG AND KENTARO YANO

[September

where a is a certain scalar. From (7.2), we find

- ViRijkl

(7.3)

=

Vpkl,

where (7.4)

CYkl

=

- vla;k.

vl;Ly;1

We first suppose that A p = l ( b ) X L ’ . Then equations (3.8)l should be satisfied by any ti and t i ; j satisfying (1

(7.5)

+

nb)Xvj

= (1

+

b)ta;avj.

We see that conditions

put together are stronger than ( 7 . 5 ) . Hence any ti and [ ‘ ; j satisfying (7.6) and ( 7 . 7 ) must satisfy (7.5) and hence satisfy (3.8)l. The group being that of affine collineations, the covariant differentiation and the Lie derivation are commutative, and consequently, from (7.2) a n d (7.6) we find Xa=O, which shows, the group G being transitive, that a is a constant. Thus the integrability conditions (3.8)*should be satisfied by any tiand satisfying (7.6) and ( 7 . 7 ) and consequently there must exist functions F Z , k l and Gijnlasuch that (7.8)

Ri,jlcl;a =

- aGijklCVcv,

and (7.9)

&(bRajkl

- 64RibkI - G;Rijbl

-

6YRR”jka = 6 > i j k i

+

vbGijkla.

From (7.9), after some calculation, we can conclude that (7.10) Ri j .k i = k v j ( v , J ’ i - V 1 6 ; k ) , where k is a constant. Thus equations (3.8)1 become .YRijkl

=

PR’jjel,

ovi.

where 0 is given by Xvj= When 1 +b#O there exists X such that p#O and thus we have Rijkz=0. When l + b = O then Xv,=O and thus (3.8)1 is really satisfied by all the infinitesimal transformations X of the group G. 8. The case in which A p is conjugate to I(b)X L’ or L’ and G is intransitive. Let us consider the invariant variety through P. All the points on this invariant variety being equivalent under the group G, isotropic groups a t points of this invariant variety are conjugate to each other. Thus the invariant variety should be ( n - 1)-dimensional, because the hyperplane tan-

208

19551

.A CLASS OF AFFINELY CONNECTED SPACES

83

gent t o this invariant variety a t a point should be left invariant by the isotropic group I ( b )x L’ or L’ a t this point which fixes one and only one liyperplane. Take a point Q not on this invariant variety. If the isotropic group a t Q is one of the groups hitherto examined except I ( b ) X L ’ and L’, then the group G should be transitive. Thus the isotropic group a t Q should be also I ( b )XL’ or L’. Consequently, passing through every ordinary point on the domain under consideration, there exists an ( n- 1)-dimensional invariant variety whose tangent hyperplane is fixed by the isotropic group a t the point of contact. We denote this hyperplane a t Q by VQ. T h e isotropic group I ( b )x L’ a n d L’ being respectively with dimension n2-n and n 2 - n - 1, and the invariant varieties being ( n - 1)-dimensional, the group G is respectively with dimension n 2 - 1 and n 2 - 2. Thus if we denote by f(x)

(8.1)

constant

=

the family of invariant varieties and put

xvi = aj//axi,

(8.2)

then, using the so-called adopted frames, we can prove t h a t (8.3) p k

(Xvj);e =

(Xvj)pk

+ (Xv~)pj,

being a certain covariant vector. On the other hand, we know t h a t

Xj = 0,

= 0,

X(XVj)

X(Xv,.);L.= 0

and consequently, from (8.3), we find

xpa = 0. But the hyperplane represented by o j is the only one hyperplane fixed by the isotropic group and consequently, we should have

PA

(1/2)ave

=

where a is a certain function off. T h u s substituting this into (8.3), we get (8.4)

(hVj):k

=

CY(hvj)(hPk)

from which (8.5)

viRijr,l = 0.

We first suppose t h a t A p = l ( b ) XL’. Then equations (3.8)l should be satisfied by a n y ti and ti;j satisf).ing

209

84

HSIEN-CHUNG WANG A N D KEN'I'ARO YANO

+

(1

(8.6)

P2b)XVj

= (1

[Septembcl-

4-b ) [ % J j .

We see that conditions

= X v a ( a = 0, xv, = ( a v , j ; o + , $ a ; j v , xj

(8.7) (8.8) (8.9)

[a;a

=

0,

= 0

put together are stronger than (8.6). Hence any ti and t i ; j satisfying (8.7), (8.8), and (8.9) must satisfy (8.6) and hence satisfy (3.8)1. Equations X(Xvj)= O and Xvj=O show t h a t x X = O and consequently that X is a function off. Thus, from (8.2), we can see that we can suppose X = 1. Thus equation (8.8) can be written as

xv. = p . . v

(8.10)

)--

I

J

=~ 0

by virtue of (8.4) and (8.7). Thus the integrability conditions (3.8)1should be satisfied by any ti and t i ; j satisfying (8.7), (8.9), and (8.10) and consequently there must exist functions E i j k l , F i j k l and G i j k p such that (8.11) (8.12)

Rijkl;a 8iRajkl

- GRibkI - 6ZRijbl

=

EijklV,,

- 6 Y R i j k b = 6>ijkt

+

Gijl;pvb.

From (8.12) we can conclude that the curvature tensor of the form (8.13)

R ij.k l =

i kZlj(Vk61

Rijkt

should be

i

- V16k).

But since we have X R i j k l =0, XZJ, = 0 , we find from this X k = 0 , which shows that k is a certain function off. Thus equations (3.8)1 become XRijkl

=

PRijki,

where fi is given by X v j = p v j . When l + b # O , there exists X such that p # 0 and thus R i j k l = O . When l + b = O , then X v j = O and thus (3.8)l is really satisfied by all the infinitesimal transformations X of the group G. T h e case A p = L ' is characterized b y (8.7), (8.8), and (8.9) also and consequently the above discussion shows that when A P = L', the space has also the curvature tensor of the form (8.13). 9. Theorems. Gathering all the results in $03-8, we have

THEOREM 2. If an n-dimensional space with a symmetric afine connection admits a group of afine collineations with dimension greater than n 2 - n + 5 , then the isotropic group G p at a point P , the dimension of G p , the groups of a@ne collineations G I the dimension of G , and the structure of the space should be one of those on the opposite page:

210

19551

is0tropic group GP

H, H;

85

.\ CLASS OF AFFINELY CONNECTED SPACES

dimension of G p n2

group of affine collineations G transitive

?I2

dimension of G n2+n n2+n

(I

I(

n2+n- 1

II

122-

KXM

n2- n+ 1

((

n2+ 1

K X Al’

n2-n+1

((

n2+ 1

KXL KXL’

n2- n

<(

n2

-n

16

n2

I ( b )X L 1:

It? - ri

L(

n2-n- 1

((

n2 n2- 1

I ( b )X I , ’

I$-

112

I1

affinely flat

l(

P,

1

structure of the space

(1

(i) l+b#O, R f k j = 0.

112

(ii) l + b = O , v j ; k =(YVjVk,

R ij.k l =

i i kvj(vk6l-Vlbk),

a, k : constants.

intransitive

n2- 1

(i) l + b # O , Rijkl=

0.

(ii) l + b = O , vj;k= avjvk,

transitive

it2-

intransitive

21 1

1

n2-2

86

HSIEN-CHUNG WANG AND KENTARO YANO

[September

10. Determination of ?ln when G is transitive.

THEOREM 3. Let %, be a simply-connected, n-dimensional manifold with a symmetric a f i n e connection. Suppose that 3, admits a group G of a f i n e collineations with dim G 2 n 2 - n + 5 . Let 23, denote the entire coordinate space w i t h co. . . , xn, and let i, j , k, . . . be indices r u n n i n g f r o m 1 to n ordinates xl, 9, while a, p indices r u n n i n g f r o m 2 to rc. T h e n %, i s equivalent to one of those on the follom‘ng page. Moreover, the a f i n e l y connected spaces listed above are nonequivalent. Before proving the theorem, we shall firstly give three remarks and establish a lemma. REMARK 1. Let G* be the group of all affine collineations of PI,. Then with respect t o a suitable topology, G* forms a Lie group(’j). Thus the closure of any subgroup of G* is a Lie group. For this reason, we can always assume the group G in Theorem 3 to be a connected Lie group, for otherwise, we can take the identity component of the closure of G instead of G. This does not effect the transitivity and the dimension restriction. REXLIRK2. Let G, be the isotropic subgroup of G a t a point p , and G, G, be the Lie algebras of G, G, respectively. Since 21n is simply-connected, the space 91, as well as the action of G on 2[, is uniquely determined by the pair (G, GiJ. 3. Let E be the quotient G/G, in the sense of linear space, and REMARK 4: G-+E the natural linear mapping. Denoting respectively by “Ad” and “ad” the linear adjoint representations of G and G over G , we have

Ad (Gp)(Gp) C Gm ad (Gp)(Gp) C G,. Thus Ad and ad induce linear representations\k and respectively. In fact,

and G, over E

$ ( 4 ( b ) = 4 ad (44-’@),

N . U ) ( b ) = 4 Ad ( u ) 4 - Y b ) , u EG,,

rl/ of G,

xEGp,

b E E.

On the other hand, there is a natural 1-1 linear correspondence between E and the tangent space of a t $, U p t o this correspondence,

lP an

W P )

=

m,).

11. A lemma.

LEMMA. Let P1denote the L i e algebra of the real special Linear group of degree t , and S a semi-simple L i e algebra containing P, a s a subalgebra. I f dim S S r 2 + 2 r + 2 , r 2 4 , then the least ideal S,,, of S such that P,CS, i s either P, itself or a P,+’. (6) Cf. S. Kobayashi: Groupe de transformations qui laissent invariante une connexion infinitbsimale, C. R. Acad. Sci. Paris vol. 238 (1954) pp. 644-645.

212

87

A CLASS OF AFFINELY CONNECTED SPACES

19551

2

I

%,,-origin

II

n2

I

1

Spaces

3

!

I

Connections

braxima1 group

i k yi=cb+ckx,

G* of collineations -

dim G* Rijkl

0

0

complete

not

I

Completeness

n2

not

-

Types

I

5

4

I

Spaces

1

23n

%?i

I

1

Connections

1

a

a

r l l = 2 k , I ' ; ~ = ~ S ~rll=% ,

others = 0, k = constant 2 O

others = 0, k = constant 2 0

~

Iaximal group

G* of collineations

Rijkt

-

t

rll= 2k, ryB= &, ryl= - xu

Q

Q

y =CO cosh xl+Cy sinh x1

+c;xB

(k

2

-

1 i l l)Sj(&&-

Sls:)

1

i l

(k2+1)6j(61ek-

i l

8h.61)

complete when k = O

213

88

HSIEN-CHUNG WANG AND KENTARO YANO

[September

Proof, Evidently S m is semi-simple and dim S m I r 2 + 2 r + 2 . Let S: be the complex form of S,, and P,* the subalgebra of S: which corresponds t o P,. We know that (1) S: is semi-simple, (2) P,? is the complex simple Lie algebra of class A and rank r - 1, (3) no proper ideal of Sc can contain P,*. Suppose that S: is not simple. Then we can write it as a direct sum

s,*

=

s:: G3 sz*

of two semi-simple nontrivial ideals. The intersection Sl*n P,? must be zero, for otherwise, we obtain from the simplicity of P? that P,*CSS which is impossible. Similarly, SPA p,* =O. I t follows then t h a t a*(p,*) = p,*, T Z ( p?) = P,*,where 7 1 :S:+SS, a 2 : S:-&* are the projections. Thus dim S$ 2 dim P,* = r2 - 1,

dim S,*2 dim P,* = r2 - 1, whence 2r2 - 2 5 dim S:

+ dim S: = dim S,*5 r2 + 2r + 2.

But this contradicts our hypothesis that r 24. Hence S: must be simple. Since P,*CS:, we know that rz

- 1 I dim S*,Ir2 + 2r + 2,

*

rank (S,) 2 r

- 1,

I

h 4.

A survey of the list of complex simple Lie algebras tells u s that S: has only : is of class A and rank r - 1 : (ii) S: is of class A and three possibilities: (i) S rank r ; or (iii) S: is of class B or C and of rank 3 and r =4. But it is well known that the complex simple Lie algebras of class B and C of rank 3 cannot contain Pt. Therefore, only cases (i) and (ii) can happen. Now we return t o the real Lie algebras P, and Sm.In case (i),

*

dim S,,, = dim S, = r2

- 1 = dim P,

whence S, = P,. In case (ii), S, is one of the real forms of the complex simple Lie algebras of class A and rank r . These real forms have been completely determined by Cartan [2]. We can see immediately from the list that PT11 is the only such real form which can contain P,. The lemma is thus proved. 12. Proof of Theorem3. Firstly, we observe that the maximal group G* of collineations of the five types of spaces listed in Theorem 3 are not isomorphic. T h e G* corresponding t o the first two types can be easily distinguished from the other. As for the G* corresponding to the spaces of types 3, 4, 5 , we can distinguish them by comparing the radicals of their Lie algebras. Thus two spaces belonging t o different types in Theorem 3 are not equivalent. However, in each of the later two types, there are infinitely many spaces depending on k. We find, b y a direct calculation, R i j k ~= ; ~

i

- 4kSmR’jki.

214

89

A CLASS OF AFFINELY CONNECTED SPACES

19551

From these equalities and the expression for Rijkz, i t follows that the absolute value of k is a local scalar invariant. Thus two spaces of the type 4 or 5 with different k (k 2 0 ) are not equivalent. 'The last sentence in Theorem 3 is proved. Now we shall show that each 71, satisfying the hypothesis of Theorem 3 is equivalent to one of the listed spaces. Let $ be a point of a,, and let G I G,, G, G,, E , T , !PI3/ have the same meaning as in §lo. Since 8, is simplyconnected and G connected, G, must be connected. Therefore, r ( G p )=A , . We have determined all the possible linear groups A,. T h e discussions for the various cases are similar. We select only three cases t o study as patterns. I. 7 ( G p )= A , = H,;t or P,.In this case, T(G,) is transitive over the nonzero tangent vectors at 9 , and thus G is transitive over the nonzero 1-elements of 8,. I t follows that 71, is complete in the sense of affine connection. From Theorem 2, 8, is affinely flat. We know that a complete, simply-connected, homogeneous affineIy connected space is uniquely determined by its local properties [9, p. SO]. Hence 8,zmust be equivalent to the space 8, with

r;,(x)

= 0.

11. 7 ( G p ) = A p = L . In this case, dim G=n2-1. The hypothesis dim G 2 n 2 - n + 5 then implies n 2 6 . Since 7 is a faithful representation, G, has a Levi-decomposition of the form

G, = P,-l

+ R1,

R1

=

radical of G,,

dim R1 =

IZ

-1

where P,t+,is the Lie algebra of the real special linear group of degree n - 1. Choose a maximal semi-simple subalgebra S of G such that P,-ICS. Let Sn, be the minimal ideal, of S, which contains P,-I. By the Lemma in $11, either S,,,= P,Lor S, = P,,-l. We shall discuss these two cases separately. CASE II1. S , = P , . Then dim Sn,=n2-1=dim G whence S,,,=P,=G. Passing the results in Theorem 1 to Lie algebras, we know that, up t o an automorphism of P,, P, has only two subalgebras of dimension n2-n-1, i.e., the Lie algebras L,L' of L and L' respectively. If G,= L', then i t is easy to see that 3/(G,) and hence !P(G,) does not have any invariant vector. This contradicts the fact that !P(G,) =T(G,) has an invariant vector. Therefore, G,= L. The real special linear group P, acts transitively on the space % ' , = 8,origin. Its isotropic subgroup a t the point (1, 0, . . , 0) is the subgroup L. Since n > 2, %Yl, is simply connected. B y Remark 2 in 810, we can regard 8, to be iDI,,and the group G to be P,. Using the coordinates XI,. * , xn in' , we find, by a direct calculation, that the only affine conherited from % ' !, invariant under P, is given by I'jk(x) =O. This gives us the nection over % space of type 2 in our theorem. Here we shall be a little brief and omit the tedious CASEI12. S,,= Lie algebra arguments. Taking account of the fact 9(G,) =T(G,) =L,we first show that S= P,,-l, R1belongs to the radical R of G and that Gp is reduc-

-

215

90

HSIEN-CHUNG WANG AND KENTARO YANO

[September

in G. Then, by rather elaborate Lie algebra arguments, we can prove that G has a basis n-1

1

a

1

a

, en-1, ee, ea, eo, eo

-.

( a # p ; a, p = 2 , 3 ,

*

(s, t , u , v = 0, 1, 2,

* . * ,

with the multiplication rule 1

u t

%

[e,, e , ]

=

t ' U

6*ev - ayes

n)

such that G, is spanned by

Now let us consider the group G' of all transformations of 1

y = x

1

+ c:xa+c:,

ya

=

c;xa + c:,

anof the form

IC,"( = 1.

This group G' is transitive over S nwhose isotropic subgroup G; at the origin consists of transformations: 1

1

1

y = x +C.X",

y

a

=

coax8

We see a t once that, up t o an isomorphism, (G, G,) = (G', GA).By Remark 2 in ill, we can regard W n = G = GI. A direct calculation shows t h a t the only affine connection over Dn invariant under G' is given by I'&(x) =O. This is the space of type 1. 111. r ( G p )= A , = L'. I n this case, we first show that Pn-l is at the same time a maximal semi-simple subalgebra of G, and then we can show that the pair (G, G,) has only the following three possibilities: CASE1111. G has a basis 2 e2

n

- en,

*

-,

n-1 etl.-l

n

a

a

a

a

1

- en, es, eo, e l , hea - eo, a! # P ; a , @= 2 , 3 , * .

*

, n.; x

# 0

such that G, is spanned by 2

n

e2 - en,

In this case, we can regard formations of the form:

n-1

. . . , en-1

n

- en, e

a

Q

~ el ,

a,, = Snand regard G to be the group of all trans-

I t follows then that the affine connections over 2$, invariant under G must be of the form: ( 6 ) A subalgebra L of a Lie algebra G is called reductive if there exists a linear subspace R of G such that G - L f R , L n R s O , [L,R ] C R .

216

91

A CLASS OF AFFINELY CONNECTED SPACES

19551

i

+

i l

=

r j m

k(6jSm

a

1

k

6mJj)S

constant.

=

When k = O , the space is of type 1. When k # O , we find t h a t two connections corresponding to different k's are affinely equivalent (in the global sense). Thus we can assume k = 1, and obtain the space of type 3. CASEII12. G and G, are spanned, respectively, by 2

n

e2 - en, *

n

n-1

a

a

a

. , en-i - en, es, el, eo,

x(e,O

*

+ e h + 4 + e;,

a

zP

and 2

n

n-1

e? - e,,, . . . , en-1

We can regard '& to be form 1

y = x1 - t ,

n

a

a

- en, eg, el.

Sn, and G t o be the group of all transformations of the

y a = C," cosh x1

+ CPsinh x + Cox , I ($1 a 8

1

=

exp ( n - 1)Xt.

T h e invariant affine connections are given by

I'fs(x)

I':l(x) = 2 k ,

=

kS;,

k

I'yl(x) =

- xu,

other

r

=

0,

= constant.

But the affine connections corresponding t o k and - k are equivalent. T h u s we can assume k z O , and get the spaces of type 4. CASEIIls. G and G, are spanned, respectively, b y 2

e2

n

n

n-1

a

a

a

- en, . . * , en-i - en, ea, el, eo,

and 2 e2

We can regard form yl

=

U n t o be

x1 - t , y'

=

n

- en, .

n-1

'

. , en-l

n

a

(I

- en, eB, el.

anand G t o be the group of transformations of the

~ t c o x1 s

+ Cfsin x1 + cox , 1 c,"I a 8

=

exp (rc - 1 ) ~ .

T h e invariant connections are given by I'il(x) = 2 k ,

I'yo(x) = Mi,

r:l(x)

=

xu, other 'I

=

0,

k

=

constant.

J u s t as in the above case, the connections corresponding t o k and - k are equivalent. Thus we can assume k20, and obtain the spaces of type 5. T h u s we know t h a t each ?In satisfying the restrictions in Theorem 3 is equivalent t o one of the five types. T h e completeness, curvature tensor and the maximal group of affine collineations of these five types of spaces can be obtained by a direct computation.

217

92

HSIEN-CHIJNG WANG .-\AID KENTAKO Y A N O

BIBLIOGRAPHY 1. E. Cartan, Les groupes projectifs continus rdels qui ne laissent invariantes accune multiplicitd plane, J. Math. Pures Appl. vol. 10 (1914)pp. 149-186. 2. -Sur certaines formes riemanniennes remarquables des gdometries d groupes fondamentales simples, Ann. &ole Norm. vol. 44 (1927)pp. 345-567. 3. , Sur la structure des groupes de transformationsfinis et continus, 2d ed., Vuibert, 1933. 4. I. P. Egorov, On the order of the group of motions of spaces with afine connection, C. R. (Doklady) Acad. Sci. URSS vol. 57 (1947)pp. 867-870. 5. , On the groups of motions of spaces with asymmetric a$ne connection, ibid. vol. 64 (1949)pp. 621-624. 6. , Collineations of projectively connected spaces, ibid. vol. 80 (1951)pp. 709-712. 7. --, A tensor characterization of A,, of nonzerocurvature with maximum mobility, ibid. VOI. 84 (1952)pp. 209-212. 8. , Maximally mobile L, with a semi-symmetric connection, ibid. vol. 84 (1952) pp. 433-435. 9. C. Ehresmann, Les connexions infinittsirnaks dans un espace jibrd differentiable, Colloq. de Topologie, Bruxelles, 1950,pp. 29-55. 10. G. Fubini, Sugli spazii che aminettono un gruppo continuo d i movimenti, Annali di Matematica Pura ed Applicata (3) vol. 8 (1903)pp. 39-81. 11. J. Levine, Classification of collineations in projectivcly and afinely connected spaces of two dimensions, Ann. of Math. vol. 52 (1950)pp. 465477. 12. D.Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. vol. 44 (1943)pp. 454570. 13. Y.Muto, On the a$nely connected spaces admitting a group of a$ne motions, Proc. Imp. Acad. Tokyo vol. 26 (1950)pp. 107-1 10. 14. K. Nornizu, On the group of a$ne transformations of a n a$nely connected manifold, Proc. Amer. Math. SOC.vol. 6 (1953)pp. 816-823. 15. K.Yano, Groups of transformations in generalized spaces, Akademia Press, Tokyo, 1949. On n-dimensional Riemannian spaces admitting a group of motions of order 16. --, n(n-1)/2+1, Trans. Amer. Math. SOC.vol. 74 (1953)pp. 260-279. ALABAMA POLYTECHNIC INSTITUTE, AUBURN,ALA. 'rOKYO UNIVERSITY, TOKYO, JAPAN.

218

EINSTEIN SPACES ADMITTING A ONE-PARAMETER GROUP OF CONFORMAL TRANSFORMATIONS BY KENTAROYANO

AND

TADASHI NACANO

(Received July 25, 1957)

The purpose of the present note is to prove the following m a i n theorem: Let M be a connected complete Einstein space of dimension n > 2 and of class C- and suppose that a vector field on M generates globally a oneparameter group of non-homothetic conformal transformations. Then M i s isometric to a simply connected space of positive constant curvature. I n particular M i s homeomorphic to the sphere S”. Let (h, i,j , * = 1 , 2 , - * . , n ) as2 = g j L ( E ) d € J d € ~ (1) be the positive definite fundamental form of M and ~ ” ( 6be ) the vector field on M which generates globally a one-parameter group of non-homothetic conformal transformations. Then we have [5]‘

+

g j l = vj v 1 vlvj = 2$gJi (2) where 2, denotes the Lie derivative with respect to v h , vj covariant $ 0

derivative with respect to the Christoffel symbols

{i}

and

(G

a non-

constant scalar. From ( 2 ) , we get [5]

S.{Fi}= 4944;

(3) and (4

+

+ $LA? - Sji@ +

- (Vk$”)gjl (vJ$“)gkL where c/)~ = vj+,4h = ghL$l and K,,l is the curvature tensor of M. From (4),contracting with respect to h and k and taking account of ( K = gJJKjl = const.) KILjL =hK j L= (K/n)gJL , (5) we find (6) Vj$t = k4gJL where K k = = const. (7) n(n - 1) We consider first the $ 0

= - AFVj@i

KLJL”

A?vL$L

9

Y

___1 2

-

All t h e quantities appearing in t h e discussions are supposed to be of class C-. See the Bibliography at the end of the paper. 451

219

452

KENTAROYANOANDTADASHINAGANO

(A) Case in which k = 0. In this case, we have, from (6), vjg, = 0, which shows that (pL is a parallel vector field. Thus, the space being complete, by a theorem of de Rham [4],the universal covering apace i@ of M is the product of a one-dimensional Riemannian space MI and an ( n - 1)-dimensional Riemannian space Mn-'. Thus we can choose a coordinate system ( E l k ) for M in such a way that ( E ' ) is a coordinate system for MI and ( p )( p , q , r , = 2, 3, .,n) is a coordinate system for Mn-,. Moreover, we can assume that the vector +lk has the components 9" = 8: for this coordinate system. In this coordinate system, we have g,, = const., g,, = 0, {;i}={;l} =o, and consequently, OIL+ = vh+= 41L= i3pgaIL = g , , which shows that 4 is a function of E' only. Thus we have from (2)

...

V4VP

where

+

vpvq

= 24(E')Q*PP

vnv,can be regarded as covariant derivative of

Mb-, defined by

--

v, in a subspace

= const.

The subspace ML-l is isometric to Mn-*and is complete and moreover admits a one-parameter group of non-isometric homothetic transformations. On the other hand, we have THEOREM(S. Kobayashi [3]). If M i s a n irreducible and complete Riemannian manifold of class C", then A ( M ) i s equal t o I ( M ) , except the case M i s the one-dimensional euclidean space, where A ( M ) ( I ( M ) )is the group of a f i n e ( i s m e t r i c ) transformations of M. THEOREM(J. Hano [l]). Let M be a simply connected complete Riemannian manifold of class C" and M = Mu x MI x x M, be the de R h a m decomposition of M . Then. the group A , ( M ) is isomorphic to the direct product A,(M,) x A,(M,) x x A,(M,) and the group I,(M) i s isomorphic to the direct product Io(Mu)x &(MI)x xI,(M,.), where A,(M) and A,(M,) (Iu(M)and I,,(Mi))are the connected components o f the identity in A ( M )and A ( M J ( I ( M )and I ( M J ) respectively. Combining these two theorems, we get THEOREM.If a simply connected complete Riemannian m a n ifold of class C" admits a one-parameter group of non-isometric homothetic transf ormations, then it i s a euclidean space. Thus our space M;-l should be a euclidean space and consequently the space i@ is also a euclidean space. But a euclidean space cannot admit globally a one-parameter group

-.

.- -

a

220

453

EINSTEIN SPACES

of non-homothetic conformal transformations. Because a global oneparameter group of conformal transformations in @ can be, as is well known, regarded as a group of transformations on a sphere which fix the point a t “ infinity ” and change circles into circles. By this a straight line in corresponds to a circle passing through the point a t “infinity ”. But conformal transformations, carrying circles passing through the point a t “ infinity ” into circles passing through the same point, carry a Thus they are projective. straight line in into a straight line in The transformations, being a t the same time conformal and projective, are affine and consequently homothetic [ 5 , p. 1671. Thus the case (A) cannot happen. We next consider the (B) Case in which cI # 0. In this case, we have, from (6),

a

a.

(8)

$&Jt

vJ@i + vi$,

2k$’81L

-

Thus, from ( 2 ) and (8), we have $ w g f l = vJwl f vlwJ= (9) where w,= - kv,. Equation (9) shows that w,is a Killing vector. On the other hand, we have

THEOREM(S. Kobayashi [2]). A Killing vector in a complete Riemannian space of class C” generates globally a one-pavameter g w m p of motions. We note that the conformal transformation group is a Lie group and if kv, and w,generate its one-parameter subgroups, so does cPL =kv, + wi . Thus @, = kv, + zuL generates globally a one-parameter group G, of conformal transformations. Let T, be a transformation of G, corresponding to the canonical parameter t and [ ( t )= T,E,,for an arbitrary fixed point E,,. Then LEMMA1. W e have @(E(t)) = - a tanh (ak(t tu)), (10) where tois given by c$(.&) = a tanh(kt,) and a is a constant. PROOF.By the definition of the canonical parameter t , we have

+

from which

221

454

KENTAROYANOANDTADASHINAGANO

On the other hand, we have, by virtue of (6), Vj($~i$~-

I@)

= 2k$$

I

-

2k44j = 0 ,

from which +L+'

-

k@ = kc ,

where c is a constant. Thus d4- = k(@

dt

+ c)

,

We shall prove that the constant c is negative. Because, if we suppose that c = 0, then (11)becomes

The case 4 = 0 being excluded, there exists a point at which + # O . Consider the trajectory passing through this point. The uniqueness of the solution of the differential equation d+/dt = k$' tells us that 4 never vaniRhes on the trajectory. Thus we have along the trajectory - -1- = k(t

4 Thus, for t +. - t,, we have I 4 I +

+ to) ,

(tu= const.)

00 and consequently +(T,E,)4 f 00, which contradicts the fact that T , defines a global one-parameter group. Next suppose that c > 0, then, putting c = a', we have from (11)

9 = k(@ + a2), dt from which $ = a tan ak(t

+ to),

(to= const.)

+

which gives $ 4 + ~0 for t t, +. n/2ak. This contradicts also the fact that T , defines a global one-parameter group. Thus the constant c should be negative and if we put c = -a2, equation (11) becomes

3 = k(#? dt

-

a2)

,

(a > 0).

Thus writing vh,$, $a,t instead of ad', a$, a&, t / a respectively from the beginning, we have (12)

&?= k($2 - 1)

dt

222

.

455

EINSTEIN SPACES

Thus if

+ f + 1, we have

Since erk(t+to)is 1 for t = - t,, we have 1- 4 = ezk(t+t")

I++ from which

9

+ = - tanh k(t + t,) .

We suppose a = 1in the sequel. From Lemma 1, we have

LEMMA2. ( i ) @51. ( ii ) A point at which +( E)" = 1 i s a $xed point : T,(E ) = f . (iii) If $(€)%< 1,the function +(E(t))= +(T,(f,)) is a monotone fiinction of t. Its upper limit is 1 and its lower limit i s - 1.

+

LEMMA3. The constant k i s negative, the trajectories of G are geodesics

-= unless zero. d - k-

and their length i s given by -

PROOF. Let s denote the arc length along a trajectory which is not a point, then from

we get (13)

which shows that k

< 0 and consequently K > 0.

From

and Vl@

= k4At I

we have

which shows that the trajectory is geodesic. From (13) we find

223

456

KENTARO YANO AND TADASHI NAGANO

Q,= - c o s I / - k s , where we assumed that s = 0 corresponds to the point lim,+mTt(€). Equation (14) shows that the length of trajectories is always equal to 7t

v'-k

'

LEMMA4. A trajectory has no conjugate points on i t except two points. PROOF.From

v,Ct)*= k$gl,, we find

- K k J i b $ ' L = k($kgII - $ j g k L ) * Thus the sectional curvature with respect to a plane determined by and 4'' which is perpendicular to 4'' is given by

$h

E j t l L

- p K k J ~ l ~ t $ ? ~ ! & = - k,

4*4b$a$a which is a positive constant. Thus t h e Lemma follows from the classical argument.

LEMMA5. There exists at least one point at which we have $(€) = 1 or +(€) = - 1respectively. PROOF.This follows from the completeness of the space and Lemma 3. LEMMA6. The point set M I = (6,e M ,

= 1) i s discrete in M.

PROOF. Since f 1 are extremal values of Q>(t), we have BJ$ = 0 a t E,. Thus the expression of the function $ ( E ) a t El has the form

+

+

+

$(F1 h) = f 1 kgl,(El)hjh6 O(h3), which shows that a sufficiently small neighborhood of the point E, cannot contain the point of MI.

LEMMA7. The point El i s one of the end p d n t s of the trajectory which passes through any point t suficientlg near to 6,. In other words, a n arbitrary geodesic issuing from the point t , i s a trajectory at least in a sufjiciently small neighborhood of 6,.

224

457

EINSTEIN SPACES

PROOF. Following Lemma 6, M I is discrete and consequently, for a sufficiently small s l , an arbitrary point E # E l , whose distance from El is less than sl, does not belong to M,. Suppose that $(El) = - 1 (The same method applies to the case C#I(€~)= 1). Take a positive number 6 such that I + -(- 1)I = I cosI/-k s - 1 I < a implies s < s1/2. Since the function + ( E ) is continuous, we can find a positive number s,( < 4 2 ) such that

+

Distance ( E , E,)

< s, implies I (I, - ( -

1)1

< a.

Now take an arbitrary point E in the s,-neighborhood and put 1imt+-- T(E)= E-

.

Then since distance ( E , El) < s, we have ] +(() - ( - 1)] < 6 and consequently s < s1/2, s being the distance between E and €- along the trajectory, thus distance ( E , E - ) < s,/2. On the other hand distance (El, E ) < s, <s,i2 and consequently distance ( E - , El) < s, from which we can conclude El = E - . The last part of the Lemma follows from the fact that a sufficiently small neighborhood of E, does not contain a conjugate point El. LEMMA8. When ths number (I,1 is su&iently close to c#I((,) = + 1, the surface defined by +(E) = +I in a suflciently small neighborhood of El is homeomorphic to a sphere.

PROOF. This follows from Lemma 7. ( 6 , El) = constant. LEMMA9.

If + ( E ) =

The M I consists of two points.

MI = { E + , E - } ,

+(E,) =

+ 1,

then the distance

Thus w e can p u t (I,(€-) = -

1.

This means that, if E .f; E,, E - , then the trajectory of the point E i s the geodesic joining the points E, and 6-and passing through 6.

PROOF. Following Lemma 5 , there exists a point E, such that +(€,)= $1. Following Lemma 8, we can assume that the subspace S , : +(€) = ++( < 1) is homeomorphic to a sphere for a real number ++ sufficiently near to 1. For an arbitrary point E on S,, we put 1im6+-.-T,(E)=E-. Then, according to Lemma 7, there exists a neighborhood U of €- such that E- is an end point of the trajectory of any point in U. Thus when t,, is negative and has sufficiently large absolute value, we have, for the fixed E, Ttu(E)e U , thus, the neighborhood T;JU) fl S , is carried by Truinto the U, consequently lim+- T,(T,JU)n 8,)= E-.

+

225

458

KENTAROYANOANDTADASHINAGANO

This shows that the mapping : S+ MI given by t + lim+m T,(E) is continuous. On the other hand, S , is connected and consequently its continuous image is also connected. But Mlis discrete. Thus we have limt+-J"(S+) = E-. If we put MI€+,6-1 = { € ; 1imt4+Tt(E) = E , or €-} , then M{E+,5 - } is an open subset of M. Because, it is evident that M {€+, 6-} contains the neighborhoods of E , and €-. For the point 5 other than E , and 6-, we have only to consider T; ( U ) for a to such that T,U(E) e U, U being a neighborhood of E, or E-. Thus T , J U ) is open and is contained in M{E+, E - } On the other hand, M{E,, E - } is obviously compact and a fortiori closed in M. M being connected, M coincides with M{E+, E - } . Now consider a sphere Snwhose scalar curvature coincides with that of M. We fix a point F- in S" and consider the partial differential equations in S" :

.

under the initial conditions : Since S" is of constant curvature, this system of partial differential --equations is completely integrable and admits a solution ( E ) and c$~(€). -Since S" is compact, $*(€)generates globally a one-parameter group of conformal motions. Moreover since S" is an Einstein space, every thing stated above is valid in S". We denote by (3,say, equation in S" corresponding to equation (2) in M. We now define a homeomorphism from a neighborhood U of 6-in M to of ?- in S". a neighborhood S" For this purpose, we consider normal coordinate systems valid in U c M and in S" - {?+} with polesT- e M and E- e S" respectively, and to an arbitrary E in U , we make correspond a pointTin S" which has the same coordinates as E. This is possible because distance ( E , E - ) < --r and distance 1/- k

+

{r+}

(cz)<

We denote this map by h : E into S".

37.

226

Then h is a diffeomorphism of U

459

EINSTEIN SPACES

LEMMA10.

We hava

T,

0

h = ho T,

0%

U n T;'(U)

for an arbitrary t . PROOF.For E E U , we have c$(€)=&o h(E) according to the properties of normal coordinates and (14). Following (iii) of Lemma 2, there exists a real number C, and a point _ E, E M such that E = T,,(€,)and $(€,,) = 0. Define 5 by h(E) = T,,(€,,), then we have, from (iii) of Lemma 2 and (p(6)= 0. Thus we have

(n),

T,,(€d = € h T,,(€"o)= Tt,(W Similarly this relation being true also for t + tosuch that Tt+Lu(€,,) E U, 7

O

= R+'"(G)*

h O Z+',(€O)

Thus, for € E U

n

T;'(U), we have

h 0 T,(E)= h 0 T,T,,(E,)= h 0 T,+,,(€,)

-_

=

-

T,+,"(FJ = T,T,,(€,)= T,h(€).

We enlarge h to the whole M in the following way, for E = E+ h(E) = F+ for E f € + h(€)= F;' 0 h 0 T,(E)

for T,(E) E U.

In the second case, h(€)does not depend on t according to Lemma 10. LEMMA11. The mapping h i s onto and is o m to one. PROOF.For an arbitrary E S n , if i=i+, then by the definition of h, h(€+) = If F # g , for a negative t with sufficiently large absolute -value T,(E)is very close to ?-, and consequently it is an image of some E by h, that is, T,(gj = h(E) = h(T,T;'i)= T, 0 h 0 T,-'(E), h 0 T;'(E)=

c.

r.

Suppose that h(E)= h(€'). There exists t and C' such that T,E, T,$ E U , thus we can assume T , Z E U n T;?,,U (i.e., t - t' S 0). By the definition of h, we have -

T;'

0

h o T,(E)= F,'

o

h o T,,(E')

ho T,(E)= F,-,,0 h 0 T,,(E')= h o T 6 - , 0, T,,(E')= h 0 T,(E'). But h is a homeomorphism in U , and consequently T,(E)= T,(€'),

€=El.

LEMMA12. The mapping h is a homeomorphism.

227

460

KENTAKOYANOANDTADASHINAGANO

PROOF.Take an arbitrary point E E M . If E # E,, then denote by V a neighborhood of E which does not contain E,. For a sufficiently large t , we have T,( V )c U. Thus h = F;’ o h o T,is a homeomorphism on V. Because T , and Ft are both homeomorphisms, and h is also a homeomorphism on U. If E = E,, we take a point 7 in a neighborhood of E + , then distance ( E + , 7 ) = distance ( h ( E , ) , h(7)), because distance ( E + , 5 ) is the length of the geodesic joining two points E + and 7 and geodesics and their length are invariant under la according l o (10) and (14). Thus h is a homeomorphism in the neighborhood of E , too.

PROOFOF THE MAIN THEOREM. The mapping h is a diffeomorphism on M - { E , } and M - { E , } is a neighborhood on which a system of normal coordinates can be defined. Thus = 7 0h and at the points corresponding to each other under h, we have

+

vJ” = k$A:

= k+A$ = v,+“

.

Thus

or

This equation shows that the parallel displacements along the trajectories of @ and cph are the same for the two spaces. This means that the parallel displacements along the trajectories of @ and $h are commutative with h. A vector at E- of M is transformed into a vector a t F- of S” which has the same length. Thus h is an isometry on M - { E + } . Hence M - { E , } is a space of constant curvature and consequently M is also of constant curvature. Thus M and S” are isometric. TOKYO INSTITUTE

OF

TECHNOLOGY BIBLIOGRAPHY

1. J. HANO, O n aflne transformations of a Riemannian manifold, Nagoya Math. J., 9 (1955), 99-109. 2. S. KOBAYASHI, Groupes dc transformations qui laissclzt iizvarianle 2cng cmnexion in$nithimale, C . R. Acad, Sci. Paris, 238 (1954), 644-645.

228

EINSTEIN SPACES

461

, A theorem on tl@ a f i m transformation group of a Riemannian manifold, Nagoya Math. J., 9 (19551, 39-41. 4 . G. DE RHAM, Sur l a r8ductibilite’ d’,un espace de Riemamn, Comment. Math. Helv., 26 3.

(1952), 328-344.

5. K. YANO, Theory of Lie derivatives and its applications, North Hoiland Publishihg Co., Amsterdam, 1957.

229

A N N A ~OF U

MATIIHMATlCr)

Vol. 69, No. 3. May, 1959 Printed in Japan

HARMONIC AND KILLING VECTOR FIELDS IN COMPACT ORIENTABLE RIEMANNIAN SPACES WITH BOUNDARY BY KENTAROYANO (Received January 5, 1958)

The purpose of the present paper is to generalize some of the results on harmonic and Killing vector fields stated in Yano [5] and Yano and Bochner [6] to the case of Riemannian spaces with boundary. The study of harmonic tensors on such Riemannian spaces was begun by Duff and Spencer [3] and Conner [2]. Nakae [4] also studied the relations between curvature and relative Betti numbers in Riemannian spaces with boundary. 1. Stokes' theorem

We consider a compact manifold M which is the closure of an open submanifold of a n n-dimensional orientable Riemannian manifold V, of class = c' (r 2 2) with a positive definite metric ds9 = g,A(E)dE"dEA ( K , X, p, 1 , 2 , --., a ) and is represented, in a neighborhood of each point on the boundary B of class C' by En 2 0. It follows that B is an (n - 1)-dimensional compact orientable submanifold. (See, Chern [l],Theorem 6.1, p. 85). The boundary B is represented locally also by

p=EK(vh)( h , i , j , k , - . - = 1 , 2 , - . . , n - l ) in U ( P ) n M , U ( P )being a coordinate neighborhood in Vn of a point P on (1)

B. We put (2)

&K

= a,p

(6, = 6/87)')

(3) 'gjt = B.YBiAgpA and denote by N" the unit normal to B such that N" and B;",Big,. form the positive sense of M . Then we have (4)

.- , B;-;

g,AN'BiA = 0, g,,NILNA= 1

1/ A[N",B;"1 =V"g

where 9 and '0 are the determinants formed by g,, and ' g j t , respectively. Now denoting by '0, the covariant differentiation of van der Waerden-Bortolotti with respect to 'gjtalong B, we have 588

230

HARMONIC AND KILLING VECTOR FIELDS

589

(5)

'y,B;" = H,,N"

(6)

'T,N" = - H;'B;" (equations of Weingarten)

(equations of Gauss)

where H,, is the second fundamental tensor of B with respect to the normal N". We also use the notations (7)

B,, = B;'QA,

Bf, = B$'"''gg,<

Then we have

NAN"

(8)

+ B!ABE"= At

At(= 8:) being the unit mixed tensor. Now Stokes' theorem can be stated as follows :

for any ( n - 1)-form (0 in M . Now taking an arbitrary vector field u" in

M , we put (1)

=

(n- l ) !

.

*(2bAdEA)

Then we can easily show that daJ = t ? ~ ( / ~ ] u ' ) dAE 'dE' A

*.-

A&"

on M

- A du,"-'

on B .

and fiJ

= (u,N")I/'g dq' A dy/?A

- a

Thus Stokes' theorem can also be stated as follows :

STOKES'THEOREM.We have, f o r an arbitrary vector Jield u"in M , (9) where

IMVAu"dn=

s

uANAd'(T,

/<

vh denotes covariant difl'erentiation with rmpect

symbols

{ !Fk} formed by g,,

(10) do = 1J 9 dE' A d t 2 A

to ths Christoflel

and

~

*

*

dE", d'a = v"!)dq' A dv' A

- - - A dr,"-' .

It is easily seen that, for an arbitrary vector field v' in M , v p

{vh(Vhv')

- U p ( VAV")) =

+

Kfi,4V"nu" (v'vA) (vA2(&) (vpv")(VA.")

Thus applying Stokes' theorem, we find

23 1

*

590

KENTAROYANO

2. Non-existence of harmonic or Killing vector fields

We now assume that the vector field v‘ is tangential to B, that is,

(T)

8%=

&“v‘

or vKNK =0

on B .

Differentiating v A N A= 0 along B, we obtain Bj”(vpv,)NA - vAH:Bt“ = 0 by virtue of the equations of Weingarten. Multiplying this by ’ v J , we obtain (1%

(c~vA)v”LN”

= Hji’v’ ’v‘ ,

where (and in the following) the subscript T ( N )indicates that the formula or theorem concerns a vector or a tensor field tangential (normal) to B. Thus from (11) and (12),,.,we obtain (1317

1 {Kp~v’v~ Y

4- (r’”u^)(C~v,J - ( v p v ” ) ( V A v A ) ) d a

=

1

H1~’d’vid’Q

for a vector field vKtangential to B. This formula can be written in the following three forms :

232

591

HARMONIC AND KILLING VECTOR FIELDS

v&vA- VAvp = 0, V A V A = 0 V p V A f VAvp = 0 v&vA v A v p = (2/n)gpAVava* We have from these equations, 9

9

+

I,.. If, in an M with boundary B, the Ricci curvature is posiTHEOREM tive (negative) definite and the second fundamental f o r m is negative (positive) semi-definite, then there does not exist a harmonic (Killing, conformal Eilling) vectoy field tangential to B other than the zero vector field. We next assume that the vector vKis normal to B, that is, vK= aN“ or v,B;” = 0

(N)

where a is a scalar on B. Differentiating v” = cwNAalong B , we find

.

Bi”(V,V“) = ( ’ ~ 1 a ) N” nH;“BiA

Multiplying this by Bt, and taking account of (8), we obtain

(A! - N A N p ) ( v p v = A ) - aHi‘ , or (vA((vAv”)- V”(VAvA)))Np = n‘H: . (17)lv Thus from (11)and (17)N,we obtain

for a vector field v<normal to B. Equation (18),,,can be written in the following three forms : (19)N

\

dl

+

{KpAvyU* ( v p v A((v,vA) ) - i(v’va

-

Vhvp) (V+vA- VAv&)

- (V,v”)(vAvA)jda =

233

5

a2HH;,”d’o,

592

KENTAROYANO

From (19),, (20), and (21)N,we have

THEOREM .2, I f , in an M with boundarg B, the Ricci curvature i s positive (negative) definite and the mean curvature of B i s negative (positive) or zero, then there does not exist a harmonic (Killing, conformal Killing) vector field normal to B other than the zero vector .field. 3. Necessary and sufficient conditions for a vector field to be a harmonic, Killing or conformal Killing vector field

We now apply Stokes’ Theorem to vA(vKvK) and obtain

or

234

HARMONIC AND KILLING VECTOR FIELDS

Forming finally (26),

593

+ (16)T,we find

But

and consequently

Thus the above equation gives

From (27),, we have

THEOREM 3,. A necessary and suficient condition fw a vector field v‘ tangential to B t o be harmonic i s that in M on B.

(30)T

From (28),, we have

THEOREM 4,. A necessary and sz4ficient condition for a vsctor j e l d v K tangential to B to be a Killing vector i s that V 0~ in M g’A~p~A~ +u KFvA ‘: = 0 , V ~ zz (311, on B. ( v ~ v A vAv~)v”N~ =0

+

From (29),, we have

THEOREM5,. A nacassary and sz&ient condition f o r a vector field v” tangential to B to be a conformal Killing vector i s that

235

594

KENTAROYANO

( v ~ v A

+ v,~v,)v’N~= 0

on B.

From Theorem 4,, we have

THEOREM 6T. An infinitesimal afline transformation vKtangential to B and satisfying (vpvA vAv,)v”N~= 0 on B i s a motion. In fact, from e u { ;A} = vpVAvKf K;;iKvv= 0 we find

+

+

vAvA = const.

gpAVpvAvKK:vA = 0 and

On the other hand SMV,+’VAdO. = V’NAd’a = 0 SB

and consequently VAvA= 0

.

We next assume ( N ) . Then from ( 1 7 ) N ,we have {NA(vAvN) - N“(VAV~)}V, = or

+ a2Hh? .

(331, (t7Aw,)v”NA = a(VAvA) Substituting this into (22), we find

Forming (34)N - (19)N,we find

Forming (34),

+ (20)N, we obtain

Forming finally (34),

+ (21),,,,we obtain

236

HARMONIC AND KILLING VECTOR FIELDS

595

and consequently

Thus the above equation becomes

=2ln{

CY(VAV^)

+ C Y ’ H ; ~ } ~. ’ C

From (35)Np we have 7,. A necessarg and sufident condition for a vector field v* THEOREM normal to B to be harmonic i s that gpAvpVAvK - KiKVA= O in M (3% VAvA= 0 on B.

Suppose that vKis a Killing vector field normal to B : VpVA

+

VAVp

=0*

On taking account of v A = aNA on B, from B;”Bi’(VpVA

we have

V ~ v p= ) 0

+

9

BiA‘ v , ( ~ N J BjA’~a(aN*) =0 - 2tuHjl = 0 . Thus we have

231

9

596

KENTAROYANO

THEOREM 8., If an M admits a Killing vector jield normal to the bozmdary B and vanishing only on a nowhere dense set of points on B, then the boufidary is totally geodesic. From (36),, we have THEOREM 9,. A necessary and suficient condition for a vector jield v' normal to B and vanishing only on a nowhere dense set of points on B to be a Killing vector is that

+

in M on B.

gPhVpVhVr KiKvA= 0 , V ~ V '1 0 Him= 0

(39)N

THEOREM 10,. An injinitesimal afina trann.qformationvKin M which is normal to tha minimum boundary B and does not change the volume is a motion. Suppose that vKis a conformal Killing vector field normal to B :

+

vpvh V ~ V ,= (2/n)g,~v,v" . On taking account of v A = aN, on B, from

+

B;"BiA(~pv, ~ h v p = ) (2/n)'gj,(Vava)

9

we have B i h ' ~ j ( a N J BjA'VL(aN2) ( 2 / 4'gjt(VavU") - aHjt = ( l / n )'~~L(Vav") , from which n - i (vavm) aHia = 0 . n Thus we have

+

+

11; If an M admits a conformal Killing vector Jield normal THEOREM to the boundary B and vanishing only on a nowhere dense set of points on

B, then the boundclry is umbilical. From (37)Nfwe have 12,. A necessary and su.ficient condition for a vector Jield vK THEOREM normal to B to be a coqformal Killing vector is that

TOKYO INSTITUTE OF TECHNOLOGY

238

HARMONIC AND KILLING VECTOR FIELDS

597

BIBLIOGRAPHY 1. S. S. CHERN,Lecture note on differential geometry I. Chicago, 1955. 2. P. E. CONNER,The Green's and Neuman's problems f o r differential f a r m s on Ricmannian manifold, Proc. Nat. Acad Sci. U.S.A., 40 (1954), 1151-1155. 3. G. F. D. DUFF and D. C. SPENCER,Harmonic tensors on Riemannian manifolds w i t h boundary, Ann. of Math., 56 (1952), 128-156. 4 . T. NAKAE,Curvature andrelativc Betti numbers, J. Math. SOC.of Japan, 9 (1957), 367373. 5. K. YANO,On harmonic and Killing vector jields, Ann. of Math., 55 (1952), 38-45. 6. K. YANO and S. BOCHNER,Curvature and Betti numbers, Ann. of Math. Studies, 32, 1953.

239

This Page Intentionally Left Blank

Offprint of Commentarii Mathematici Helvetici. vol. 36, fasc. 3, 1961

Projectively Flat Spaces with Recurrent Curvature To the University of Hong Kong on its Golden Jubilee in 1961 By YE"-CHOW WONGand KENTARO YANO

Introduction Let AN be an affinely connected N-dimensional space with a symmetric connection (i.e. a connection without torsion). AN is a projectively flat space, or simply, a PN if there exists a coordinate system in terms of which the finite equations of the paths are linear. AN is of recurrent curvaturel) if the covariant derivative of its curvature tensor is the tensor product of a nonzero covariant vector and the curvature tensor itself. The purpose of this paper is to determine all the projectively flat spaces with recurrent curvature. For convenience we shall denote such a space by P:, It is found that the space P: (or rather its connection) depends on 2 arbitrary functions of one variable or on 3 arbitrary functions of one variable according as its RICCItensor is symmetric or non-symmetric. The actual construction of the connection of the P; depends on the solution of a differential equation of the RICCATI type and on the solution of a completely integrable system of differential equations. Projectively flat space with covariantly constant curvature tensor (i.e. projectively flat symmetric space) is also considered. We prove that it is a well-known type of projectively flat space characterized by its RICCItensor being symmetric and covariantly constant.

1. Preliminariesa) Throughout this paper, each of the indices a , h , i , j,. . . runs through the range 1 , . . . , N ; a, denotes partial differentiation with respect to the k t h coordinate ; and a repeated index implies summation. Let AN be a linearly connected N-dimensional space with a symmetric connection = Ti.). The curvature tensor, the RICCItensor, and (i.e. the tensor P j i are defined respectively by:

R~~~~=

+

- a,r& r&r;- r,?yii, Rgi = RZji ,

(1.1) (1 * 2)

l ) RIEMANNian spaces with recurrent curvature have been studied in great detail by H.S. RUSE[2] and A. G . WALKER [3]. Certain classes of nOn-RIEMANNian spaces with recurrent curvature have been studied by Y. C. W o ~ o[4]. See EISENHART [l], but our notation is slightly different from his.

24 1

224

YUNQ-CHOW WONQ @ENTARO

YANO

It follows from (1.3) that P,, is symmetric iff the RICCItensor R,, is symmetric. By definition, AN is of recurrent curvature if its curvature tensor satisfies the condition V1Rkjih= TIRkjih ,

(1.4)

where V denotes the covariant differentiation and r, a non-zero covariant vector. If vlRkjih = 0 , (1.5) the space AN is said t o be symmetric. AN is projectively flat iff there exists a coordinate system in terms of which the components I$ have the form

I'$ = 0,Af where At is the KRONECKER delta and

+ 0,A; ,

(1.6)

0 , a set of N functions. The special 'I is preserved by affine transformations of coordinates. Exform (1.6) for : pressed in terms of P l i , a well-known necessary and sufficient condition €or AN to be projectively flat is

For N > 2 , (1. 7)2is a consequence of (1.7), ;for N = 2 , (1.7), is an identity. There exists, for (1.6), a function 0 such that o5 = 8,0 iff P,,= P i , . Since on account of (1.7) a projectively flat space with P,I = 0 is flat, we shall always assume that p,i # 0 (1.8) 2. Some necessary conditions By definition, a P: is a non-flat projectively flat AN with recurrent curvature and therefore it is characterized by the conditions (1.4), (1.7) and (1.8). To find all the P: 's, we first derive a set of necessary and sufficient conditions and then determine the functions 0, in (1.6) which satisfy these conditions. From (1.4) we obtain V I R5, = r, R,i, and consequently vkPjt

= rkpji

242

(rk

# O)

*

(2.1)

225

Projectively Flat Spaces with Recurrent Curvature

It is easy to see that conversely ( 1 . 4 ) is a consequence of (2.1) and ( 1 . 7 ) , . On account of ( 2 . l ) , equation ( 1 . 7)2 becomes

where p i is some non-zero vector (since P j i # 0). Using ( 2 . 2 ) in ( l . 7 ) l , we see that the curvature tensor of a P : form Rkjih

=

- (Atr, - A f r k ) p i

+

-r

(rkpj

.

jp k ) At

has the (2.3)

On account of ( 2 , 2 ) , equation ( 2 . 1 ) becomes

+

(vkrj)pi

= rkrjpa

r5(vk13i)

(2.4)

>

from which it follows that

for some vectors s j , t , . These equations show that ri and p i are of the form

ri = Ati

(tf= sit+ 0 ) ,

Pi = p q i

( ~ = i

aiq

+

0)

3

for some scalars l , 17, 2 , p . Substitution of ( 2 . 6 ) in ( 2 . 3 ) and ( 2 . 5 ) gives

where y , 01,?! , are some scalars. We now proceed to consider the integrability conditions of the differential equations ( 2 . 8 ) . For any covariant vector u i , we have the RICCIidentity VkVj Ui

-vjvkui

=

- RkjihU,.

Using ( 2 . 7 ) in ( 2 . 9 ) and putting ui = t i , we get, since (tkO1j

-

i.e. which is equivalent to

ij&k)

tk(&j

=

- v ( tk q 5

-

tjqk)

+ vqj) = + + YTJr= et5 f5(.k

a5

243

(O1j

v q k )

=

(2.9)

ti + 0 , 3

YUNG-CHOW WONQ1 KENT~LRO YANO

226

for some scalar get

e.

Similarly, using ( 2 . 7 ) in ( 2 . 9 ) and putting ui= q i , we

Br - 2Yt, = wi for some scalar 0. Equations (2.10) tell us that The scalars

OL,

8, y , e , 0

(8, = a i m

(2. lo),

can be expressed as functions of 6,q alone. (2.11)

On account of ( 2 . l l ) , equations (2.10) can be re-written as (2.12)

a&

a&

a = -, etc. Two cases arise according as whether or not at q aV q is functionally dependent on 8 , i.e. whether or not q i and Et are proportional. Now, by ( 2 . 2 )and ( 2 . 6 ) ) q i and ti are proportional or not according as the tensor P,i (or the RICCItensor) is symmetric or not. We shall consider these two cases separately in $ 3 and $ 4 respectively. where

016

= -,

3. P:

with symmetric RICCItensor

In this case, q = q(E) so that (2.12) reduce to

(q- e)

+ (a, + Y ) $ = 0

>

(86 - 2Y) + ( B q - 4 r ' = 0 >

where q' = d y / d E . Since these equations just determine the unspecified functions e , 0 of E which appear in ( 2 . lo), the integrability conditions of ( 2 . 8 ) reduce to the mere fact that O L ) fi and y are functions of 6 . On account of q = q ( E ) , equations ( 2 . 7 ) and ( 2 . 8 ) become

Rkjih= y r ' ( 4 Es VjE, =OLEjEi

Vj Ei

=

- J.4;

E7J ti

7

)

(87' - q/'/q')E 5 t i

-

Hence, we have the first part of the following

Theorem 3.1. For a P; with symmetric R I C Ctensor, ~ there exists a scalar ( and functions 8 and

OL

of E such that

where E, = a, 6.

244

227

Projectively Flat Spaces with Recurrent Curvature

[The integrability condition of ( 3 . 2 ) i s identically satisfied.] Conversely, a n AN satisfying this condition is a Pz (or a symmetric PN) with symmetric RICCItensor. Proof, We need only prove the last part of the theorem. It follows from ( 3 . 1 ) that Pji = - e t j t i (3.3) 9

and so P j i is symmetric. On account of ( 3 . 3 ) and ( 3 . 1 ) , condition ( l . 7 ) l is satisfied. Furthermore, a simple calculation involving ( 3 . 3 ) and ( 3 . 2 ) will show that V P j i = rkP j i, with

It follows from this that ( 1 . 7)2 is satisfied and that V l Rkiih= rl Rkiih. The AN is therefore 2 P : or a symmet,ric PN according as O,/e 2a # 0 or = 0. Hence our theorem is completely proved. I n order to construct the rjh, of a P; with symmetric RICCItensor, we choose a coordinate system in which TA have the form

+

r;i = o j A ! + o i A ; ,

(3.5)

where oj are N functions of the coordinates. With respect to the by ( 3 . 5 ) , the covariant derivation of a cova,riant vector ui is

vjui= a,ui

- O,ui - O i U j ,

r$ given (3.6)

and the curvature tensor has the components Rkjih

= - Ak0ji

where 0ji

+ A:0ki +

(0kj

- 0jk)Af .

(3.7)

= ajO, - 0 j 0 i .

(3.8)

Now identifying ( 3 . 7 ) with ( 3 . 1 ) and rewriting ( 3 . 2 ) by means of ( 3 . 6 ) , we get

a,oi = 0 ~ -0 e~(8) i jti, aj ti = ojti oi t, a (5) t jti ,

+

+

(3.9)

where t i = sit. It is easy to verify that' for any functions e ( 5 ) and a ( [ ) of 6, the integrability conditions of the differential equations ( 3 . 9 ) in the N 1 unknown functions P)~,8 are satisfied, on &count of the equations ( 3 . 9 ) themselves. Now the solution of a completely integrable system of differential equations contains a finite number of arbitrary constant,s, while the special form ( 3 . 5 ) for the conneckion of a PN is preserved by a.ffinetransforma,tions of coordinates

+

245

228

YUNO-CHOW WONQ/ KENTARO YANO

which depend on a finite number of constants. But neither of these would alter the fact that I'; as given by (3.5) depends on two arbitrary functions of one variable, namely, the functions 0 , a of E . Hence (cf. (3.4)) we have

Theorem 3.2. If we take any functions 0 , a ( &#

of one variable 6, and any solution o i , E of the completely integrable system ( 3 . 9 ) , then the connection I'h = ojA? oiA; defines a P$ with symmetric RICCItensor; and any P$ with symmetric RICCItensor can be constructed in the way. T h u s the most general P$ with symmetric RICCItensor depends on 2 arbitrary functions of one variable, and the actual construction of the connection of such P: depends on the solution of a completely integrable system of differential equations. - 05/20)

+

4. P: with non-symmetric RICCItensors Let us now return to (2.12). Since q i , ti are not proportional, (2.12) are equivalent to

&=*,

a[=@,

ol,+yl=o

,

p5-2y=o.

The first two equations merely determine the unspecified scalars which appear in ( 2 . 1 0 ) .Therefore,

.,+y=o,

Q, u ,

pt-2y=o

are the integrability conditions of equations ( 2 . 8 ) . From this and (2.7), ( 2 . 8 ) and ( 2 . l l ) ,we have the first part of the following

Theorem 4.1. For a P i with non-symmetric RICCItensor, there exist functionally independent scalars [ and q and functions y , a , /3 of and 7 such that REjih = Y [(A:~j - A; t k ) q i - ( t k q j - tjrk)AZ1 (4.1) vjti

= atjEi

>

vjqi

= BTjBi

3

where ti = ail, l;li = sin. [The integrability conditions of (4.2) are &,+ly=o,

Br - 2 y

= 0.1

Conversely, any AN which satisfies the above condition is a P: symmetric RICCItensor.

(4.3)

with non-

Proof. We need only prove the last part of this theorem. It follows from (4.1) that (4 * 4) p5i = - ly'55r1,

246

Projectively Flat Spaces with Recurrent Curvature

229

end so P,i is not symmetric. Now (4.4), together with (4.1) and (4.2) show that condition (1.7) for a PN i s satisfied. Furthermore, a simple calculation involving (4.4) and (4.2) will show that V k P j i= r k P f i , and hence v z R k j i h= T I R k i i h , with

This vector rk is not zero. I n fact, since q k , t, are not proportional, rk = 0 would imply that a =0, (1% $46 (log y), B = 0*

+

+

Differentiations of these give Dl,l =

Bg

9

from which and (4.3) it follows that y = 0 . But this gives a flat space. Hence our theorem is completely proved. I n order to construct the I'h of a P : with non-symmetric RICCItensor, we proceed as in the case of symmetric RICCItensor, and have (3.5), ( 3 . 6 ) , ( 3 . 7 ) and (3.8). Identifying (3.7) with (4.1) and rewriting (4.2) by means of (3.6), we get = 0j0i ~ ( 5q ,) t j q i

+ + nit, + & ( 5 , q ) 5 , q i 0jqi + 0iqj + 17) 5jqi 9

= 0,ti

ajqi =

P ( t 9

>

(4 * 4)

>

where ti = ai6, q i = aiq. From Theorem 4 . 1 it follows that an AN with connection (3.5) is a P : with non-symmetric RICCItensor if the functions 0, together with two functionally independent scalars t , q and some functions y , a ,P of [ , q , satisfy the differential equations (4.4). We know already (from Theorem 4.1) that the integrability conditions of (4,4)z,3 are (4.3). It is easy to verify that on account of (4.4), the integrability condition of (4.4), reduces to =0* (Y, Y B ) ( 5 k V , -

+

Since 17 # q (t) , this is equivalent to

Hence combining this with (4.3), we obtain the following complete set of integrability conditions of (4.4):

24 7

230

YUNQ-CHOW WONG/ KENT~RO YANO

Rewrite these as

From the last two equations, we obtain

i.e. Integration of this with respect t o

t

gives

where f (7) is an arbitrary function of q . If fying ( 4 . 6)1,then in virtue of ( 4 . 5 ) , y and as follows:

is any function of 6,q satiscan be expressed in terms of 9,

B

where h ( 6 ) is an arbitrary function of t . Equations ( 4 . 5 ) are equivalent to equations ( 4 . 6 ) . Since equation ( 4 . 6 ) , is of the RICCATItype, the most general function ( t, 7) satisfying it is of the form

where ,!I1(q), . . . , p4(q) are certain functions of 7 and g(6) is an arbitrary function of 5 , Hence, with an observation similar to that immediately before Theorem 3 . 2 , we have

Theorem 4.2. W i t h a n y functions f ( q ) and h ( t ) of one variabEe q and t respectively, a n y solution /?( 5 , 7) [cf. ( 4 . 7 ) ] of equation ( 4 , 6 ) ,, and the functions y ( E , q ) , a ( : , q ) given by (4.6)2, ( 4 . 6 ) 3 , the system of differential equations ( 4 . 4 ) is completely integrable. If oi, 6,q are a n y solution of the completely

248

Projectively Flat Spaces with Recurrent Curvature

231

+

integrable system (4.4),then the connection I'i = OjAt 0iA7 defines a Pg with non-symmetric RICCItensor; and a n y such P: can be constructed in this way. Thus the most general P: with non-symmetric RICCItensor depends o n 3 arbitrary functions of one variable, and the actual construction of the connection of such Pg depends o n the solution of a differential equation of the RICCATI type and o n the solution of a completely integrable system of differential equations. 6. Projectively flat symmetric N-space

We now consider the AN'Scharacterized by

Rkjih= - AkP,,

+ A ) P k i + ( P k j- P j k ) A t

vlRkjih = 0 .

(5.1) (5* 2)

On account of (5.l), the condition (5.2)is equivalent to (5.2')

VkPji = 0. Substituting (5.2')into the RICCIidentity vl

VkPji - VkVgPji = - RzkjaP,i

- RlktaPja

and taking account of (5.l), we obtain

and consequently, (6.1) becomes

Rkjih= - A t P , ,

+A)Pki.

(5* 4)

Since, conversely, (5.1)and (5.2)are easy consequences of (5.4),(5.3)and (5.29,we have

249

232 YUNQ-CHOW WONQ/ KENTAROYANO Projectively Flat Spaces with Recurrent Curvature

Theorem 5.1. An AN is a symmetric PN,iff Rkjih= - AtPji

+ A;Pw

,

VkPfi= 0. P.. 3a = Pi5, The properties stated in Theorem 5 . 1 characterize a well-known type of projectively flat space; in particular, if Pji is of rank n , it is RIEMANNian and of constant curvature (cf. EISENHART [l], p. 97 and p. 166). We can easily prove that in contrast, a PN with recurrent curvature, i.e. a P;, is never a RIEMANNian space, Obviously, a P; with a non-symmetric RICCItensor cannot be RIEMANNian. If a P; with a symmetric RICCItensor is RIEMA"ian, let g,* be a fundamental tensor whose CHRISTOFFEL symbol oomponents Rkfih

177

is equal to I'ti. Then the covariant

= Rklia

gha

of the curvature tensor satisfy the well-known equation Rkfih

= Rihk5

t

which on account of ( 3 . 1 ) reduces to g kh 65 t i = gi5 l

ht k '

But this contradicts the fact that g,i is of rank N R1EMA"ian.

> 1 . Therefore,

no P; is

University of Hong Kong Tokyo Institute of Technology REFERENCES [ 11 L. P. EISENHART, N O W R I E M A N Geometry. N~~~ New York (1927). [2] H. 5. RUSE,A c2u.aaificatio-n of K*-spaces. Proo. London Math. Soo. (2), 53 (1961), 212-229. 131 A. G . ~ A L K E ROn , RusE'a apace of recurrent curvature. Proc. LondonMath. Soc. (2), 62 (1950), 36-64. [4] YUNQ-CHOW WONQ,A cluaa of N O W R I E M A NK"-apuce. N~ Proc. London Math. Soc. (3). 3 (1963).

(Received August 1 1, 1960)

250

Reprinted from the TENSOR (New Series), vol. 14, 1963 p. 99-109.

ON A STRUCTURE DEETNED BY A TENSOR FIELD f OF TYPE ( 1 , l ) SATISFYING f 3 + f = 0 . Dedicated to Prof. Dr. Akitsugu Kawaguchi on the occasioa of’ his sixtieth birthday. By Kentaro YANO.

3 1. Let AI, be an 71-dimensional differentiable manifold of class Cwand let there be given a tensor field f 0 of type (1,l)and of class C“ such thatl)’%

+

J ”f=

(1.1)

0.

Theorem 1. 1. For a tcmor $eld f # O (=-f’,

(1.2)

satisJj,ing (l.l),the operators

77Z=J’2+1,

1 r?enoti)ig the identity operator, cGplied to the targent space at a point of the manifold arc co ~ ~ ~ p l ct n en tprojection aiy operators.

Proof.

We have

I + 71L = 1 and l 2

=J’” = -f’=

I,

1)2*=flf2f2+1= - y + 2 , + 1 = f 2 + 1 = ? ) 1 by virtue of (1.L), which prove the theorem. Thus if there is given a tensor field f # O satisfying (1.l),then there exist complementary distributions L and iL1 corresponding to the projection operators 2 and 111 respectively. If the rank o f f is constant and is equal to r, then the dimensions of L and Af are 7- and 12-r respectively. We call such a stmcture an f-strzictzrre of rank r.

Theorem 1.2. For j satisfying (1.1)and I , riz d&md by (l.3),we have (1.3)

J’I=If=f,

f??L=??2f= 0 ,

f’l=-I,

f m=0 ,

that is, f acts on L as an almost complex structure operator and on M as Received by the editor October 1, 1962 1 ) A brief summary of the paper has been p~lblishedin [5]. Numbers in brackets refer

to the references at the end of the paper. 2 ) A similar structure appears also in A. G. Walker [4].

251

100

K. Yano.

a null operator.

Proof.

fI = lf =f(-f) = -f” =f , f’l =f” = -1,

f m = mf yn1=

=f c f ‘ +

1)= 0 ,

0.

If the rank o f f is n, then l=1 and m=O and f satisfies

f+1=0 and consequently the f-structure of rank n is an almost complex structure and n must be even. If the rank of f is n-1, then L is (n-1)-dimensional and nil is one:, 1,h and mch(It,i, j , ..., =1,2, ...,n) dimensional, consequently if we denote by f the local components of f, l and m respectively, then mirkshould have the form : m;

= viu”,

where z, and u are covariant and contravariant vector fields respectively. From the relations f”+1= m , fm=O, mf=O and m2=nz, we have

f,”fi“ + SI;= VjUh , fku4 = 0 , fehVh = 0

vdui= 1 .

,

Thus an f-structure of rank n-1 is what S. Sasaki calls (f,u,v)-structure [3] and is equivalent to an almost contact structure.

Theorem 1.3. F o r f satisfying (1.1)a n d Tn defined by (1.2), we have (1.4)

(nt+f)(m-f)= 1 ,

fm=mf

=0.

Proof. We have, taking account of (1.1)and of (1.2), (m+f)(m-f)

=

(f+f + 1)Cf-f

+ 1)= f ’ + f + 1= 1

The relations f.z=mf=O have been already proved in Theorem 1.2. Theorem 1.4. Suppose that there is given a projection operator na a n d there exists a tensor field f such that (1.4) are satisjied, then f satisfies (1.1). Proof. From the first equation of (1.4), we have, taking account of the second equation and m2=m,

n i - f = 1. Applying f to this equation and taking account of fm=O, we have

-f” =f , which proves the theorem.

Theorem 1.5. F o r tensors p and q defined by

252

On a structure defined by a tensor field f of type (1,l) satisfyingf3+f=0.

(1.5)

p

= vz

+f=f'+f+ 1,

4 = m-f=f"-f+

101

1

w e have

(1.6)

Pq=qP=1,

p?-p+ 1 = q ,

(1.7)

q2-q

+1=p,

and conseqzrently

p'

(1.8)

= q2.

ProoJ: First of all, we have, from ( l . 4 ) 7 pq=qp= 1 .

We then have from (1.5)

p 2 = i ? ~ + J " , q2=nL+fLl and hence

p'-p+l q2-q

=(??L+f)-(nz+J')+1=f'-j+1=q,

+ 1 = (7?2 +J'")-(

n&--f)+ 1=y+f

+ 1 =p .

Theorem 1.6. If tliel-e are given in the manifold two distinct tensor fieZds p and q both o j type (1.1)satidying (1.9)

p q = qp= 1,

p'-p+1= q

1

then we can $nd out (in f + O such that J" +f = O and p and y coincide with those given by (1.5). Proof. Applying q to p'-p + 1= q from the right and using pq= qp= 1, we find 4'-q + 1= p and consequently p 2 = q 2 . We now put (1.10) then

f is

f

1 P-4)

=4

2

not zero by assumption.

3

W e have from this

(1.11)

and hence

f'+f=O. On the other hand, we have, from (1.10) and ( I . I I ) ,

p=f'+f+17

y=y-f+l.

253

102

K. Yano.

Theorem 1.7.

For the tensors p and q dejined by (1.5), w e have

(1.12)

J Pz=f, I p'Z= -z,

pna = m , p'na = m ,

(1.13)

i

qm=m, 4% = n 2 ,

q z = --1, q'Z= fy

that is, p and q act on L as almost complex structure operators and on ill as the identity operator. ProoJ We have only to substitute p=f'+J+1, I= -f? ,

q=J"--J'+l, ni=f+l

into these equations and to take account of J3+f=0.

$j2. We now introduce in the manifold a local coordinate system and denote by f:, Z,", m," the local components of the tensors .f, Z,n i respectively. We also introduce a positive definite Riemannian metric in the manifold and take r mutually orthogonal unit vectors u$ (n,b,c;..,=1,2;..,7-) in L and 72-7mutually orthogonal unit vectors zc: (A, B, C, ... = r+ 1,..., n ) in Ad, we then have

\ ILhu;= , 1 ?7L,hu:= 0 ,

,

ZLhZLL = 0

24;

(2. 1)

m,hu; = u;, .

From f t n = O , that is, f6hnajZ=0, we find, contracting with ui: and taking account of the last equation of (2.1),

(2. a)

j,"uz, = 0 .

If we denote by (vz,v,")the matrix inverse to (u;, ui),then v: and v: are both components of linearly independent covariant vectors and satisfy (2.3)

,

v:24: = 8;:,

v:u;

vau; = 0 ,

vdu;,= 3;

=0

and

v:u; + V,'Zljl

(2.4)

=

sl" .

Now from (2.1) and (2.3), we find J

(l'hZ)'L) h GI, z - 00 "R ,

I

(.zihvh")u:= 0 ,

( l , " ~ ~=) ~0 {, ~ j (nz'";)u;, = 4; ,

which show that

I;v;

= v; ,

nz,"v;f= v,",

254

On a structure defined by a tensor field f of type (1,l) satisfying f3+f=0.

103

from which J

(2.5)

Z,“V;=V;,

t??v;=O,

nztv; = 0 ,

?n:vi

= v:

.

From mj”=O, that is, fihmhJ=O,we find, contracting with v: and taking account of the last equation of (2.5),

fi”.v.;l= 0 .

(2.6)

On the other hand, from Zjhui=zi:, we find

z,nv:u:

= v;u:

,

= v;u; ,

zJ”(6+v;u.{)

that is,

z;

(2.7)

= v;u;

by virtue of (2.1) and (2.4). Similarly we get (2.8)

’F1lifL= V $ U ;

If we change ut and z& formations

into

?z;

n h *- C>,LC, ,

zS.h

(2.9)

,

and a; respectively by orthogonal trans-

a;

= c;u;

,

where

c:c;

=

,

c;c;

then va and v: are changed into 6; and 6;= &);,

(2.10)

fj;

pi’

=

a,,. ,

respectively by the rules:

,

= cz ;:,

and we have

a;%; = v p ; ,

fj$:

= ,;v;

.

Consequently if we put nj, = v;v; + VAVA ;i L

(2. 11)

9

then aji is a globally defined positive definite Riemannian metric with respect to which (zit, ZL;) form an orthogonal frame and such that (2.12)

,

vj”= aj&, ,

21; = aj,zi:,

Zj, = Zjluli ,

inji = ’F?)2jiari ,

as we can easily verify it. If we put (2. 13)

we find, from (2.7) and (2.8),

(2. 14)

lji = Zj;Vy ,

712ji

255

VJV)”

104

K. Yano.

by virtue of (2.12). Consequently (2. 15)

= a,,

L,,+7?Ljl

,

that is, I,, and m,, are both symmetric and their surn is equal to u j L . We can easily verify the following relations: (2. 16)

I j f l ~ a= t aI,, , IJlrn,'a,, = 0 , nzjtmzSaf,= ?

.

) L ~ ~

If we put 1 gj, = -@jl.

(2. 17)

2

+ f , f f 1 4 8

+ 1TZid) >

then g,, is again a globally defined positive definite Riemannian metric which satisfies (2.18)

v: = g,uR

and nijL= niJLgfr.

(2.19)

Equation (2.18) tells us that the distributions L and ill which were orthogonal with respect to a,, are still orthogonal with respect to g,, and u2 which were mutually orthogonal unit vectors with respect to a j Lare still mutually orthogonal unit vectors with respect to g j L . We can easily verify that the tensor g,, satisfies (2.20)

+ ntj, = g,,

f,%SfY

by virtue of ?+ 1= m . The relation f'-nz= -1 may be written as fjyft-??i; = 4;.

(2.21) If we put

h'ga1 = hf,

(2. 22)

we have, from (2.20) and (2.21), and fjtfil-wij,

f j t f t f + n i j , = g,,

=

-gjL

respectively and hence f j f

(hf+fiJ

=0

*

The rank o f f being r and 72-r linearly independent solutions of f,'vt=0 being given by v f , these equations give

hi +A6 = 4 4 for certain wt,from which w,4=0, or (2. 23)

A 1

+A, = 0

256

On a structure defined by a tensor field J’ of type (1,l) satisfying f3+f=0.

105

by virtue of fi:u:

=A‘zf

=0

\

and

-f&:

= (f:juZ)g,r = 0

*

Thus f,, is a skew symmetric tensor of rank r, hence r must be even. Gathering the results, we have

Theorem 2.1. I f , in an n-dimensional manifold, there is given a tensor jield f f 0 of rank r satisfying f” -tf = 0, then there exist complenaentary distributions L of dimension r and M of dimension n-r and a positive dejinite Riemannian metric g with respect to which L and M are orthogonal and such that f;sz”gls

+ na,tY:a

= g,,

9

where

T h u s the rank r off must be even. Take a vector uh in the distribution L, then the vector f:ud is also in L and orthogonal to uh, and moreover has the same length as uh with respect to the metric g5,. Consequently we can choose in L r= 2m mutually orthogonal unit vectors ~ 4 ; such that u;,l

=f t u :

)

u;,, =f

t d , ..., u2”,=f:u:z

Then with respect to the orthogonal frame have components : Em 0 0

(ut,ZL;), the tensors

g,, and f,<

Em 0

0

0 E,, 0 f = -E, 0 0 0 0 0) ‘, 0 0 E,z-zm) ’ Em denoting na x m unit matrix. We call such a frame an adapted frame of the structure f. Now take another adapted frame (a,”, 22;) with respect to which the metric tensor g,, and the tensor f,,have the same components as (2.24) and put (2.24)

g=

)

ah

- rzu;,

a;

=

r&.d

then we can easily see that the orthogonal matrix

257

)

106

K. Yano.

must have the form

i

?’=

1.

-B, A, 0 Am 0 Bm 0 o,,-*, O

Thus the group of tangent bundle of our manifold can be reduced to U( m) x O(n-2m). Conversely if the group of tangent bundle of the manifold can be reduced to U(m)x O(n-2m), then we can define a positive definite Riemannian metric g and a tensor f of type (1,l) and of rank 2m as tensors having (2.24) as components with respect to the adapted frames. Then we have

f j,” -;; I, f+: -EnL

-Em 0

0

0

and consequently

f”+f=O. Thus we have Theorem 2.2. A necessary and suficient condition for an n-dimensional manifold to admit a tensor jield f#O of type (1,l)and of rank r such f = O is that r is even: r=2nz and the group of tangent bundle of that f”+ the nianifold be reduced t o the group U(nz)x O(n-2m).

Q 3. The distribution L is defined locally by (3.1)

?npd:6 = 0

or by

(3.2)

v:dcd = 0 ,

where Ed are local coordinates in the original manifold. The integrability condition of (3.1) is given by (3.3)

lj*l:(a,m,”-a,??a,“) =0 ,

(a, = slap).

We assume that the distribution L is integrable, that is, (3.3) is satisfied. Then denoting by (3.4)

( E) = const.

7iA

equations of integral manifolds, we can choose vf in such a way that (3.5)

v:

= &v”.

If we represent one of the integral manifolds by parametric equations

258

On a structure defined by a tensor field f of type (1,l)satisfying f 3 + + f = 0 .

(3.6)

t" =

107

:n(q&) ,

7" being parameters, wc have

B b h z=~0~,

(3.7) where

(3.8) in such a way that the matrix inverse to ( B t ,zi;)

Thus we can choose is (BnL, and we have 21,')

( 3 .9)

BfLLB; =GJ,

BRk~Lil= 0 , v,1B!,'= 0 ,

V ~ Z=L6,;~

and

I,"

(3.10)

,

.

= B'kbB,"

)?L&" = z l , " ~ ?

If we put

y7: = L?,"B",,S,h,

(3. 11) we can easily verify that

'L~J'fb[fi = -",;

(3. la)

and consequently the subspaces v = const. admit an almost complex structure. We now introduce a symmetric affine connexion in the original manifold, the thing which is always possible. For example we have only to introduce a Riemannian metric in the manifold and denote by the connexion given by the Levi-Civita parallelism. Since we have ) I - ? - linearly independent vectors 21: which are transversal to subspaces zjA=const., we can induce a symmetric affine connexion on the subspaces and are given by

rtk

r:,

'rr6

+ U?,Bt)B", .

= (B,!B,"r:,

(3. 13)

Denoting by 8, and 'V,. the covariant differentiation in the enveloping manifold and van der Waerden-Bortolotti covariant differentiation along one of the subspaces ZJ'= const., respectively, we have

(3. 14)

1 'VcB," = 1

I

h I

, ' V c B B= ", - h P , a ~ , ZL

' 8 , ~ :=: h,",B,h-h$,~~;,

'V,v:

=

-?i,,z'B",+h$jz~~ ,

as equations of Gauss, Weingarten and Ricci, where

(3. 15)

i

ll,;f

+

= (BcjB,"r,", &,B,")

74 ,

I L ,= ~ (a,B,-B;Bnr:&) ~ $ , 1 h,A,= (a,~$-B2virjA) ZL;~ .

Now the Nijenhuis tensor 'Nca for the almost complex structure 'Jb" is

259

108

K. Yano.

given by (3. 16)

IN,:

= ' f c " ' V d I f ~ - ! " r ' ' V , 1 ' f c " - ( ' ~ c ~ ~ ' - ' V 2 , ~ ~. ' ) ~ ; 1 R

It is well known that the tensor does not depend on a special choice of symmetric a f h e connexion involved and we can replace the covariant differentiation by partial differentiation. Substituting (3.11) into (3.16), we find

IN,:

(3. 17)

= B,;'B,"B"hNj,h,

where

N5a =f;Vlf6" -AZVZfk-( v 5 f 6 " - 0 , f ~ " . f ~ h * When the distribution L is integrable and the almost complex structure

(3. 18)

induced on the integral manifold by the f-structure is also integrable we say that the f-structure is partially integrable. Suppose that our f-structure is partially integrable, then we have (3.3) and

BdB,"B"hN5,h= 0

(3.19)

by virtue of (3.17). Now from (3.18) we have

N5tmzh= -f5mf6z(Vmm,h-Vzm,~), and consequently, if the distribution L is integrable and v: are chosen as (3.5), then mlh being given by mt=q'u$, we have

(3.20)

N5rzmrh =0 .

Thus from (3.19) or B:BbmPZN,,d= 0 , we have, by contraction with

B,BbrB,h, (3.21)

IjnlrmNn,h =0

by virtue of (3.20). Conversely suppose that our f-structure satisfies (3.21), then we have by definition (3.18) of N5ih,

z~1:(v,.~z-v8f,",f,h =0 , which is equivalent to (3.22)

ljfl~(Vrmsh-V8m~) =0 .

Thus the distribution L is integrable and we can induce an almost complex structure on the integral manifold. For the Nijenhuis tensor of this almost complex structure we have

x,"

'No: = B:BbiB"hN51"= 0

260

On a structdre defined by a tensor field f of type (1,l) satisfying f3++f=0.

109

by virtue of (3.21). Thus we have Theorem 3. 1. A necessary and suficient condition f o r an f-structure to Be partially integrable is that the Nijenhuis tensor (3.18) satisfy (3.21). Equation (3.21) can be also written in the form :

(6?--n~,")(6~-m,")N,,~ =0 or

N,dh-mm,",,h + mdtN,j"i m9/m:Nl,"

=0

.

For an arbitrary tensor field f of type ( l , l ) the , tensor of the type (3.33)

H,,"

=

N,,n-nnz,tNt~'k+m,lNt,"+m,lm,~Nt,"

has been first introduced by J. Haantjes [I], so we call H,,' tensor. Then the above theorem can be restated as follows:

the Haantjes

Theorem 3.2. A necessary atid su.cient condition f o r an f-structure to be pur-tially integrable is the oanishing of the Haantjes tensor of .f. Department of Pure Mathematics, Ilniversity of Southampton.

REFERENCES . Akad. W e t . Am[I] J. Haantjes : O n S,-forrning sets of eigenvectors, Pror. K O HNed. &?erd~t71, A 58 j2) (1955), 158-162. [ Z ] A. Nijenhuis : X,-l-forming set of eigenvectors, Ihid, A 54 (2) (1951), 200-212. [ 3 ] 5 , 3a3aki : On differentiable manifolds with certain structures which are closely related to almost contact structure I, T6hok11 A h t h . Journ., 12 (1960), 459-476. [4] A. 1.; \Talker : Almost-product structure, Proceedings of the Third Symposium i11 Pir7-e it4atliettia~ii.sof the A.tiuiw'rciii Matlieitiatz'ral Society, 3 (1961), 94-100. [5] li. Yano : O n a structure .f' satisfying f3++f'=0,Tl'eclrnicnl Report, No. 12, Jiiize 20, 1961, U)ii.r.ei-sil-yof ~ I ' m l i i i g t o i ~ .

26 1

J. Math. SOC. Japan Vol. 18, No. 2, 1966

Prolongations of tensor fields and connections to tangent bundles I - General theory By Kentaro YANO*)and S h o s h i c h i KOBAYASHI**] (Received Dec. 13, 1965)

1. Introduction and notations. The purpose of this paper is to define a natural derivational mapping (called the complete lift')) of the algebra s(M)of tensor fields of a manifold iM into the algebra T(T(i1f)) of tensor fields of the tangent bundle T(1Zf)of M , to associate with each affine connection 0 of 111 a n affine connection pc (called the complete lift of 0)of T ( N ) in a natural way and to derive basic formulas and properties of the complete lift. To define the notion of complete 'lift, w e introduce also that of vertical lift and transvection a s well a s a more familiar notion of Lie derivation. T h e notions of complete l i f t and vertical lift have been already defined for tensor fields of special kinds by several authors, [5], [7], CSl, C91, C141, [15]. IJsing t h e notion of complete lift we shall show that such familiar G structures a s a pseudo-Riemannian structure, a n almost complex structure and a symplectic structure on ,\I induce similar structures on the tangent bundle T ( M ) . A n unexpected but perhaps more interesting result is that each pseudo-Riemannian (resp. affine) symmetric space structure on ,\I induces a pseudo-Riemannian (resp. affine) symmetric space structure on T(,\J). T h i s suggests us a method of producing a large class of a f i n e symmetric spaces. Let *-l be a local algebra of the form .-I=R+I where R is the field of real numbers and f is the maximal ideal of . I such that dim I < (x)and I"0 for some h. Weil has shown [lo] that such an algebra -4 defines a fibre bundle A(h1) over M, generalizing the construction of the tangent bundle T(.\f). (If dim f = 1 and 1 2 x 0 , then .l(.\I)is nothing but the tangent bundle T(.\f) of ,\I.) A successful generalization of our theory to i1(,2f) would furnish a usef u l tool for the differential geometry of higher order contact and yield a large number of a f i n e symmetric spaces. ~~~~~

.

~

* ) Supported b y NSF G r a n t GP-3990. **) Sloaii Fellow, partially supported b y NSF G r a n t GP-3?S2. 1) Perhaps, n a t u r a l l i f t '' is iiiorc appropriate. But in c u i l f u r n l i t y \\.it11 o t h e r authors, \ye use t h e term conipletc l i f t ' I

99.

262

195

Notations We shall generally follow notations and terminologies of [S] ; in particular, components of curvature tensors a r e written in the same a s in [S] including signs. We list below notations used often in this paper. 1. T ( M ) = U T,(Al) is the tangent bundle over a manifold -11 with prox .M

Similarly, T*(Al) denotes the cotangent bundle over ill. 2. T;(M) is the space of tensor fields of type ( r , s), i. e., contravariant degree r and covariant degree s, on Af. An element of Tt(Al) is a function and is denoted often by f. An element of T:l(.4f)is a vector field and is often denoted by X or 1'. An element of T~(A1) is a 1-form and is often denoted by w. 3. T(A1) = X T ; ( A f ) = T*(Af)&Ts:(Af), Lvhere T*(A1) 2 !T;(hf) and T*(Af) 7,s r = ~T:(!\f). jection

T.

S

4. An affine connection is often denoted by its covariant differentiation symbol p. 5. Given a local coordinate system s', ... , s" we denote by XI, , x'', y ' , ... , y n the local coordinate system in T(,U) induced a s follows: If 9 = C h f ( 8 / 8 x i )E T,.(Al) and x is a point with coordinates a ' , ... , a" with respect to x', ... , xn, then f has coordinates ol, ... , an, O', ... , / j 7 L with respect to x ' , ". , xn, y , ... ,?.'". 6. T h e so-called Einstein's summation convention is used. ' 1 .

2. Lie derivations and transvections. with respect to .,, Let X be a vector field on Af. 'The Lic tlcriiSaliou LAY is a linear endomorphism of T(M) characterized by the following properties :

(r.1)

L',(S5.T)--(1',S)@TT-l S g , ( r , T )

(L.2)

L,f= xj-

for

, f yli(.Z1) ~ ;

(L.3)

r.Yd/-=dL,f

for

, f Ti;(Jf) ~ ;

(L.4)

1 ' , 1 7 = [dY,I - ]

for

1 . g:,(L\f). ~

S,T E T(,\l);

for

It follows that L.Ypossesses also the following properties:

(x.5)

r.Y is type

(r.6)

L, commutes with every contraction of

preserving, i. e., &.v(is.:(!\f))~

g;(dl); ; I

tensor field.

For the preceding and other properties of d',.,see [3] and [ll]. T h e skezo s y i n n ~ c t r i ctransccrtion i.Y by X is a linear endomorphism of T((nf) characterized by the followirig properties :

263

196

K. Y A N Oa n d S. KOBAYASHI

x(S@ T )= ( f x S )@ T+(-l)*S@ ((XT) for S E T ~ ( M )and T E E T ( M ) ;

(f.1)

f

(f.2)

rxf=O

(1.3)

fxdf=Lxf=Xf

(f.4)

ixY=O

It follows that

f X

for for

f ~ s ( M ) ; for f~5”!(M): YETXM).

possesses also the following properties :

(f.5)

f x ( T ; ( M ) ) CT;-l(M), in particular, r,(Tg(M)) = 0 :

(4

1x 0 f x = 0 .

When applied to differential forms, ix is often called the interior product, [S]. From (i.1) we obtain also (1.7) ixK=O if K E z ( M ) is a symmetric covariant tensor field. The symme’tric trdizsvection ax by X is a linear endomorphism of ET(M), characterized by the following properties : (0.1)

~x(S@T)=(axS)@T+S@(o,T)

(0.2)

uxf=O

(0.3)

u x d f = L x f =X f

(a.4)

oxY=O

for for

for

S, T E T(A4):

f~s;(Ad): for

f

E

%(Ad) ;

YETA(M).

It follows that ax possesses also the following properties : (a.5)

ax(T;( M ) )c T:-,(M), in particular, a,(T;(M)) = 0 ;

(0.6)

ax o ay = avo a,.

In contrast to (f.7),we have

(a.7) axK= 0 if K E Tt(A4) is a skew-symmetric covariant tensor field. We now fix a positive integer k. Then, for s 2 k, every vector field X defines a linear mapping T, : 9; ( M )-T;-,(M) such that y,(S@w,@

..*@Wk@

”.

@ w 8 ) = s @ w , @ ...@Wk(X)@

“. @ u s ,

where S E T;(M) and w i E z ( M ) for i = 1, ... , s. In terms of a local coordinate system X I , , x n of Ad, let Kj:::;; be the components of a tensor field KET;(M) and Et the components of X. Then yxK is the tensor field of type (r, s-1) with components K $ ; : : : x . . . j s [ j k . If s = 1 so that k = l necessarily, then yx coincides with tX and Considering yx for all k , 1 5 k 5 s, it is easy to express both l X and g X by means of yx. Since ~ ~ and, T, abehave ~ in~ a similar manner, they will be denoted by ax when the distinction is not necessary. PROPOSITION 2.1. For X , Y E Tb(izf) tile have

264

Prolongatioiis of tensor B e i d s a n d coiiriectioris

JYI

197

= -fca,yi ;

(1)

CJx,

(2)

C-C’,Y, arl = a[.Y,Yl ;

(3)

-CXw= d o c . ~ ~ + c , o d w

f o r a n y differential f o r i n w

,

PROOF. Since S(iL1) is generated by f~ Tt(M), d f E T!(&f) and Z E S~(izr), it is sufficient to prove (1) and ( 2 ) when the both sides are applied to f, d f and 2. T h e verification in the three special cases are straightforward and are left to the reader. Similarly, it is sufficient to verify (3) in the cases where W = f and w = d f . For (1) and (3) we refer the reader also to [3 ; p. 32 Q. E.D. and p. 351. 3. Vertical lifts. Let .Y E T,(A/). T h e projection T : T(i\l)-pAl induces a surjective linear mapping T*:TF(T(M))+T&U), called the differential of T a t X. Its dual mapping x* : T,T(i\f)+T:(T(:\l)) is injective. Clearly, can be extended to a unique isomorphism of the covariant tensor algebra T,(x) a t x into the covariant tensor algebra T&) a t ,T. This gives rise to an isomorphism, called the vertical l i f t , of the algebra S,(M) of covariant tensor fields of ,If into the algebra T*(T(M)) of covariant tensor fields of T(iL1). For a covariant tensor field f{E S,(M), its vertical lift will be denoted by K V . T h e following is immediate from the definition. PROPOSITION 3.1. (1) f“ =f 0 T f o r f E Ti(l\f) ; (2) For K E T;(hl) considered a s a mzdfilinear iizappiizg T,(M) X ... >: TL(.\/) -+ R”, its vertical l i f t K V :T;(7‘(A4))x .-.x T;(T(M))- R” satisjies

KV(,?,,

... , .qq>= ~(x.*.?,,... , ~T,R,> f o r

X,E T ; ( T ( M ) );

(3) T h e vcrtical l i f t m a p s the algebra LD(hl) of di.rfcr-eiitia1 . ~ O T J I I S of j\f isomorphically i n t o the algebra D(T(.II)) of differential f o r m s of T(.\/). T h e vertical lift LD(Jf)-G7(T(hf)) is usually denoted by T*. To introduce the notion of vertical lift to the algebra of contravariant tensor fields, we define two linear mappings c and u of T&Lf) into T+(T(;\f)) which are similar to i X and ux, T h e mapping I : T,(,\f)+ T,(T(L!l)) is a linear mapping characterized by

((.l)/

l(S @ T)= ( r S )@ TV+(-l)QS “@(rT)

S E T:(iZl) and T E TX.(A/);

for (c.2)’

tC.3)/

rf=

0

~ ( d j=) d f

f E T;(hf) ;

for for

f E T:(Af),

where d f on the right hand side is considered a s a function on T(A1).

265

K. YASO and S. KOB.~E..-\SIII

198

T h e mapping a : T*(,\f)+T%(T(M))is a linear mapping characterized by

(0.1)’

u ( S c T ) = ( u S ) @ T Yt S Y @ ( o T )

(a.2)’

0f=0

(0.3)’

a ( d j )= [If

S,T E T,(Af);

for

f~ T;(,!f); for f~ V) ,

for

where dj on the right hand side is considered a s a function on T(AI). Later these mappings .r and u will be extended to linear mappings of T(M) into T(T(JJ)). We note that if w =fL,rlXl

in terms of a local coordinate system

XI,

I ( @ )= a(w)

... ,

of -11, then

=f,y

in terms of the induced local coordinate system X I , ... , ,I”, y’, ... , J ) of ~ ~T(i\f). As a first s t e p to extend the vertical lift to the algebra T(>\f),we define a vcrticnl l i f t X v of a vector field X of 111. It is a vector field on T ( I \ f ) characterized b y (.)

for

Xi’(!(dj))=(Xf)v

{E

T;(I1)#

In terms of a local coordinate system s’, . , r” of \f, if

then

in terms of the induced local coordinate system X I , ... , I”, y l , ... , j~12 of T(\f). T h i s proves the uniqueness a s well a s the existence of XF‘satisfying (*). By (*) the vertical l i f t T~l(j\f)-+T~~(T(!\f)) is clearly injective. It should be warned however that it is not a Lie algebra homomorphism. We extend the vertical lift T:(Af)- T;(T(‘\f)) to a unique algebra isomorphism of y*(2\f)into T*(T(,U)). By tensoring the two vertical l i f t T&f) --T,(T( If)) and Tr(‘\f)- TX(T(.If)) w e obtain an algebra isomorphism of T(,U) into s ( T ( d f ) ) ,which is called the vevtical / i f t . In resumk w e m a y s a y that the vertical l i f t is a linear mapping of 5Y.U) into T(T(.If)) characterized b y the following properties : (11.1)

(SaT)V=SVQTV

(u.2)

fv=f

(u.3) (u.4)

for fET{(.\f);

(df)v= d(fv) X v ( r ( d f ) )= ( X f ) v

for S , T = % A I f ) ;

for

for f E s!(.\f) ;

X E Tb(A\f)and f~ T:(Af) .

266

Prolongations of tensor fields and coizizections

199

We are now in position to extend c and a to linear mappings of ~ ( I M ) into s(T(A1)). They can be characterized by the following properties : r(S@ T ) = ( r S ) @ T"+(-1)4S "@ ( I T )

(r.1)"

SET;(M)

for (r.2)"

r/-=

0

r ( d f ) = df

(r.3)"

TEY(A~);

and

j~ Y:(A/f) ;

for for

f~ Tt(h/f),

where df on the right hand side is considered a s a function on T ( A f );

fX=O

(r.4)"

XEY;(A~).

for

It follows that r ( Z (Af)) c Y;-dT(Af))

(c.5)*

,

Similarly,

(a.1)"

o(S@T)= ( u S ) @ T ' + S " @ ( o T )

(02)"

af=O

(a.3)"

o(df) = d./

for

for

S,T E Y(M) ;

f~%(Af);

for

f E T'g(Al),

where df on the right hand side is considered a s a function on T(Al);

uX=O

(u.4)"

XEYXAJ).

for

It follows that a(T;((nf))c T:-l(T(M)).

(u.5)" Evidently,

(=a

on

S(Ad)

and I

As an example defined to be r I = o f , In terms of the local by a local coordinate

rs

=o=O

on

Y*(M).

we mention the caiioizical vectoy field on T ( A l ) ; it is where r ~ Y i ( A 4 is ) the field of identity endomorphisms. coordinate system x', ... , x", J ~ I , ... , 3'" of T ( M ) induced system .I-'... , , xn of Ad,

We now fix a positive integer k . Then in a similar manner a s we defined in !j 2, we define a linear mapping r : T;(Ad)-+T: l(T(Af)) for s 2 k by 7(S@U1@. ' . @ U A @.'. @oJs)=Si'@oJy@ .'. @c(w*)@

.'. @w:

where S E T;(A!) and w, E $(,\f). In terms of a local coordinate system X I , ... , S" of M and the induced local coordinate system x', , xn, yl, , y" of T ( A f ) ,

267

K. Y:xxo and S. K O B \ Y . \ S I I I

200

I f s = 1 so that 12 = 1 necessarily, then y coincides with c and 0. Considering y for all Ir, 1 ~ 1 ~ ~it . iss easy , to express both c and u by means of 7. Since I , u and y behave in a similar manner, they will be denoted by a when no distinction is necessary. 4. Formulas on vertical lifts. Throughout this section, X is a vector field on ,If and Ii is a tensor field on LU. We recall that a y (resp. a ) stands for any one of cx, ox and yx (resp. I , a and y). PROPOSITION 4.1.

(1)

(2)

L,V(K")

=0;

a.yv(IY) =0 .

PROOF. Since the vertical lift is an algebra isomorphism of ~ ( h l into ) T ( T ( M ) )and since Ir(ib1) is generated by f E T:(.bl), df E G'(h1)and 1. E st(Al), it suffices to verify the formulas above in the special cases where K = j , K = d f and K = Y. T h e verifications in these special cases are trivial if one writes X=t'(a/ax') and Xv=['((a/ay') in terms of the local coordinate system XI,... , xn, yl, ... , y n of T(i\l) induced by a local coordinate system XI, ... , .r" of Ill. Q. E. D. PROPOSITION 4.2. L 1 v ( a K ) = (a~yzT)v. PROOF. We shall prove the formula for the case a = u . position 4.1 we have

By (1) of Pro-

Since the vertical lift is an algebra homomorphism, we have (a,(S @ T ) ) V = (rr,S)" @ T " f S

@ (@ ,T)"

.

Hence it suffices to prove the formula in the special cases where K = f , K = d,f and K = Y for the same reason as in the proof of Proposition 4.1. If K = f or I<= Y , then the both sides of the formula vanish. If K = d f , we write X = EL(a/ar7)and df =f , d n l . Then ~ ( r l f =fip ) and X v = ;i(a/ayL). Hence both the left and the right hand sides are equal to Q.E. D.

fiet.

268

201

Complete lifts of tensor fields.

5.

T h e complete / z f t is a linear mapping of T(I1)into ir(T(\ I ) ) characterized by the following properties. If we denote the complete lift of ICE T( \f) by A", then

(c.1) (c

( S ~ T ) C = S C ~ T 1 ' t - S 1 ' ~ for T C S,T e s(i\l),

.a

fC=

(df)c = d ( f C )

(r.3) (c.4)

for f~ z(llf) ;

r(df) = o(tlf)

X c ( f c ) = (Sf)C

for f ' TX,\f) ~ ;

%(M)

for f~

and

A' EI T:,(Af).

I t follows t h a t

K E 2'; (if), then K C E T ;(T( 11)). In terms of the local coordinate system x', ... , x",y', ... , y" of T( 11) i n (c.5)

if

duced by a local coordinate system xl, ... , ,I'~of A{, we obtain easily fr=f1yi

where f i = a f / a r l ,

( d t j r =.~~d.t;+.f,d\*~ where

1,= a.//ail

T h e exisence and the uniqueness of the complete lift m a y be proved readily from these formulas in local coordinate system. REMARK. It is also clear from the formula above of X c ' that the set of all X", where XET,&ZI), gives the whole tangent space a t each point of T ( M ) except a t the zero points. T h e zero points of T(Al), i.e., the zero tangent vectors of 111, may be identified with ill in a natural manner. Since T(M)-Af is dense in T ( M ) , a continuous clapping of T(T(iZI)) into a vector space sends T ( T ( M ) ) into the zero element if it sends each S C into zero. T h i s remark is useful in simplifying later the proofs of certain formulas. PROPOSITION 5.1. Let X E Ti(A1) a n d KEY(!\{). Let a (resp. staiid f c r aiiy o n e of I , o aiid 7' (iwsp. cx, o.y aizd ys). Then = (L.yIL-)" ;

(1)

L.yc(KC)

(2)

L.yc(lL-I')=

(3)

L.rr(1P) = (L,yI<)r';

(4)

1'.yCO

= Lt'

(L.rlc)l-;

0

269

L.r ;

K.

202

YANO

and

s. K O B A Y A S l l l

It is sufficient to verify these formulas in the special cases where K = f ,

K = df and I ( = 1'. The verifications in the special cases a r e straightforward and a r e left to the reader. 6. Complete lifts of special tensor fields.

Let

K E T;(A!) and consider it a s

a multilinear mapping of Ti(hl)

x ...

x Th(Al) (s times) into Ti(&') under the natural identification 5';(A!)=T:(Al), @T,O(k!)= Hom (Ti, 9;).From (5) of Proposition 5.1 we obtain PROPOSITION 6.1. If ICE g ; ( M ) , tlzeiz IcC(XF,... , Xf) = ( K ( X l , ... , XJ)'

for Xi E TA(A4) .

From Proposition 6.1 and from the remark made in § 5, we m a y conclude that if Ir' E T ; ( h l ) is a symmetric (resp. skew-symmetric) multilinear mapping of Tt(iL!) x .'. x Ti,(Af) into s;(hl), then K C E s ; ( T ( M ) ) is also a symmetric (resp. skew-symmetric) multilinear mapping of Tb(T(A4))x ..+x T$J'(A4)) into T;(T(M)). In particular, if y is a p-form on M, then 'pc is a p-form on T(M). PROPOsITlON 6.2. For an)' differelitin1 f o r m 9 and 0 of A{, we h a w

(1)

(2)

( y A $J)" = p" A p,+'Pv A p ; (dro)" = &roc)

PROOF. From Proposition 6.1 it follows that the complete lift commutes with skew-symmetrization. Hence (1) follows from (c.1). (2) follows from (l), (c.3) and the fact that the vertical lift also commutes with d. Q. E.D. We shall now study the complete lift I P of a tensor field of type (0, 2) a little in detail. If we write K=

KfjdXi@CIXf

in terms of a local coordinate system 2,... , xn of Ad, then (c,l), (c.2) and (c.3) imply that

in terms of the induced local coordinate system xl, ... , x7',311, ... , 3.'" of T(Af). If we express K by an ( n X 7 1 ) matrix (I&), then K C may be expressed by a (277 x 212) matrix :

From this w e may derive a number of properties of KC.

270

Prolongations of' tensor fields and connections

203

PROPOSITION 6.3. Let g be a p s e u d o - R i e m a n n i a n metric o n M. T h e n g c i s a pseiido-Riernannia?i iizetric o n T ( M ) (with n positive a n d n negative signs). PROOF. If we take a normal coordinate system, then a t the origin ag,j/axk vanish. Our statement is now obvious f r o n the m a t r i x expression for Kc' given above. Q. E. D. REMARK. If we write ds2=gL,dx1d.d for g, then g c m a y be written 2gt,6y'dxJ, where Sy' = d y L + c k d x J y k . PROPOSITION 6.4. If p i s a 2-forin d e f i n i n g a n (alrizost) synzplectic ture o n -\I,then pc defines a n ( a l m o s t ) syinplectic structure o n T ( M ) . PROOF. From the expression above for K C , i t is clear that if 'p maximal rank, so is pc. If 'p is closed, so is pc by Proposition 6.2. We shall now study tensor fields of type (1, 1). PROPOSITION 6.5. L e t -4, B E T&U) aiid consider tlzein a s f i e l d s of endoliiorphisiizs of tangent spaces of ,\.I. Let I be the j e l d of identity f o r m a t i o n s of taiigeiit spaces of .\I. Tlieii ('4

0

struc-

is of

linear traizs-

B)c= -4' 0 B C ,

Ic = tlie field of identity aiitornorpliisiris o f tarigciit spaces of T ( M ) . I n particular, if P is a polynoinial of one variable, tlzeiz

P(#)

= (P(A))C.

PROOF. By Proposition 6.1, we have (-40 B ) C ( X C ) = ((A 0 B)(X))C = (--1(B(X)))C= A"(BC(XC)) = ( A C 0 BC)(X?

By the remark made in Q 5 we may conclude that (A a IC(XC)

.

= ACoBc. Similarly,

=( I ( X ) ) C = X C,

and by the same remark we m a y conclude that Ic is the field of identity automorphisms of tangent spaces of T ( M ) . Q. E. D. Let -1, B E st(JI). Then the torsion t A , B ( X ,Y ) of A and B is a tensor field of type (1, 2) defined by (cf. [3])"

t&Y,

Y ) = [ ; l X , B Y ] + [ B X , AYI+zlBCX, Y I S B A C X , Y ] -,-l[X, BY]--.I[BX,

Y]-B[X,

A4Y]-5[AX, Y ] for X , Y E S;,(Lf) .

From the definition of the torsion, Propsitions 6.1 and 6.5 and (1) of Proposition 5.1, we obtain PROPOSITION 6.6. L e t *i,U t T;(,\f). Tlieii 2) T h e notion of torsion f . 4 , ~ ( S , U ) is due t o Nijenhuis although o n l y t h e special case + ~ J , J ( -YY ,) ( w h e r e J is a n almost complex s t r u c t u r e ) is widely known as t h e Nijenhuis tensor. See, A. Nijenhuis, ,Y,-,-forming s e t s of eigenvectors, Indag. Math., 13 (1951), 200-212.

27 1

204

K. Y A N O and S. KOUAYASIII (t",LJC = t,4c,B::

From Propositions 6.5 and 6.6 we obtain PROPOSITION 6.7. If J E S;(M)d c j i n e s ati almost coirip1c.l- structiire 01%XI, 1 so does J c o n T(i\J). 1 f l v J = tJ,J denotes the Nijcnhiiis teizsor of J , (Nj)' IS the ivijciiliziis tciisor o f J c : (iV,)" = iVJc. If w e write Il E $ ( J J )

in terms of a local coordinate system

XI,

... , x" of ,\I,

then (c.1-4) imply that

, y" of T(iL1). in terms of t h e induced local coordinate system 3c1, ... , s",yl, !f we express A by a n ( 1 1 x I ! ) matrix (--l;), then /Ic m a y be expressed by a (2ri x 211) matrix :

It is clear from this expression for 11" that if A is of rank r a t each point of .\I, then 11" is of rank 2r a t each point of T(A1). As another example we mention a tensor field F E $(!\I) satisfying F3+F = O ; such a structure has been studied in [lZ], [13]. Then FC E Tt(T(iZ1)) satisfies (Fc)3++r;c = 0. From (1) and (3) of Proposition 5.1 we obtain PROPOSITION 6.8. If K is (1 tetisor jicld o n M urid X is a vector j c l d 011 AI sutislyiiig LAr1i= 0, therz

-C.,c(K") = 0

Ulid

L ~ Y V ( 1 P= )

0.

In particular, we have PROPOSITION 6.9. If X is n Killirig vector j e l d of u pseiido-Ricnianiiiaii riiaiiijoltl 121 zuilli irietric g , theri both X " uiid X v are Iiilliiig vector fields o f t h e pscudo-Riertinn,Iiall ~~iunif'old T(,\l) ioith rrielvic 6". Going back to a general vector field X on Al, we shall consider X c a little more geometrically. Let exp t X be a local 1-parameter group of transformations of it1 generated by X. From the coordinate expression of X c it is not hard to see that the induced local 1-parameter group (exp t X ) , of transformations of T ( M ) (where (exp t X ) , denotes the differential of exp t X for each fixed t ) coincides with a local 1-parameter group exp t S C of transformations of T ( M ) . Hence we have

272

Pr oloiigai ioii s of t r ii sor ./ie/ds

a 11 ti coii ii r c l i 011 s

205

PROPOSITION 6.10. I f a i w t o i - Jiclti A' oil Al is coiiij)lcir i i i t h e s c i i s e that it g e i i c r a f e s a globnl 1-pal-niiieter groiip o f Ii.oiisfoi.i!intioii.~ o f Al, f l i c i i S c is also n coiiiplcle i'ccioi- ,Geld o i i T(.ZI). R E M A R K . From the coordinate expression for ST.we see immediately that is complete whether A' is complete or not. From Propositions 6.9 and 6.10 and the remark above we obtain PROPOSITION 6.11. If A l is a hoiiiogeiieoiis psciido-Ririiiciiiiiinll iiiniiifold uiih iiictvic g, so is T(A4) with iiwfvic g r . Similarly we have PROPOSITION 6.12. If A l is N homogriiroi/s ( ~ l l i i i o s i )coiiiplcs iizaiiifhld uifh (aliiiosi) coiiiples s f n i c t i r r c 1,so is T ( A l ) iLii/li ( c i i i / i o s i ) coiiiplrs sfi-ireiiii.c 1 '.

7. Complete lifts of affine connections. Let j- be the covariant differentiation defined b p an affine connection of Ad. Then there exists a unique a f i n e connection of T(i\l) whose covariant differentiation satisfies

rc

j-f;c( Yc)= (j-.yl.)r

for S,1- E Y;(Af)

.

Our assertion m a y be verified by a simple calculation using connection components. Let be the connection components for r with respect to a local coordinate system .TI, . . . , a?'. With respect to the induced local coordinate system .rl, ... , x'', y ' , ... , y " of T ( A f ) ,we set

r:,

1''

jP

=Ti,' .1k

qczo,

I:& =o. Jk

p:.xov Ik

where the indices with bars refer to y', , J . Then the i"s a r e the connection components for PROPOSITION 7.1. //* T a n d R ni'c /lie f o r . > i o i i a n d i h c c u i . i t o i i i i ~ c tensor fields f o r r, ihcii T r n i i d Rr a i t~h e ~t o i w o i i a i i d tlir cirrzvtirre i o i i s o i . ]ielcls for qn

rr.

rc. PROOF. Our proposition follows from the following formulas :

T C ( X r ,IYC)=(T(.X,).))r~(r\~~-r,.,Y-[.Y, - j-Tf, J'r-j-T.rAYc-['YC, I"] . RC'(*YC,J-r)Zc= ( R ( X , 172)('= ( [ r \ j-,.]Z-rc,,,.,Z)c , = [r'ir,r C r l 7 r - r f \ r , I . ~ l Z f. PROPOSITION 7.2.

For

aiiy

tciisor ,field /i

Af

011

Ad, we hniic

(1)

j- Cc(Kr)=

(r,

273

;

oiid a i i y i w f o i '

Q. E. D. licld X on

K. YANOand S. KOBAYASHI

206

PROOF. A s usual it suffices to verify these formulas in the special cases where K = f = T!(M), I(= dffr T : ( M ) and K = Y E TA(M). ) ~I<= . I-, (1). If K = f , then j7;c(fC) = X c f c = X C , c j c= (L,f)" = ( ~ ~ f If then the formula to be verified is nothing but the definition of gc. If K = d f or more generally I<= w E $(M), then (j7$CcwC)(I'")

= pPc(w( I'))"-o"((g'sI-)c)

= pEc(w"( I'C,)-wC(j7$cI'C)

I')Y

= (rs(w(Y)))"-(w(P.r = ( F . y w ) C ( 1'")

= ((rsw)(l'))c

.

Hence, r.ycwc = (g,w)".

( 2 ) . T h i s follows from (1) and Proposition 5.1 a s follows: r . y r ( r C K c )= r.:c(Kc)

( B . ~ K=)(~T , ~ K = ) ~r , y c ( ~ I < ) ~

1

and hence

y C K r= ( V K ) ~ . (3). If K = f , then by Proposition 5.1 we have

r.Fc(f')

=L , C ( f " )

=( L ' x f ) V = (P.yf)V.

4f I<= Y , then write X=i;"(d/axi) and Y=vi(i3/dxi). Then

.

= (g.yl-)"

Either by a similar calculation using a local coordinate system o r b y a calculation similar to the one in the proof of (l),we obtain r.yC(d')

= (r.rw)v

for every w

E T,'(AI).

(4). T h e proof is similar to that for (2). (5). T h i s may be proved in the same way a s (3). Or (7) of Proposition 5.1 and (2) m a y be used as follows:

274

g,@(IP)= ygV(gCIP) = ysv((j7K)C) = (y.yVI<)v = (p.yK)Y . Q. E. D. (6). T h e proof is similar to that of (5). PROPOSITION 7.3. Let d l be a riianifold iuitii aijiize coiinectiuiz p. Then a Jacobi field alorig a geodesic o j 151,considered a s a czirue in T(.\I), is a geodesic of T ( M ) w i t h vespect to the ajjine coniiection uiad vice cersu. PROOF. From the coordinate expression for the equation of a geodesic, we know that the geodesics of 111 (resp. (T(.U))a r e determined by the symWe may therefore metric part of the connection components of r (resp. j?). assume without loss of generality that r (and hence also re)is torsionfree. Let XI, ... , ,P, y ' , ..+,y'l be the local coordinate system in T(.U) induced b y a local coordinate system s', ... , x'~of &\I. Then a geodesic of T ( L U )is given b y the following set of equations.

re,

From (1) We see that a geodesic of T(.\l) projects upon a geodesic of .\f. We transform ( 2 ) a s follows.

(3)

If we denote by

dt

the covariant differentiation in t , then (3) m a y be writ-

ten a s follows.

T h i s shows that y i ( t ) is a Jacobi field along the geodesic x'(t) of >\l. Q.E.D. As a n immediate consequence, we have PROPOSITION 7.4. I f .\f is complete i~litiirespect to nii a.ijiiie coiiiiectioii r , tlien T(d1) is coiiiplete icitli respect to ,?j a n d i>ice versa. T h e following result relates the complete lift of a n affine connection with Proposition 6.3. PROPoSITION 7.5. f f is t h c Rieiiintiiziaii cotziu?ctiori u/ .\I iuitli ~ e s p e c tt o a pseiido-Riei~iaiiiiiaiimetric g, theit pc is t h e Rieniniiiziaii coriizection o/ T(,Lf) iuith respect to the pseudo-Rieniaiiiiinii metric g c . PROOF. Since the Riemannian connection is a unique torsionfree connection f o r which the metric is parallel, our proposition follows from Proposition

215

Q. E. D. 7.1 and (2) of Proposition 7.2 applied to K = g . PROPOSITION 7.6. Let p be a n ajiize connection on M . I/ X i s an injinitesiiiznl afline transforinntion of M, theii both X c a n d X v are infinitesimal a j i i i e traizsforiiiatioiis oj’ T ( M ) ioitlz respect to rc. PROOF. A necessary and sufficient condition for X to be a n infinitesimal affine transformation of h.l is that l‘,? 0 rr-pr

= pcs,u,

0

for every

I’ E s:(M).

Making use of Propositions 5.1 and 7.2 we verify easily

or = lTr’.From in the following special cases: .?= XC or = XV and .I’.= the coordinate expressions for 1 - C and I.’’’ we see that the formula above is valid for a n arbitrary .I’.. This proves that both X C and X ‘ are infinitesimal affine transformations of pc. Q. E. D. From Propositions 6.10 and 7.6 we ‘obtain PROPOSITION 7.7. I f t h e g r o u p o f a j i n e traiisformations o j A1 ulitli p is trtiiisitivc 011 Af, t h e i i f i l e gl-oiip of a,ljiiie trciizsfol-inations of T(,Zf)w i t h i.espect to p c i s tran.sitiz3c 011 T(A4). From Propositions 7.1 and 7.2 we obtain PROPOSITIOS 7.8. Let T aiid R lie t h e torsion aiid t h e ciii.i,atiire tensor $elds o j a n a j i i i e coizizectioii p o f A l . Accordiiig a s T = 0 , p T = 0 , R=O or p R = 0 , w e h a v e TC=O, r C T C = QRe= , 0 01’ rCRC=O. I n particrilar, i f hl is locally ajjiize symiiietric w i t h yespeci to p, so i s T ( M ) w i t h respect t o rC. From Propositions 7.5 and 7.8 we obtain Pi
If A f is of constant curvature h , then the curvature RC of gc m a y be calculated a s follows.

RC(.XC, Y c ) Z c= ( R ( X , J-)Z)c= f<(g(Z,I’)X-g(Z, S)l’)’

“+( g ( 2 , Y ) ) V X C - ( g ( 2 ,X))G‘I.’v- (,g(%, x +g ’ ( 2c, I’ XC-gC(ZC, XC) I’ “-‘Y

= k(( g(2, I*))CX = I?( g C ( Z c, I

~

C)

C)

V

Define a tensor field E of type (1,1) on T(A2) by

276

S))“ I.’

”(ZC,

s C) Y . C)

Prolongations of teizsoi- f i e l d s aird coiiricctiorzs

209

in terms of the local coordinate system s ' , ... , .Y", yl, ... , y J Linduced by a local coordinate system x', ... , ,yn of Xf. ( E m a y be defined without local coordinate systems.) Then EYC= S v and g'(*, E a ) = g C ( a , a). Hence

--g"(Z", XC)EYC--g"(Z, E.Yrj1.C) . Using this formula it is not hard to prove that the complete lift gCof a metric & of constant curvature k has constant curvature if and only if k = 0 . Similarly, the complete lift g c of a n Einstein metric g is again a n Einstein metric if and only if g has vanishing Ricci tensor. T h e operations such a s Is, a , vertical lift and complete lift a r e all natural in the sense that they commute with a n y diffeomorphism of one manifold onto another. In particular, if h is a transformation of Ad leaving a connection y invariant, then its differential h , : T(,Z/)-T(Af) leaves the connection pc invariant. Similarly, if 12 leaves a tensor field K on i1.f invariant, then 17, leaves K C invariant. Applying this reasoning to a pseudo-Riemannian (or affine) symmetric space ,If and its symmetries, we obtain PROPOSITION 7.10. I i A f is n pseudo-Rieiiiaiziziciii (resp. a j i n e ) synziiielric spncc zcith metric g (resp. coiziiectioii p), f h e i i T(.\l) is also ci pseudo-Rieiizniiiziniz (rcsp. a-ljine) syirlnietric space wifh metric g C (resp. coriizcctioiz ~ c ) . R E ~ I A R K .T h i s proposition is related to the following known result, [S]. If G / I f is a reducible a f i n e symmetric space with G simple, then it is a tangent bundle over a compact Riemannian symmetric space. See also 1111, 141 for fibrations of affine symmetric spaces. I\'e shall make a concluding remark about generalizations of results in this paper. One possible generalization is, as we have already pointed out in the introduction, to lift tensor fields and connections to the bundle A(:\f) defined by a local algebra .A. Another generalization is to lift a wider class of geometric structures of .If to the tangent bundle T(.\L). Fop instance, would i t be possible to associate with each C-structure on .\f a naturally induced G'-structure on T(.\f), where C' is a certain subgroup of GL(2/1; R ) ? \Ye have seen that a pseudo-Riemannian structure on .\f gives rise to a pseudo-Riemannian structure on T(.\Ij. lIoreaver, T(\f) carries a n obvious iz-dimensional distribution. T h i s means that for G = O(ii), a correct G' m a y not be SU(iz, iz) but a certain subgroup of SU(iz, iz). Similar comments may apply to almost complex structures, symplectic structures, etc.. It seems Lhat a conformal structure on .\I does not immediately induce a conformal struct u r e on T(.lL). T h e question of associating a correct G'-structure on T ( M ) to a G-structure on .\l is probably related to the follo\ving question on holorelated to the holonomy nomy: how is the holonomy group of T(.\L) tvith

rr

277

210

K. Y A N Oand S. KOU.AYASIII

of hJ with 0. Another related question is this. In [a] it was shown that if P is a principal fibre bundle over h J with group G and if Z‘ is a connection in P, then there exists a naturally induced connection T ( r ) in the bundle T ( P ) over 7YU) with group T(G) and the restricted holonomy group @ ( T ( r ) ) of T ( r ) is isomorphic in a natural w a y with T ( O o ( r ) )where , @O(Z’) is the restricted holonomy group of (The last statement is essentially the socalled Ilolonomy Theorem of Ambrose-Singer.) If P is the bundle L(.\I) of linear frames of .\I, then the connection T(r)seems to be related to the complete lift of I’.

r.

Tokyo Institute of Technology and University of California, Berkeley

Bibliography M. Berger, Les espaces sy m6triqucs noii compacts, A n n . Sci. Ecole Norm., Sup., (3) 74 (1957), 85-177. S. Kobayashi, T h e o r y of conncctioiis, Ann. M a t . P u r a Appl., 43 (1957), 119-194. S. Iiobayashi a n d K. Nomizu, Voundations of differential geometry, Interscience T r a c t No. 15 (1963). S.Koh, On a f i n e s y m m e t r i c spaces, ’Trans. Amer. Math. Soc., 119 (1965), 291309. A. J. Ledger a n d li. Yano, T h c t a n g e n t bundle of a locally syiiinictric space, J. London Math. Soc., 40 (19G3), 487-492. T. Nagano, Transioriiiatioii groups on s y m m e t r i c spaces, t o appear i n T r a n s . Amer. M a t h . Soc.. S. Sasaki, On t h e differential geometry of t a n g e n t bundles of Riemannian manifolds, TGlioku Math. J., 10 (1958), 338-354. S. Tanno, A n almost complex structure of t h e t a n g e n t bundle of a n almost contact manifold, Tdhoku Math. J., 17 (1965), 7-15, P. Tondeur, S t r u c t u r e presque kiihlbrienne iiaturelle stir la librb dcs vccteurs covariants d’une varibth riemannienne, C. R. Acad. Sci. Paris, 254 (19G?), 407-408. A. Weil, Thkorie des points proches sur les varibtks diffbrentiables, Colloque d e g ho in 6 t r i e d i f i &rc i i t i el 1e, S t ras bo u rg, ( 1933), 111-117. I<. Yano, The theory o f Lie dcrivativcs and its applications, Norih Holland Pub. Co., iir11sterdan1, 1937. K Yano, On a s t r u c t u r e f s a t i s f y i n g f:’+/=O, Technical Report, University o f VV’ash i n g t on, 1961. I<. Yano, On a s t r u c t u r e defined b y a tensor field f of type (1, 1) satisfying ,iJ+f=O, Tensor, 14 (1963), 99-109. 1i. Yano and 15.T. Davies, 011the t a n g e n t bundle of 1:insler a n d Rieinannian manifolds, Rcnd. Circ. M a t . Palcrmo, 12 (1963). 211-228. K. Yano a n d A. J. Ledger, Lincar connections on tangent bundles, J. London hlath. Soc., 39 (l%iI), 193-300.

278

Nomzu, K., and K. YANO Math. Zeitschr. 97, 29-37 (1967)

Some Results Related to the Equivalence Problem in Riemannian Geometry * KATSUMNOMIZU and KENTARO YANO Received October 2, 1965

A classical result on the equivalence problem in Riemannian geometry is the following. Let M and M ' be two analytic Riemannian manifolds with metrics g and g', respectively. If F is a linear isomorphism of the tangent space T,(M) onto the tangent space T,#(M')which maps g, into g$ and (V" R),into (V'"R'),, for all m = 0, 1,2, ... ,then there exists an isometry f of a neighborhood U of x onto a neighborhood U' of x' which induces the given mapping F as the differential at x. Here V"R denotes the m-th covariant differential of the curvature tensor field R and (P'"R), its value at a point x of M , an similarly for V'"R' and (V'"R'),,. A global conclusion can be drawn if M and M' are both simply connected and complete. In this case, given F as above, there is an isometry f of M onto Ad' which induces F. (For these results, see [ I , Chapter 61.) The purpose of the present paper is to prove a different version of the equivalence problem and other related results. We assume that a certain transformation f of M onto M' is given and that f maps the tensor field V"R into V'"R' for all m ; then we conclude that the given mapping f itself is indeed homothetic. (Iff is furthermore isometric at one point of M , then f is an isometry.) More precisely, we shall prove Theorem A. Let M and M ' be both irreducible and analytic Riemannian manifolds. I f a diffeomorphism f of M onto M' maps P"R into V'"R'for allm, then f is a homothetic transformation of M onto M ' . Corollary. Let M be a complete, irreducible, analytic Riemannian manifold. Then a strongly curvature-preserving transformation (that is, a transformation wlricli preserves V"R f o r all m ) is an isometry of M . By irreducible, we mean that the restricted homogeneous holonomy group is irreducible and non-trivia!. The corollary is a consequence of Theorem A and a theorem of KOBAYASHI (cf. [ I , Theorem 3.6, p. 2421). We also note that the infinitesimal version of the corollary was proved in [ 2 ] . The following result related to Theorem A is of independent interest.

Theorem B. Let M be an irreducible, locally symmetric Riemmanian manvold of dimension 2 3. Iff is a diffeomorphism of M onto another Riemannian mani-

* This paper was prepared under partial support by NSF grants G24026 and GP-4251.

279

30

K. NOMIZU and K.YANO:

fold M' which maps R into R ' , then f is a homothetic transformation (and, as a result, M' is also locally symmetric). Corollary. Let M be an irreducible, locally symmetric Riemannian manifold of dimension 23 with metric g. I f g ' is a Riemannian metric on M with the same curvature tensor field R as g, then g' = c g, where c > 0 is a constant. Theorem B was proved earlier by the first author and OZEKI(unpublished). The corollary shows that for a differentiable manifold M of dimension 2 3, an irreducible, locally symmetric Riemannian metric, if it exists, is essentially unique within the class of all Riemannian metrics with the same curvature tensor field. The proof of Theorem A is based on the following theorem, where no analyticity assumptions are made.

Theorem 1. Let M be n Riemannian manifold with metric g, where dim M Z 3. I f a conformal change g' = p 2 g of the metric preserves R and VR, then either R=O or p is a constant. The 2-dimensional case of Theorem A will be handled separately. Theorem 1 is to be compared with its infinitesimal version given in [2, Theorem 21. The proof of Theorem B is based on the following result, although a more direct proof could be given.

Theorem 2. Let M be a locally symmetric Riemannian manifold with metric g, where dim M Z 3. r f a conformal change g' = p 2 g of the metric does not change R, then either R=O or p is a constant.

1. Preliminaries For the terminology and notations, we generally follow [ I ] .On a differentiable manifold M (which we assume to be connected in all cases), let g and g' be two Riemannian metrics on M and let V and V' be the corresponding Riemannian connections. We define a tensor field K of type (1, 2) by K(X)=

- Vx ,

where X is an arbitrary vector field. Indeed, for any A', the derivation V i - Vx of the algebra of tensor fields on M is actually induced by a certain tensor field K ( X ) of type (1,l) (cf. [ I , Proposition 2.9, p. 1241). Moreover, K ( f X)= f K ( X ) for any differentiable functionf, which shows that K defines a tensor field of type (1,2) which associates to X the tensor field K ( X ) of type (1,l). We have K ( X ) Y = K ( Y ) X . For the metric g, we shall denote by oxthe 1-form which corresponds to a vector field X , namely, w x ( Y ) = g ( X , Y ) for any vector field Y. For a vector field X and for a 1-form n, X @ n will denote the tensor field of type (1,l) which maps a vector field Y into n ( Y ) A'.

280

Some Results Related to the Equivalence Problem in Riemannian Geometry

31

We shall prove a few lemmas.

Lemma 1. Let K ( X ) = V i - Vx, where V and V' are two linear connections. I f R = R ' and V R = V ' R ' , then K ( X ) R=O f o r any vectorfieldX. (Here and in thefollowing, R and R' denote the curvature tensorfields of Vand V', respectively.) Proof. This is immediate from

v; R'=(Vx + K ( X ) )R = vx R + K ( X ) R = v; R ' + K ( X ) R

*

Lemma 2. Let g'=p' g , where p is a differentiable function > O (that is, g and g' are conformal). Let K ( X ) = V i - Vx . Then (1)

K(X)=a(X)I+XOa-UOw,,

where I is the identity transformation, a is the l-form d(log p ) , and U is the oector field such that mu =a. Proof. From Vx g = 0 and V i g' =0 we have

p x +K ( X ) }

*

(P' g) =o

and hence K(X)g= - 2 W ) g ,

where a = d log p . This means g ( K ( X ) y, z )f g( K ( X ) z, y ) = 2 4 X ) g( y, Z )

for any vector fields X , Y , and Z. We have also and

g ( K ( Y )x,z)+ g ( K ( Y )z, x )= 2a( Y ) g ( X , Z ) -g(K(ZjX, Y ) - g ( W ) Y J > = - 2 a ( Z ) g ( X , Y ) .

Adding these three equations and noting K ( X ) Y= K ( Y ) X , we obtain g ( K ( X ) r, z)= 4x1 g(Y, Z ) + @ ( Y g(X, ) Z ) - a(Z>g(X,Y ) .

Since 4 Z ) g ( X , Y )= g(U, Z ) O x ( Y ) = g ( w x ( Y ) u,z )3

we have g ( K ( X ) y, z) = g ( G ) 1:

z )+ g(a ( Y )x ,z)- g ( w x ( Y 1 u,z ).

This being valid for any vector fields Y and Z , we have K ( X ) = a ( X )I +

x0

c(-

u 0 w x,

Lemma 3. Ifg'=p' g as in Lemma 2, then the curvature tensor field R and R' of g nnd g' are related by R ' ( X , Y ) Z = R ( X , Y ) Z - / ? ( Y, Z ) X + p ( X , Z ) Y(2)

-

g ( Y , Z ) B ( X ) + g ( X , z>B ( Y )

281

9

32

K. NOMIZU and K. YANO:

where p is a symmetric bilinear form defined by

P ( r , Z ) =(Vy a>( Z )- a( Y )a(Z>+4 .(u) g ( r , Z )

(3)

and B is the linear transformation associated to (4)

p by

g(B(X), Y)=P(X, Y ) .

Proqf. We have for any vector fields X , Y, and Z

v.; v; z =(vx +K ( X ) ) (vy+K ( Y ) )2 =( Vx + K ( X ) )(Vyz + a( Y )2 + a ( Z ) Y- g (y, Z ) u)

by (1). Thus

v.; v; z = Vx v y z +(VX a ) ( Y )z+ a(Vx Y )z +a( Y )vx z + + ( V x a ) ( Z ) Y+a(VxZ>Y+a(Z) VX Y -g(~xr,~>~-g(~,~xZ>~-g(y,Z)~x~+

+a(X)V y z + a ( v y z ) x - g ( x , V y Z ) u+

+ a( Y ) { a( X )z+ a ( Z )x - g ( X ,Z) U } + + a ( Z ) { a ( X )Y + a ( Y ) X - g ( X , Y ) U } - g ( r , Z ) ( a ( X ) U + a ( U ) X - g ( X , U)U } . Alternating this in X and Y and using a similar equation for V/x, y l , we obtain

Using p and B defined by (3) and (4), we obtain (2), because B ( X ) = - g ( X , U ) u+vx U + + a ( U ) X , as can be easily verified.

Lemma 4. Let n=dim M 2 3 . If a conformal change of the Riemannian metric g' = p2 g does not change the curvature tensor field, then the symmetric form P in ( 3 ) is identically 0. Proof, By Lemma 3 we have

282

Some Results Related to the Equivalence Problem in Riemannian Geometry

33

where X,Y, and 2 are arbitrary vector fields. Fixing Y and Z , let S be the tensor field of type (1,l) which maps X upon the left-hand side of (5). Using a general fact that, for any 1-form cp, the trace of the linear mapping X+rp(X) Y is equal to cp ( Y ) , we see that the traces of the linear mappings : X +p ( X , Z ) Y and X + g ( X , Z ) B ( Y ) are equal to p ( Y, 2)and g ( B ( Y ) ,Z),respectively. In fact, p ( Y , Z ) = g ( B ( Y ) ,Z ) by (4). Thus the trace of S i s equal to n p ( Y , Z ) + g ( Y , 2) x trace B - 2 P ( Y , Z). Hence we obtain

+

( n - 2) p( Y, Z ) g (Y, Z ) trace B = 0 ,

(6)

which can be written as ( n - 2 ) g ( B ( Y ) ,z)+g((traceB) Y , z ) = O .

Since Z is arbitrary, we have (it

- 2) B ( Y )+ (trace B ) Y= 0.

Taking the trace of the linear mapping which maps Y upon the left-hand side, we find 2 ( n - 1)(trace B ) = 0 , that is, trace B=O. By (6), we have ( n -2 ) p ( 1: Z ) = 0 .

Since n 2 3, we obtain

p( Y , Z )= 0, where Y and

Lemma 5. I f p= 0 in (3) and thefunction p is a constant).

2 are arbitrary.

if a vanishes at a point, then a =0 on M

(and

Proof. Since the set of zero points of a is closed, it is sufficient to show that it is also open (note that M is connected). Let a=O at a point X ~ E and M let 2, .. . , x" be local coordinates with origin x, . For any points with coordinates (a', ...,a")yconsider the curve x ( t ) = ( r a ' , ... tan)and let Y be the family of tangent vectors of x ( t ) . Then the equation (Vy

a>

(a-a ( Y > + (+I.(U)

g(Y, Z) =0

becomes a system of ordinary differential equations for the components ai(x(t)) of the form a along x ( t ) . By the uniqueness of the solution, we see that avanishes along x(t). Thus a vanishes in a neighborhood of xoyproving our assertion. Lemma 6. Let dim M>=3. If a conformal change g' = pz g of the Riemannian metric does not change the curvature tensor field, then we have

R ( X , Y)a=O J'or all vector fields X and Y, where cc=d(log p ) . Proof. By Lemma 4, we have (7) 3

VY a=

a ( Y )a-

(4)

Math. Z., Bd. 97

283

4U Y

K.NOMIZU and K. YANO:

34

2. Proof of Theorem 1

In Lemma 2, take X = U in (1). We have K(U)=a(U)I+Uoa-Uoc=a(U)Z,

so that K ( U ) * R = a( U ) ( I * R )= -2a( V )R ,

where I . R means that the identity transformation I operates on R as a derivation. By Lemma 1, we have a( U )R =O . If a( U )=g(a, a) = 0 at a point, then by Lemmas 3 and 4, we see a= 0 on M . This means that p is a constant. If a ( U ) = g ( a , a) is not 0 at any point, then R=O at every point. This concludes the proof.

284

Some Results Related to the Equivalence Problem in Riemannian Geometry

35

3. Proof of Theorem 2 By Lemma 6, we have R ( Y, Z ) a = 0 for any vector fields Y and 2. Taking

VX under our assumption Vx R=O, we have

R ( y, Z ) ( V X a) = 0.

By Lemma 4,we have

p=O, that is, (7). Hence

R(l-, Z ) (VX 4 = @ ( X IR(Y, Z )

4 3 ) da,a ) R (Y,Z ) w,y

= -(*) g(a, @ ) O R (Y,Z) X

Y

again using R ( Y, 2)a =O. Thus we get w R (Y. Z) X = O *

If a=O at a point, then ct=O on M and hence p is a constant as before. If 01 never vanishes, then g(a, a)+O and R ( Y , Z ) X=O at every point. Since X , Y , and 2 are arbitrary, we have R=O.

4. Proof of Theorem B Let g" be the metric on M which is the transform of the metric g' on M' by the transformation f . It is sufficient to prove that g" is conformal to g: g " = p 2 g; indeed, the curvature tensor of g" is equal to the curvature tensor R of g and hence, by Theorem 2, either R=O or p is a constant. Since M with metric g is irreducible by assumption, we exclude the case where R=O. Thus p is a constant, that is, f is homothetic. In order to show that g" is conformai to g, consider the holonomy algebra h, of g at a point. Since g is locally symmetric (VR=O), h, is generated by all linear transformations R(X, Y ) , where X,Y are tangent vectors at x. On the other hand, the holonomy algebra hi of the metric g" is generated by R ( X , Y ) and other linear transformations that arise from the covariant derivatives of R (with respect to g") of all orders. (cf. [ I , Section 9, Chapter 1111). Thus h, is contained in h:. The metric tensor g! at x, which is invariant by h i , is therefore invariant by h,. Since the restricted homogeneous holonomy group Y, of g (of which h, is the Lie algebra) is irreducible, we see that gl is a positive scalar multiple of g,. Since this is the case at each point x, we see that g l = p z g,, where p>O is a function.

5. Proof of Theorem A We first prove Theorem A for the case where dim M 2 3 by using Theorem 1. Let g" =f * g' be the metric on M which is obtained from g' by the given diffeomorphism f . As in the case of Theorem B, it is sufficient to prove that g" is conformal to g. By assumption on f , we know that V"R= V""R" for all m, where V''"R" denotes the m-th covariant differential of the curvature tensor R" of g". Since g is analytic, the holonomy algebra at X E Mis spanned by all 3'

285

K. NOMIZU and K.YANO:

36

linear transformations of the tangent space of the form (V”R) ( X , Y ; V 1; ... ; V,), where X , Y, V , , ... , V , are tangent vectors at x (cf. [Z, Section 9, Chapter 1111). Each of these transformations is contained in the Lie algebra of the infinitesimal holonomy group of g”, which in turn is contained in the restricted holonomy group of g”. Thus the restricted homogeneous holonomy group Y (x) of g is contained in that of g”. The metric g: is therefore invariant by Y (x), which is irreducible by assumption. Hence g: is a scalar multiple of g,. This being the case for each point x, we see that g” is conformal to g . We shall now prove Theorem A for the case where dim M = 2 . As above, we may assume thatf is conformal. Since dim M=dim M ’ = 2 , we may write the Ricci tensors S and S‘ of M and M‘ in the form S = A g and S’=A’g’, where 1 and A‘ are functions on M and M ’ , respectively, which are not identically zero (if I is identically 0, then R will be 0, and similarly for A’). Sincef maps R upon R‘, it maps S upon S‘. This means that for every point X E M we , have A(x>=A‘(f(X>)p2(X),

where p is a function such thatf* g = p 2 g. Thus we have

A= (A’0 f)p z .

(8)

By assumption, f maps V R upn V’R‘. Thus f maps V S upon V‘S’, This implies that for any X E Mand for any tangent vectors X , Y , and Z at x, we have (VX

S ) (Y,

a=v;( X )S’) (f Y,fZ>.

Since ( V x S ) ( Y ,Z ) = ( X I ) g ( Y , Z ) and similarly for V’S’, we have Thus we have

( X 4g ( r, Z ) = (f(N 2’)p 2 ( X I g( Y, Z ) .

x i =(f(X)A’)p 2 .

(9)

From (8), we have

x I = X(A’ o f ) p 2 +(A’ of) 2 p x p =(f(X)A’)p2+2(A’of)p. xp. Comparing this with (9), we obtain that is,

If (A’of)+O at a point xo of M , then in a certain neighborhood U of x o , we have A‘o f =+O and hence

xp=o,

286

Some Results Related to the Equivalence Problem in Riemannian Geometry

37

where Xis an arbitrary tangent vector at XE U.Thus p is a constant in a neighborhood of xo, and, by analyticity, on the whole manifold M . If I ' o f = O at every X E M ,then I'=O on M ' , contrary to the assumption that R' is not 0. This proves Theorem A. Added in proof: An abstract of this paper, together with comments and a direct proof of Theorem B, has appeared under the same title in the Proceedings of the United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965 (pp. 95- 100).

Bibliography [I] KOBAYASHI, S., and K. NOMIZU: Foundations of differential geometry, vol. I. Interscience Tracts, No. 15. New York: John Wiley & Sons, 1963. [2] NOMIZU, K., and K. YANO: On infinitesimal transformations preserving the curvature tensor field and its covariant differentials. Ann. Inst. Fourier (Grenoble) 14, 2, 227-236 (1964). Brown University and Tokyo Institute of Technology

Druck der Universitiitsdruckerei H. StUrtz AG.. Wlirzburg

287

This Page Intentionally Left Blank

J. Math. SOC. Japan Vol. 19, No. 1, 1967

Vertical and complete lifts from a manifold to its cotangent bundle By

K. YANO and E . M . PATTERSON (Received A u g . 10, 1966)

§ 1.

Introduction.

Let ,\I be a differentiable manifold of class C and dimension 11, and let '7'(.!f) be the cotangent bundle of rZI: that is, the bundle of covariant vectors in .\I. T h e n T(.\f)is also a differentiable manifold of class C ' ; the dimension of T ( . \ I ) is 211. In this paper we consider methods b y which certain types of tensor fields in J f can be extended to T ( , \ I ) so as to give useful information about the relationships between the structures of the two manifolds. We call extensions of this kind lifts of the tensor fields in '21 and consider two main types of lifts, which we call vertical lifts and complete lifts respectively. Our main interest focuses on complete lifts of vector fields, tensor fields of type (1, 1) and s k e w s y m m e t r i c tensor fields of type (1, 2). In each of these cases we define the complete lift to be a tensor field of the same type as t h e original. In general, the vertical lift of a tensor field does not have the same type as the original ; nevertheless the construction is a useful one. Our methods enable us to examine the structure of T ( M ) in relation to that of \I. In particular, we show how almost complex and similar structures on \I can be extended to T(.\f).We also examine lifts of affine connections in \I, using the idea of a Riemann extension ([4], [ 5 ] , 161). T h e methods used and the results obtained a r e to some extent similar to results previously established for tensor fields in the tangent bundle of a differentiable manifold ([l], [a], 171, [S], C91, 1121, 1131, C141, 1151). However there a r e various important differences and it appears that the problem of extending tensor fields to the cotangent bundle presents difficulties which a r e not encountered in the case of the tangent bundle. Throughout we use the following notations and conventions : 1. z : T ( \ f ) -\I is the projxtion of ' T ( d I ) onto ,\I. 2. Suffixes -4,B , C,D take the values 1 to 211. Suffixes a , 11, c, . . , h , ?,I, .. take the values 1 to 11 and t= L A ? ?etc.. , T h e summation convention for repeated indices is used. IVhenever notations such a s (-cc0), (in'), ( F B I ) a r e used

289

92

K. Y.\\o a n d E.M . P;\TTERSOA

for mstrices, th- suffix 0 3 th: 1-ft indicates the column and the suffix on the right indicates the row. 3. T;((nl)denotes the set of tensor fields of class C and type (v, s) in A,. Similarly T;(T(Al)) denotes the corresponding set of tensor fields in 'T(A1). 4. Vector fields in !\I a r e denoted b y X,Y , Z. T h e Lie product of X and I' is denoted by [X,Y ] . T h e Lie derivative with respect to X is denoted by LX.Tensor fields of type (1, 1) a r e denoted by F, G and tensor fields of type (1, 2) by S,T.

3 2. The basic 1-form in T(.\f). If il is a point in AT, then x-'(.l) is the fibre over /I. Any point PEZ-'(A) is an ordered pair ( A , P A ) , where p is a 1-form in A l and p1 is its value a t A. Suppose that U is a coordinate neighbourhood in ;1/1 such that A E U. T h e n U induces a coordinate neighbourhood n-'(U) in T(1l) and P E r-'(U). If rl has coordinates ( X I , 9,. . . , , Y n, relative to U and P A has components (PI, p 2 , ... , fin), then P has coordinates (XI, x2, ... , x", pl, fig, ... , $,J relative to n-'(U). If U* is another coordinate neighbourhood in A 1 containing A , then n-l(U*) contains P and t h e coordinates of P relative to x-'(U*) a r e (x*', x*', ... , x*", pik,pf, , p,T) where

the derivatives being evaluated a t A. Let p be the 1-form in T ( M ) whose components relative to n - ' ( U ) a r e (PI, ... , p,,,0, , 0). By (2.1), the components of p relative to n-'(U") a r e (pf, , $:, 0, , 0). In fact we can write

p = prd,rl = ~ T d x ". ~ We call p the basic I-farm in T ( d l ) . T h e exterior derivative d p of p is t h e 2-form given b y

dp= dp, A dx' in s - ' ( U ) . Hence, if d p =

1 2-c,,dxCAd,rR,

(where clxT=dpl), we have

where I is the unit II x 11 matrix. Since the matrix (ccB) in (2.2) is non-singular, it h a s an inverse. Denoting so that this by CB

-1

-OC,

290

1-el-tica 1 a 11 ri coiripl e t e 1i f t s

93

we have

We shall write for the tensor field of type (2, 0) whose components in r l ( U ) are This tensor field is of importance in our construction of complete lifts.

+.

$ 3. The vertical lift of a function.

I f f is a function in ,!I, we write f V for the function in cT(Af) obtained by forming the composition of T and f,so that f'=f 3 li. T h u s if ( A , p) E x - ' ( U ) , then

f V('4, p ) = (f r)(A,p) =f(il) 0

.

(3.1)

T h u s the value of f v is constant along each fibre, being equal to the value of f a t the point on the fibre in the base space. We call f v the z'ertical lift of t h e f u n c t i o n f.

$ 4 . The vertical lift of a vector field.

If X E Ti(M) (so that tion in T ( M ) defined by

X

is a vector field in ,If) we write X v for the funcX V ( A

p) = P(X,4)

(4.1)

where X , is the vector obtained by evaluating X a t A. T h u s if X h are the components of X in U a t the point A, then X v is the mapping (A, p ) + p 8 x ' . We call X V the vertical l i f t of the vector j i e l d X . We have Xr'E%?T(M)), since X v is by definition a function in T(M). We observe that if PEM, then X v ( P ) = O .

$ 5 . The determination of vector fields in cT(M). Suppose that E yA(T(M)), Then -7 is completely determined by its action on functions of class C 0 in c T ( h f ) . In $ 4 we introduced a special type of function in "T(M), namely the vertical lift of a vector field in M. We now show that any element of s;("T(M))is completely determined by its action on functions of this type. PROPOSITION 1. Let 2 and f be L a t o r f i e l d s in cT(?iI)such t h a t 2 ?

-f.zI..=

f o r a l l Z E T;(hf).

Then

-?= f.

29 1

I'ZV

K.YANOand E. M. P.~TTERSON

94

PROOF. It is suficient to show that if k Z v = 0 for all Z E 4 i ( h l ) , then is zero. If Z is the vector field with components Z h in U , then

.YP= where have

-2.

.P

X " ~ a L ( p u z ~ ~ ) + f ~ ~ ,~ a r ( f i ( L Z ~ ~ )

a r e the components of

x. Hence, if R Z V = 0 for all 2

plL+a,zQ+.Fzl- 0 for all 2. Choose 2 to be the vector field given in we get

E

4 : ( h f ) ,we

(5.1)

U by Z ' = 6;. Then from (5.1)

v-

S J= 0 .

(5.2)

Hence (5.1) becomes

p,.Y"p =0 for all 2. Let i, be fixed integers such that 1 5 i 5 n and 1s)5 11. be the vector field given in U by

za=o

z 3 = ~ ,

Then from (5.3) we get jlJt

(5.3) Choose Z to

(ar]).

- = 0.

It follows that we have N

-Y2= 0 at all points of c T ( M ) except possibly those a t which all the components p , , , p,& are zero: that is, at points of the base space. However, the components of ,Ir' are continuous (since they are of class C ) and so X " l is also zero a t points of the base space. Hence ??'=O for all points of x-'(U). This holds for each i satisfying 1 5 i 5 n. Therefore, using (5.2), Y, is the zero vector in x - ' ( U ) . From this it quickly follows that *?= 0 in T(iZI). § 6.

Vertical vectors.

Let T;,("T(M))be such that .?fV= 0 for all f~ 4 t ( M ) . Then we say that .? is a v e r t i c a l v e c t o r f i e l d . It is easily shown that Y, is vertical if and only if its components in n - ' ( U ) satisfy -. ,Y*= 0 (i = 1, 2? ... , 1 1 ) . In § 7 and $ 8 we introduce two types of vertical vector fields in ' T ( h l ) , constructed respectively from 1-forms and from tensor fields of type (1, 1) in M.

29 2

J‘erfiral a n d c o m p l e t e Iifls

93

$7. The vertical lift of a 1-form. Suppose that w + z T ; ( . \ f ) , so that w is a 1-form i n .\I. Let - 1 be a point of AI and let I;, U* be coordinate neighbourhoods containing .4. If o has components W , and w: relative to L‘ and L-* respectively, then

where the derivatives a r e evaluated a t -4. Equations (2.1) and (7.1) show that the vector which has components (0, . , 0, w l , ... , w,J relative to ;-I([-) a t a point ( . I , p ) on the fibre over A4 has components (0, ... , 0, w:, ... , (08) relative to z-‘(C*). We call the vector field determined by the vectors which have these components the i,erfical l i f t ~ r ’ of w . T h u s w V e TAVT(AU)). Clearly wI’(j-7)=z 0 (7.2)

so that

W V is a vertical vector. By Proposition 1, m y is completelj- determined by its action on functions in “T(!\I) of the form ,TI’. Since

- (P,Z’)

a

= w,ZJ ,

W1aP1

we have w’(z’-) = { w ( Z ) } “

.

(7.3)

If w , T E Z(dJ) and f~ T;(.\I), it is easily proved that

$8.

(w+T)”=wJ’+Tr,

(7.4)

(f;)‘ =f J’wJ’.

(7.5)

The vertical lift of a tensor field of type (1, 1).

Suppose now that F E T&\f). If F has components F: and F,*“ relative to U and U* respectively, then

Hence, using (2.1) and (8.1), the vector which has components (0, ... , 0,$,Flu, ... , p,FtLa)relative to r-I(U) has components (0, ... , 0, pdFP‘, ... , pZF$”) relative to n-l(U*). We call the vector field determined by the vectors which have these components the vertical l i f t F r y of F. T h e vector field F V is unlike the vertical lift wJ’of a 1-form in that the components of FY a r e not the same a t all points of the same fibre. In fact

293

K. YANOand E.M. PATTERSON

96

F V is zero a t points on the base space M. Clearly F"(jV) =0 , (8.2) so that F V is vertical. By Proposition 1, F V is completely determined by its action on functions in T(M)of the form Z V . We have

F V(ZV ) = (F(Z))' If F, G E T!(M), then (F+G)'=

FV+GV

and if f E T:(h4), then

(fF)'= f V F V . $ 9 . The complete lift of

B

(8.5)

vector field.

In $ 7 and $ 8 we constructed vector fields in " ( M ) from l-forms and tensors of type (1, l), in M . Constructions such as these can be carried out for other types of tensor field in h2, but they have the disadvantage of changing the type of the tensor fields under consideration in going from M to " ( M ) . Thus there seems to be no obvious way in which such a construction lifts a vector field in M to a vector field in " ( M ) . However, we now describe a different process by which we can lift vector fields. Subsequently we shall apply similar methods to tensor fields of type (1,l) and skew-symmetric tensor fields of type (1, a), in each case obtaining tensor fields of the same type. In our construction we use the tensor 6-l introduced in $2. Suppose that X E T;(M). Let A be a point of M and let U be a coordinate neighbourhood containing A. We have already defined the vertical lift X V of X to be a function in " ( M ) . T h e exterior derivative d X V is the 1form in T ( M ) given in n-'(U) by

We define a vector field X c in T ( M ) by XC=(dXV)&-'. In n-'(U), the components of XC are

ax. ( X I , X2," * , X", -pa -ax,,

... , -pa

ax. ax.-> -

We call X c the coinplete lift of the vector field X . We have

XCf

V=

(Xf)V

and X C Z V

= [X, 2 1 ' .

294

V e r t i c a l aizd c o m p l e t e lifts

97

By Proposition 1, X c is completely determined by (9.2). If X and Y E 5+~(Akf), then (S+ I')C = x c i E'C . (9.3)

9 10. Projectable vectors. T h e vector field X C is completely determined by its first 11 components and in particular XC is zero if these components are zero. An alternative way of expressing this is to say that X C is zero if it is vertical. If ,?E T~("T(h1)) and if there exists X E Ti@!) such that 2 - X C is vertical then we shall say that 2 is projectable, w i t h projection X . A necessary and sufficient condition for /? to be projectable with projection X is that the components of 2 a t a point ( A , p ) in T - ' ( ~ Y )are related to the components X h of X a t A by B h " Xh (I2 = 1, , 12).

xA

'.a

zh

T h u s the components are constant along any fibre. We observe that the complete lift XC of any X E T&U) is projectable with projection X , for X C - X C is trivially vertical.

$11. The tangent space of T ( M ) .

If @ denotes the algebra of functions of class Cm in " ( M ) and X denotes the @-module of vector fields in "(Ad), then a tensor field in " ( M ) of type (0, r ) (respectively (1, r)), where r is a positive integer, can be regarded a s an r-linear mapping of x r into @ (respectively x),where x r is the Cartesian product of r copies of T. (See [2], p. 26.) T h e following result, which should be compared with Proposition 1, is used frequently in the sequel. PROPOSITION 2. L e t 3, 7 be tensor f i e l d s in "(M) of t y p e (0, r ) or (1, r ) such that 3(LY(l),..* , 1%)) = 7(X1),..* XrJ I

f o r all vector fields M. T h e n

lv(s)( s = 1,... , r ) which are complete l i f t s of vector jields in s"= 7 .

PROOF. We shall consider the case of tensor fields of type (1, 2). It is easily seen that the argument extends without difficulty to the other cases. Moreover (in the general case) it is sufficient to show that if

S( for all vector fields

f,,,(s = 1,

* *

, 2(7)) =0

, r ) which are complete lifts of vector fields

295

K.

98

YA\O

and E.M.

PATTERSON

s=

in M , then 0. Let U be a coordinate neighbourhood in hi and let x - ] ( U ) be the induced neighbourhood in cT(,\l). Let 3 E Y:(eT(.l/))be such that

s"(X', IF')= 0 for all X, I'E !T:(hl). Suppose that X, I' have components S h ,Y h respectively in U. Then the components of 3 satisfy g h A

x

1

I'

h-

3?,t((p " d s ") I

x "(p a d / LI ")

h-

~

+3;,"(paa,Xa)(p,ah~~b)=

0.

(11.1)

Choose X, Y to be the vector fields given in U by X 1 = & and I ' " = a ? . Then from (11.1) we get 3.,/ = 0 . (11.2) Next choose X , Y to be given by

Xt=ao"'b . , I * " = @ , where 0, k are fixed. Then, from (11.1) and (11.2) we get Hence (11.3) at all points of T ( M ) except possibly those a t which all the components PI, ... , p , are zero: that is, at points of the base space. However the components of are continuous; hence we have equations (11.3) at all points of

"(M). Similarly we can show that Skin= 0 . h

Finally, by choosing X ,

(11.4)

I' to be given by X ' = ij;,xh,

I'h

= @,YJ

and using (11.1) in conjunction with (11.2), (11.3) and (11.4), we can show by a similar argument that (11.5) S.-" k] -0 * h

From (11.2), (11.3), (11.4) and (11.5) it follows that Hence 3 is the zero tensor field.

is zero in z - ' ( U ) .

$12. The vertical lift of a tensor field of type (1, 2). Suppose that S E T;(M) and that S has components Sjih a t a point il in a coordinate neigbourhood U. At the point ( A , p ) in z - l ( U ) , we can define a

296

Vertical and complete lifts

tensor

P

99

of type (0,2) with components given by N

N

p J. Z.- - p a S.." 32 p J:L. = o , ?

p-.: 31 = 0 .

N

N

p 31.: = 0,

T h e tensor 6-1 introduced in Q 2 is of type (2,O); hence we can define a tensor of type (1, l.) by transvecting with &-I. We write S'' for the tensor field whose components gBAin x-I(U) are given by N

SBA =

PBC&'" .

Thus (12.1) where Q is the matrix (p,S,,"). We call S'' the vertical lift of t h e tensor field S. If w E $(A{),

S"(w") = 0 and if

then (12.2)

Z E ~;(A.rl), then S"(ZC) = (S,)V

I

(12.3)

M defined by S,(X) = S(Z, X).

where Sz is the tensor field of type (1, 1) in

By Proposition 2, S y is completely determined by (12.3). Since any vertical vector a t any point is linearly dependent on vectors of the form wT', it follows from (12.2) that S V ( P ) = O (12.4) for all vertical vector fields

P.

Q 13. Identities involving vertical and complete lifts.

In this section we establish various identities concerning vertical and complete lifts, particularly involving Lie products. These are required for subsequent calculations. PROPOSITION 3. If 2, I; are vertical vectors in " ( M ) , then their Lie pro-

duct

[R, ?]

is also vertical.

PROOF. If f E %(M),then .ffV=O=

PfV.

Hence [B, I;]f"= R(?(~'>>-?(R(f'>)=O. PROPOSITION 4. If w E $(A{), then

+,

[+V, w V ]

PROOF. If Z E T ; ( M ) , then

297

=0.

K. YANOand E.M. PATTERSON

100

[+", W " ] Z " = +"(wv(zv))-w"(+"(Z")) = +v(w(z))"-wv(+(z>>v by (7.3). Since

~ ( z )+(Z) , E %(M) and

+T',

uv are vertical, we get

[$", W " ] Z " = 0 . Hence, by Proposition 1, [+", w"-J = 0 . PROPOSINION 5, If w E $ ( M ) and F E $(M),then [w", F"] = {wF}'

where o F is the l - f o r m defined b y ( w F ) ( X )= w ( F X ) . PROOF. If Z E T&k'), then [w", F " ] Z v = w"(F"(Z"))-~"V(w"(Z"))

= w"(F(Z))"= {w(F(Z))}' by (8.3), (7.3) and (8.2). But also { wF}"Z" = { ( w F ) Z }" = { w ( F ( Z ) ) } ~

so that the actions of [o",F v ] and on 2" coincide. Thus, from Proposition 1, we have [wl", F"] = { u F } V . PROPOSITION 6. If F, G E S ; ( M ) , then

[F", G"]=(FG-GF)" PROOF. If Z E Ti(A4), then, by (8.3) and (8.4),

[Fvt G V ] Z V= F"(GV(ZV))-Gv(F'(Z")) = F"(G(Z))"-G"(F(Z))" = IF (C(Z>> -G ( F ( Z ) )1 = (FG-GF)'ZV.

The required result now follows from Proposition 1. PROPOSITION 7. If w = T:(A4)and X E gA(M), then [X C , w " ] = (-c,w)"

.

PROOF. If Z E YA(M), then, by (7.3). (9.2), (7.2) and (9.1) [XC,w

y z " = xc(Wr'(Z"))-o"(XC(Z"))

= x~(w(z))"--o~'[x, 21" =

(xw(z)j)v-(ax ~ 1 ) ) ~

= I(-c,W)(z)}v

298

Vertical a n d c o m p l e t e lifts

101

(see [2], p. 32). Hence [.‘iC,

W’.]Z”

=(&xo)”zy

[ X c , w”] = (J”Xo)’. s o that, by Proposition 1, ~ K o P o s I T I O N8. If X E TA(Al) a i d F E 4i(hl), t h e n [ X c , F“] = (&,F)‘. PROOF. If 2 E ~ ; ( ~ l lthen ),

[ X C , F”]ZV= XC(F(Z))”-F’[X, 23” = [ X , F ( Z ) ] ” - { F [ X , Z]}t’ =( ( L x F ) 2 ) ” = (L,F)“ZV

(see [2], p. 32). PROPOSITION 9.

PROOF. If

If S,Y E T;(j\l), then [ X C , r’C3 = [X,Y I C .

ZET;(,\l), then, by (9.3, [XC,

YC]Z”= XCCY, Z]V-YC[X, 21“ = CX,[I-,

ZIl“-CY,cx 211‘

= “X, Yl,ZIV

b y the Jacobi identity. Hence [XC,

PROPOSITION 10.

If

I’C]Z“= [ X , Y1C.Z”.

S, T E s:(M)a n d F

E T!(,21), t h e n

S’T“= 0

sJ*Fr-= 0. PROOF. By definition, S”, T ” E T;(“T(M)). Hence S”T” is also a tensor of type (1, 1). If Z E Z’~(Jll),then, by (12.3) and (12.4), S ‘.T ’7(Zc)= S ’( T ‘(2‘))= S ”(Tz)v= 0 .

Hence, by Proposition 2, S V T V= 0. Also F V is a vertical vector field in cT(,21)and so, by (12.4), STrFv-O.

3 14. The complete lift of a tensor field of type (1, 1). Suppose now that F E S;(.\l) and that F has components Fih a t a point A in a coordinate neighbourhood U. At the point (‘4,~) in r l ( U ) , we can define a 1-form o by

299

K. YANOand E.M. PATTERSOS

102

Thus T h e exterior derivative of o is given by do = pn-aFbca ax d x c A dxb+Fbadpn A dx"

so that if we write

where r is skew-symmetric, (as before xrmeans pJ we have

r7. = J1

I:. j

731,:=

-F

I

z-.;.1L z 0

fBA

#

j

t

,

.

We write F C for the tensor field of type ( 1 , l ) in T ( M ) whose components in x - l ( U ) are given by

FBA= z

B

~

.E

~

~

~

Thus flih = Fib,

pihz 0 (14.1)

We call F C the complete lift of the tensor field F . If w E 9f(M),we have

F O(wV)= (oF)'

.

(14.2)

If Z E Y&(hd),we have F C ( Z C )= (FZ)C+(-CzF)'.

(14.3)

By Proposition 2, F C is completely determined by (14.3). T h e action of F C on vertical vectors is completely determined by (14.2). If GEY;(M), then G ' is a vertical vector in T ( M ) and F c(Gv)= (GF)' .

If R E ~ : ( C T ( M )and )

(14.4)

R(d)= (oF)"

for all o E g ! ( M ) and some F E Yi(M), we shall say that ? l is projectable with projection F. In particular, F C is projectable with projection F. PROPOSITION 11. If F E Ti(A4) and S E T;(M), then

300

Vertical and complete lifts

103

F cS " = (SF)" , where SF

E T&ld)

is dejinecl by

( S F ) ( X , Y )r z S ( X , F Y ) .

PROOF. If Z E TA(.U), then, by (12.3) and (14.4), (FCS1')Zc =F G ( S v Z C ) = F c(Sz)v

= (SZF)"

.

But

(SF)"ZC= {(SF),}" and, since { (SF),} ( 1 ' ) = (SF)(Z,I

for all Z'E

7 ,

= S(Z, F 1') = (SzF)(Y )

Tb(iM),it follows that {(SF),}'=(S,F)".

T h e required result now follows from Proposition 2. PROPOSITION 12. If F E T;(*\f)and S E T:(M), t h e n

ST'FC= (SF)"

if a n d only if S(2, F 1') = S(F2, F7)

for all 2, Y E Tb(A4). PROOF. Suppose that Z E $,(.V). Then, by (14.3), Proposition 10 and (12.3), (S"FC)ZC= S"{(FZ)C+(L,F)'~}

s

= "(F2)C = (s,z)v.

But, by (12.3),

(SF)"ZC = { (SF),} r' Now S F Z = ( S F ) , if and only i f for all I'

E

.

YL(Af) we have

Sr.zI'= ( S F ) , Y : that is, if and only if

S ( F 2 , 1') = S ( 2 , F Y ) . Since (SF,)" = (SF); if and only if Srz = ( S F ) z , the required result follows a t once.

301

104

K. YAKOand E.M. PATTERSON

8 15. The complete lift of a skew-symmetric tensor field of type (1, 2). Suppose now that S is a skew-symmetric tensor of type (1, 2 ) in M and that S has components Sj? a t a point .4 in a coordinate neighbourhood U . At the point ( A , fi) in x - I ( U ) , we can define a 2-form o by

Thus

T h e exterior derivative do of

0

is a 3-form given by

Hence, if we write

where r is skew-symmetric in all pairs of suffixes and xi means

r;iB= 0= r-/ B -h .!?cBA

-. Dzh

pi, we have

.

We write Sc for the tensor field of type (1, 2) in T ( M ) whose components in n-'(lJ) are given by

_ - sjhi, gjiX= Shij, sj? = 0 .

sj:h=

We call S c the corizplete l i f t of t h e tensor ,field S. Y , Z E 9h(A4), we have SC($",

w") = 0,

SC(w",

ZC)

= -(wSz)",

302

If $, w E Z ( M ) and (15.1) (15.2)

Vertical a n d complete lifts

105

and Scu,z,E Tl(M) is given by

From Proposition 2 it follows that Sc is completely determined by (15.3).

§IS. Theorems on structures in the cotangent bundle. We now apply our constructions of lifts of tensor fields to obtain theorems concerning the existence of certain typesof structure in "T(M). In our arguments, the torsion of two tensors of type (1,l) plays an important part. If F , G E T:(M), the torsion NF,Gof F , G is the tensor field of type (1, 2) defined by

~ N F , c ( XY,) = [ F X , G Y ] + [ G X , F Y I + F G [ X , Y ] + G F [ X , Y1

- F [ X , GY]--F[GX, Y ] - G [ X , F Y I - G [ F X , Y ] where X , Y E TXM). (See [Z], p. 37 ; we have introduced a factor convenience.) It is easily seen that NF,C=

(16.1)

1 for

-,-

NG.F

and that NF,Gis skew-symmetric. If we put F = C , we obtain the Nijenhuis tensor of F , given by

NF,F(X, Y ) = [ F X , F Y ] + F 2 [ X , Y ] - F [ X , F Y I - F [ F X , Y ] .

(16.2)

We shall abbreviate NF,o to N whenever i t is clear which tensor fields F , C are involved. If F E Tj(M) and F2= -I, where I is the Kronecker tensor field (that is, the tensor field with components St), then F is an almost complex structure on M . It is well-known that F is integrable (that is, F is obtainable from a complex structure on M ) if and only if NFIF=O. If F E T!(M) and F 3+F = 0, then F is called an f-structure on M . (See [lo], C111.) PROPOSITION 13. If F is an almost complex structure o n M and N = N F , F , then N V F C =( N F ) v ,

(NF>'Fc= - N V . PROOF. By Proposition 1.2, it is sufficient to show that

N(2,FY)= N(F2, Y ) and

303

K.YANOand E.M. PATTERSON

106

-N ( Z ,

Y ) N(FZ,F Y ) 1

for all 2, Y E ~At(~l4).This is a matter of direct verification, using F z = -1. Our next result establishes a connection between the complete lifts of two tensor fields F , G E $ ( M ) and the torsion of F and G. PROPOSITION 14. If F , G E LTt(M), then

+

F CGC+CCF = (FG GF)C+ (2N)' where N = N F , G .

PROOF. Suppose that X

E

$,(M).

By (14.3) and (14.4)

FCGCXC= Fc((GX)c+(LxC)V) = (FGX)C+(-CoxF)V+{(-CxG)F 1'

= ( F G ) C X C - { L x ( F G ) } V + ( - C ~ ~ F ) ~ ' + { ( L x C ) F } " .(16.3)

Hence

(F 'GC+CCFc, X c = (FG+GF)'XC+QV

(16.4)

where Q E st(&') is given by

Q = -CGxFS(LxG)F--C,(FG)+-C,,&

+(L',F)G -r,(GF)

By a well-known formula for Lie derivatives ([Z],

.

p. 32) we have

QY=[GX, FY]-F[CX, Y I + [ X , GFY]-G[X, F Y I - [ X , F G Y ] + F G [ X , Y ] + [ F X , GYI-GCFX, Y ] +[X, FCY]-F[X, GY]-[X, GFY]+GF[X, Y ] for any Y E Tb(M), from which it follows that

QY=2N(X, Y ) . By (12.3),

N V X C= ( N x ) V . But

2Nx( Y )= 2 N ( X , Y )= Q Y s o that 2Nx=Q.

Hence

Q" = 2 N V X C

so that, by (16.4), the actions of F C G C f G C F Cand (FG+CF)C+2NV on X " are the same. T h e required result now follows from Proposition 2. PROPOSITION 15. Zf F E T i ( M ) , then

(F c)z = (F z)c+(NF,p)vl

304

(16.5)

V e r t i c a l a n d complete l i f t s

107

This is a n immediate corollary of Proposition 14. PROPOSITION 16. I f F E s;(M), then (16.6)

(FC)~=((F~)C+(~T-I;S)"

iuhere T i s t h e torsion of F aizd I;?, aiicl AT i s t h e -Yijenhziis tensor of F. PROOF. By Propositions 15 and 11,

( F C)3 =F C ( F Z)C+F

C J V F'

= FC(F2)C+(:YF)V

(16.7)

I

By (16.3,

F C ( F 2 ) C X= C (F3)CXC+{(-CgF2)FS(-CFPXF)-I'XF3}V

so that, using (16.7) and (12.3) (FC)3XC

= ( F , ) C X C + R"

(16.8)

where

R =(r,F2)F+-CfzrF-I',yF'f(IVF)x. We have

RI'= ['X, F 3 1 7 ] - F 2 [ X , FI']+[F'X, F Y ] - F [ F 2 X , 1'3 - [ X , F 3 Y ] + F J [ X , Y]+[F?i, F 2 1 ' ] + F 2 [ X , F Y I

-F[S, FZY]-F[FX, F Y I =[ F X , F2Y]+[FZX, FY]+2FS[X, Y ]

Y ] - F ? [ S , FY]-F2[FX, Y ]

- F [ X , F"]-F[F2X,

-F[FX, FY]-FY[X, Y]+F"X, FY]+F"FX, = 2T(X, 1')-FN(X,

Y]

Y) for any Y E ~ ~ ( A l ) .

=2TX(Y)-(FN),(Y)

Hence, by (16.8) and (12.3)

(F ')'X

= (F ')'X

+(2T-y-(F1lr).r)''

-( F ~ ) C X C + ( ~ T - F ~ \ ~ ) ~ " ~ . C .

This proves Proposition 16. PROPOSITION 17. If F , C E T;(M) aizd

?i

is t h e torsion of F C a n d GC, t h e n

N

N = ,I1C

iuhere N is the torsion o,f F a n d G . This result can be proved (using Proposition 2) by means of a straightforward but somewhat lengthy computation. We come now to our main theorems.

305

K. YANOand E.M. PATTERSON

108

THEOREM 1. Let F be a n almost complex structure on M . T h e n the complete lift F C is a n almost complex structure o n cT(ilil) if and only i f F i s integrable. PROOF. Since F is an almost complex structure, we have F Z = -I. Hence, by Proposition 15, (FC)'= (-I)C+N' where N is the Nijenhuis tensor of F. Since the complete lift of I in 1 2 1 is the Kronecker tensor field r" in c T ( M ) , we have (FC)'= -7 if and only if NV=O. Since N V = O is equivalent to N = O it follows that F c is an almost complex structure in cT(A4) if and only if N = O . THEOREM 2. If F i s a n integrable almost complex structure 011 Jl, then the complete lift F C is a n integrable almost complex structure o n 'T(h1). PROOF. By Theorem 1, FC is an almost complex structure. Since I; is integrable, the Nijenhuis tensor of F is zero. Hence, by Proposition 17, the Nijenhuis tensor of F C is also zero, THEOREM 3. Let F be a n almost complex structure o n h.1, with A; the Nijenhuis tensor of F . T h e n

i s a n almost complex structure on " ( M ) . This theorem is due to Sat6 [S]. PROOF. Using Proposition 10, we have

{F

'+ +-(NF)V}

1

= ( F C)*+2-FC(NF)V+

1 -(NF)'FC 2

by Propositions 11 and 13. Since FZ= -1, we get, using Proposition 15,

1 {F'+T(NF)"}'=

I

(FC)'--NV

z=

(F2)' = -1.

1 THEOREM 4. T h e almost complex structure FC+-2-(NF)V

on cT(hil) (see

Theorem 3) i s integrable if and only if F is integrable. 1 PROOF. If F is integrable, then N=O and so F C f - 2 - ( N F ) V = F C ; by Theorem 2, FC is also integrable. Suppose conversely that FC+- 1-(NF)" is integrable. Then the Nijenhuis 2 1 tensor of Fc+-2-(NF)V in CT(A4)is zero. By a direct if somewhat lengthy computation (which makes use of the propositions proved in

306

9 13) we can

Vertical a i d c o m p l e t e l i f t s

show that the Nijenhuis tensor lqxc,

109

1 @ , of FC+-2-(.YF)T’ satisfies 1’C)

= { AyX, I V ) } C - P

where P is the tensor field of type (1, 1) in .\I given by

2 P ( Z ) = N ( Y , [ X , Z])-lY(X, [l’, Z])+.Y(X, F [ F I ’ , 21)

-K(Y, F [ F X , Z])+S([FI’, XI,F Z ) - X ( [ F X , Ir 1,F Z )

+[Y, h‘(X, Z ) ] - [ S , S ( I - , Z ) ] + F [ F Z * , AV(X,Z)]

Since

A

is zero, we get { Y(X, I

)}C

7

PT’=0 .

But this shows that the vector {LY(X,I-)}c is vertical; since the complete lift of a non-zero Vector cannot be vertical, it follows that ?i’(X, I’)=O. This Hence F is integrable. holds for all X , Y E S:(.\l) and SO iY= 0. It is of some interest to note that the expression for 2 P ( Z ) is not linear in X and Y . If we write Q(x,1 , 2) for P(Z),we find that

Q(fx’,61’) z)=fgQ(Xt 1’9 2)‘(f(zg)i-g(Zf))AY(X, I-).

THEOREM 5. Let F be a n f-structiii-e 011 l\l. Let *Vbe the A’yenliuis tensor of F and let T be the torsion o,t F and F ? . Then F C zs a n f--stviicfuve o n A 1 if a i d oiily if 2T=FS,

or, equivalently,

N(X,FI.)+K(FX, 1 7 ) + F S ( X , J - )

(16.9)

=0

f o r a l l X , Y E SA(Al). PROOF. Since F 3 + F = 0 , it follows from Proposition 16 that

( F q 3 f F C= ( F d)C+FC+(2T-FA‘)s

(2T-FiV)’.

Hence F C is an f-structure if and only if (2T--F,V)’=O,

which is equivalent

t o 2 T =F N .

To prove the last part, we simply verify that

N ( X , FY)+A‘(FX, Y)+Fh’(X, I r ) = (2T-FAV)(X, Y ) for all

X and I;.

THEOREM 6. Let F be ail f-btriictzire o n .\I, let .V be the Sijenhiiis tensor of F and l e t T be the torsion o f F and F L . Tfzeii FC“ { (Fh.-2T)(Z--i-FZ)}

307

I’

K. Y A N O and E.M. PATTERSON

110

is

f-structure 011 CT(L\l). PROOF. Write

aiz

P =(FIV-~T)(I+ 3 F')

.

(16.10)

If X E T&\l), then (FC+P ")X'= (FX)'+ ( L x F ) " S(Px)" by (12.3) and (14.3). Hence, by Proposition 10

( F '+P ')'A''

z=

(F ')'X

'+ F 'Px"+P "(FX)'

= ( F ')'X'+ (PxF )"+ (Ppx)"

and similarly ( F C +P ") 3X '= (F ')'X

+(PxF ') "+(P,.yF )"+(PFzx)'

.

Hence, by Proposition 16,

+

+

(F c P ")3xc= (F 3)cx c ( 2T- F N ~ (+P ~ 2F)

+(P>.,~F)v +( P , , . ~ ~ ) " ~

Since F S = -F, it follows that

( F '+P v ) 3 X c= -(F ' + P ")A'

'

for all X if and only if

P.Y+PxF'+P,xF+P,?.y=

(FN-2T)x

for all X. This condition is equivalent to

P ( X , Y ) + P ( X , F 2 Y ) + P ( F X ,FY)+P(F2XX,Y ) (16.11)

= F N ( X , Y)-2T(X, Y )

for all X,Y E Tb(.\!). With P defined by (16.10), a straightforward verification can be used to prove that (16.11) is satisfied. Hence (Fc+Pv)SXxc+(Fc+P1')Xc = 0 , so that (using Proposition 2 once more) we have (FC+P")3+(F'+P")=

0.

0 17. The Riemann extension and the complete lift of a symmetric affine connection in M . be a symmetric affine connection in hl. Let A be a point of A4 and Let let U,U* be coordinate neighbourhoods containing A. We write and for the components of relative to U and U * respectively. Then the tensor field of type (0,2) in T ( M ) whose components g'CB in r l ( U ) are given by

ryi

v

rT,h

h=-2P& (17.1)

308

111

Vertical a n d c o m p l e t e lifts

has components

BFB in

z-’(lJ*)given by

8%. = -2pz 31

r*a JL

9

8%. = 6 j = g*: 31 ZJ ’ g?-=0 . JL

We call this tensor field the Riemann extension of the connection denote it by (see [4], [ 5 ] , [S]). We have

vR

r and

p’R($“,w“) = 0 r”(XC, w’?)= (w(X))” rR(XC,

YC) = -(vxY+r,X)”.

rR

By Proposition 2, the tensor field is completely determined by the last of these three conditions. the Let V C be the Levi-Civita connection determined by F ~ .Lye call ?j complete lift of V . T h e components I;&, of j P in x - ’ ( U ) are given by fh.

31

- p21. ,

ph7= 0= f?.= j%:’ Ji

11

Jl

.(ahrg-ajr:,La,r~~+2r;t~rg~ , -_ -_ q;= 0 . r?:= -rib, qi= --

=p

(17.3)

-rjht,

11

PROPOSITION 18. Covariant differentiation with respect to the coniiection

vC in “T(M)

satisfies the following properties : vgvwv = 0,

pPco”=

v$vF“=z (+tF)”,

(FXW)V,

V$CF”=(~~F-(VX)F)”,

V2.Y” = -($(rY))F-,

+

Y0 = ( v x Y )C+ { ( V X ) ( V Y) (0 Y ) ( r X -K.r ) I’--KyX } where $, o E $ ( M ) , X,Y E %(W,F E TKW, I( i s the czircature tensor of v$c

v

and Kx E T ; ( M ) is given by (fY.yY)(Z)= K ( X , 2 )E’ . PROOF. These formulae can be obtained directly from formulae (17.3). An alternative expression for p$cYC is

( V X V C + { v ( v x Y s r , m - ( r x ~Y + V Y F X ) } This can be proved from Proposition 18 by using the identity j7j7xY-V.rV

PROPOSITION 19. Let

Y = (v I’)(C.X)-KxE’.

K be the curvature tensor of

309

pc. T h e n if

4, +, w

K. Y A N O and E. M. PATTERSON

112 E

g(A4)and X, Y,ZEfTb(hl),we have K(qV, $“)w” = 0,

K(#V, $ “ ) Z C =

K(XC, $“)w“ = 0,

K(XC, $“)ZC= +($KzX)“

0,

K(XC, YC)w”= -(w(K(X, Y)))“ K(XC, YC)ZC= ( K ( X , Y)Z)C

+I B(K(X9 Y ) Z > - (BK),,,,,

>

Y+ (rK),r,,X+@Z)K(X, Y 1“

where

x,Z> from the formulae for vc given in

( r m , x , z , ( w = (BK)(U,

These formulae follow

*

Proposition 18.

Tokyo Institute of Technology and University of Aberdeen, Scotland

Bibliography P. Dombrowski, On t h e geometry of t a n g e n t bundles, J. reine angew. Math., 210 (1962), 73-88. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience T r a c t , No. 15, 1963. A. J. Ledger and K. Yano, T h e tangent bundle of a locally symmetric space, J. London Math. SOC.,40 (1965), 487-492. E. M. Patterson, Simply harmonic Riemann extensions, J. London Math. SOC.,27 (1952), 102-107. E. M. Patterson, Riemann extensions which have Kshler metrics, Proc. Roy. SOC. Edinburgh Sect. A, 64 (1954), 113-126. E.M. Patterson and A.G. Walker, Riemann extensions, Quart. J. Math. Oxford Ser., 3 (1952), 19-28. S. Sasaki, On t h e differential geometry of tangent bundles of Riemannian manifolds, TGhoku Math. J., 10 (1958), 338-354. I. S a t & Almost analytic vector fields in almost complex manifolds, TBhoku Math. J., 17 (1965), 185-199. P. Tondeur, S t r u c t u r e presque kdhlkrienne naturelle sur le fibre d e s vecteurs covariants d’une variktk riemannienne, C. R. Acad. Sci. Paris, 254 (1962), 407-408. K. Yano, On a s t r u c t u r e f satisfying f 3 + f = O , Technical Reports, No. 2 (1961), University of Washington. K. Yano, On a s t r u c t u r e defined by a . tensor field f of type (1, 1) satisfying f3+f=O, Tensor, N.S., 14 (1963), 9-19. K. Yano a n d E.T. Davies, On tangent bundles of Einsler and Riemannian manifolds, Rend. Circ. Mat. Palermo, 12 (1963), 211-228

310

V e r t ica 1 a 11 d complete iift s [13] [14]

[15]

113

K. Yano and S. Ishihara, Horizontal lifts of tensor fields and connections to tangent bundles, to appear in J. Math. Mech.. K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles, I. General theory, J. Math. SOC.Japan, 18 (1966), 194-210. 11. Affine autornorphisms, ibid. 18 (1966), 236-246. K. Yano and A. J. Ledger, Linear connections on tangent bundles, J. London Math. SOC., 39 (1964), 495-300.

31 1

This Page Intentionally Left Blank

. J . DIFFERENTIAL GEOMETRY 1 (1%7) 355-368

ALMOST COMPLEX STRUCTURES ON TENSOR BUNDLES A. J. LEDGER & K. YANO

1.

Introduction

It is well known that the tangent bundle of a Cw manifold M admits an almost complex structure if M admits an affine connection [ I ] , [ 5 ] or an almost complex structure [7], [8]. The main purpose of this paper is to investigate a similar problem for tensor bundles T : M . We prove that if a Riemannian manifold M admits an almost complex structure then so does T:M provided r s is odd, If r $- s is even a further condition is required on M . The proofs depend on some generalizations of the notions of lifting vector fields and derivations on M , which were defined previously only for tangent bundles and cotangent bundles 141, [7], [8], [9], [lo].

+

2. Notations and definitions

M is a C * paracompact manifold of h i t e dimension 1 1 . F ( M ) is the ring of real-valued C” functions on M . For r + s > 0. T:M is the bundle over M of tensors of type ( r , s), contravariant of order r and covariant of order s. 7r is the projection of T;M onto M . We write TbM = T ’ M , T:M = T , M . ,F,;(M) is the module over F ( M ) of C - tensor fields of type ( r , s). We writc ./;,(M) = , T ’ ( M ) ,,=T;(M)= T 8 ( M ) , and ,F!(M) = F ( M ) . X ( M ) is the direct sum C F ; ( M ) . T , is the value at y E M of a r. v tensor field T on M , and ,Ti@) is the vector space of tensors of type ( r , s) at p . Let S E ,=T;(p) and T E . F ; ( p ) . Then the real number S ( T ) = T ( S ) is defined, in the usual way, by contraction. It follows that if S E ,Fs(M) then S is a differentiable function on T I M . A map D : Y ( M ) + .F(M) is a derivation on M if (a) D is linear with respect to constant coefficients, (b) for all r , s , D.P;(M) c 7 ; ( M ) , (c) for all C” tensor fields T , and T 2on M , D ( T , (3 TJ = ( D T , ) 0T 2 -I- T , (3 DT,, Communicated August 15, 1967.

313

356

A. J. LEDGER & K. YANO

(d) D commutes with contraction, A derivation is determined by its action on F ( M ) and F1(M). In particular, . Y i ( M ) may be identified with the set of derivations which map F ( M ) to zero. The set of derivations on M forms a module 9 M over F ( M ) . (vii) The notation for covariant derivatives and curvature tensors is that of [ 2 ] , The linear connections considered on M are assumed to have zero torsion. 3.

Vector fields on T,'M

In this section we show how vector fields on T:M can be induccd from vector fields, tensor fields of type ( r , s). and derivations on M . We first prove a lemma which, together with its corollary, will be of usc later. Lemma 1. Let p E M and S E n - ' ( p ) . If W is LI vertical vector nt S (i.e. tangential to x - ' ( p ) ai S) and W ( n )= 0 for all N E 2 - ; ( p ) then W = 0 . Proof. The vector space . Y , ; ( p ) is dual to F;(p)and hence N contains a system of coordinates on n - ' ( p ) , The result follows irnmcdiately. Corollary 1. Let W E S ' ( T : M ) . If W ( N ) =for ~ all N E f ; ( M ) then W = 0 . Proof. The assumption on W implies that for 5, E 3 - - l ( M ) and f E F ( M ) , 0=

I W(rlf203 ) = W ( ( f 7r)df013) = W ( f . 7r)tlf 0,3 . 2

-~

L

Hcncc (IxW = 0, and so W is a vertical vcctor field. Thus W = 0 by Leniina 1. thc values of W on the zcro section of .F;M being zero by continuity. Proposition 1. Let T E 3-:(M). Then there is u irniqrre C,' vector field T" on TIM siich t h t for N E J-;(M), T'(tu) = n ( T ) n

(1)

L

,

Proof. For p E M , n-I(p) is a vector spacc and so T,]determines B unique vcrtical vector field TY, on n-I(p) such that for N E c ~ - ; ( p )T;:(n) , = (y(T1,).The cross section T on T:M then determines a C' vcrtical vector field which satisfies ( 1 ) . T" will be called thc verticul lift of T , Corollary 2. Let S E r ' ( p ) , anti let T i be the valire of T' at S. Tlieri the map T,, T ; is a linear isomorphism of n - ' ( p ) (n-l(p)).,, where ( n - ' ( ~ ) ) . ~ is the tangent space to the fibre n - ' ( p ) at S . Proposition 2. Let D be a derivation on M . Tlieri there is a irniqrre Liector field D oti T;M such that for N E Y ; ( M ) --f

(2)

--f

Drr = Dru

314

,

3 57

ALMOST COMPLEX STRUCTURES

-

Proof. Let {P}(i = 1, 2, . ., n) be a coordinate system on a neighbourhood U of p E M , and {a8}(0 = 1, 2, . . . , nr+s)a basis for F ; ( U ) . Then {xz 0 n,d}is a coordinate system on r - ' ( U ) . Define D on n-'(U) by (3)

D(X1

c

n) = (DX') 0

,

71

D(wR)= D(oe) .

(4)

Thus a C" vector field D is defined on x L 1 ( U ) . Moreover, for (Y E S f ( U ) we have Dlru = Dlru. Hence, using Corollary 1, it follows that D is defined over YM as the unique solution of (2). Corollary 3. If f E F ( M ) then D(f c n) = (of) IC. Corollary 4. D is a vertical vector field if and only if D E F ; ( M ) . Corollary 5. If D,, D2 are derivations on M , and f l , f 2 E F ( M ) , then f,D, + fzD, is a derivation on M , and Q

__ _ _

f a

+ f?D, =

K)DI

(fl

t (f, n>D2 .

Thus if F ( M ) is identified with F ( M ) n IC = {f o IT : f E F ( M ) } flier1 D linear map of .QM -+ P ( T ; M ) . then for S E Ti([)), Corollary 6. If p E M and A E

A,

( 5 )

-t

D is

(I

= -(&)I, ,

where the sirfix S tletiote~evaliration at S. Proof. Let lru E . Y ; ( p ) . Then

A,(cu>= ( A t r ) ( J ) =

-(AS)L((Y) .

'I he result follows from Lemma 1 . denote Lie derivation with respect Corollary 7. Let X E T 1 ( M )arid FA, to X . Then 2,i.r a vector field on T ; M . In conformity with the notation of [4], [8], [9], [lo], we call 2, the complete Lft of X and write 9,= A''. Remark 1 . If f E F ( M ) then Y , k

where X (6)

0df

=

f 9 1

-

XOcif,

is regarded as a derivation on M . Thus

(fx).= (f

x 0clf -~

L

n)XC -

Now if T:M is the tangent bundle T ' M thcn for

LY

E

x 0df(n) = - a ( X ) d j . I-ience by Proposition 1 ,

315

*

F1(M),

358

A.

J. LEDGER 8: K. YANO

____

X Q df = - d f X v , where X" is the vertical lift of X to TIM. We then have

(7)

(fX)C = fXC

+ dfX" .

Equation (7) was used extensively in [8] but does not appear to extend to tensor bundles of high order. Equation (6) is perhaps a useful generalization. Lemma 2. Let p E M and A E FT:(p). Suppose there exist non-negative integers a and b , not both zero, siiclz that A F ; ( p ) = 0. Then A = kZ where k is some real number. If a $. h then A = 0 . Proof. We prove the lemma for the case a > 0. The proof for a = 0 and b > 0 is essentially the same with covariance and contravariance exchanged. Let S E F ; - ' ( p )be non-zero, and let X E F-'(p). Then AS @ X

+ S@AX

=0

.

Choose (0 E 2;;, ( / I ) such that w ( S ) # 0. Then ( A - kZ)X = 0, where k = - w(AS),'o(S). I t follows immediately that A = kZ. Then for T E ~ - ; ( y ) 0 = A T = I\(N

b)T.

-

Hence, if N # b then A 0 and A = 0. Remark 2. A = XI for some X is a necessary and suflicient condition for A f ; ( p ) = 0, CI # 0 . Corollary 8. Let 1) E 9 M antl si~pposcthew e x i s t non-negntii3eintegers a antl b , tzot borli ;ern, ~irclztllrit D , I ; ( M ) = 0. Then D = f!, where f E F ( M ) . If a + b then D = 0. Proof. Let Ii E F ( M ) and T E J ; ( M ) . Then

(Dh)7= 0

,

It follows immediately that D F ( M ) = 0 antl hencc D E J : ( M ) . Then by Lemma 2, D = fZ for some f E F ( M ) , and if CI # b , then f is zero by Lemma 2. This completcs the proof. Remark 3. D = fZ for some f E F ( M ) is a necessary and suflicient condition for D 5 ; ( M ) = 0, a # 0 . Corollary 9. The map D -> D of (irM --* S ' ( T ; M ) is a nioiionioryliisrn when r f s antl has kernel { f I : f E F ( M ) } wlieri r = s. Proof. This follows from Corollaries 1, 5 and 8. Corollary 10. I f r # s then TvM atlnzits N vertical vector field which vanishes only on the zero section of T ; M , Proof. The vector ficld f has the required properties. Corollary 11. Let p E M , A E f.jT:(p) and T E F ; ( P ) , r # s. Theti A = T v implies A = 0 and T = 0 .

316

359

ALMOST COMPLEX STRUCTURES

Proof. Suppose 2 = T". Then by Corollaries 2 and 6, AS = -T for all S E F ; ( p ) . Since A is linear it follows that T = 0 and A T ; @ ) = 0. Hence A = 0 by Lemma 2. Suppose now that r is a linear connection (with zero torsion) on M , and let X E P ( M ) . Then FX E Y ; ( M ) , and hence, by Corollary 4 , is a C" vertical vector field on G M . Another C" vector field on T;M is determined by the derivation V , . In conformity with [4] we write r, = X h , and call X " the horizontal lift of X . If f E F ( M ) then using Corollary 3,

r,

X / ' ( f ' n ) = r,(f n ) = ( C , f ) n = (Xf) n . Q

Hence dnX" =

(8)

x.

The horizontal lift clearly satisfies

(fxf

gy)" = ( f

G

n)xh+ (g

0

n)Yh ,

for f , g E F ( M ) and X , Y E F ( M ) . Thus the horizontal lift is a linear map of F ( M ) --t P ( P s M ) if, as before, F ( M ) and F ( M ) o n are identified. Since = 0 if and only if X = 0, the horizontal lift is a monomorphism, and so determines a horizontal subspace H , of dimension n( = dim M ) at each point S E T i M . Then C" distribution H on T i M so obtained is usually calIed the horizontal distribution determined by the connection r . If S E T:M then the tangent space ( P s M ) . yis the direct sum V,s H s , where V Sis the subspace of vertical vectors at S . Thus, if W E (T:M),ythen

e,

+

w = h(W) + V ( W ) , where h and 'u are the projections onto the horizontal and vertical subspaces at S. Clearly XIt = Iz(X[l) and Tv= v ( T u )for any vector X and tensor T of type ( r , s) at n(S). 4.

Lie brackets

We now determine, for later use, the Lie brackets of some particular types of vector fields on T;M. These results generalize some of those already obtained for tangent bundles and cotangent bundles [ I ] , [ 4 ] ,[ 7 ] ,[S], [ 9 ] ,[lo]. Lemma 3. Let T,, T , E f ; ( M ) and X , X , , X 2 E P ( M ) , and let D , D,, D p , A be derivations on M , where A E F i ( M ) . Let R denote the curvature tensor field of the connection r. Then

317

A. J. LEDGER X. K. YANO

360

(12)

[X'L, T " ] = (V 1 T)" ,

(1 3)

[X:,X:l] = R(X1, X,)

__ -_

- --

,

(14)

[ X U ]=

(15)

[x:, xi1 = [ X , , X,lC.

VIA

+ [ X I , X?l" ,

Proof. Several equations can be proved by application of Corollary 1 . If p E M then r - ' ( p ) is a vector space, and has the structure of an abelian Lie group. If S E F ; ( M ) then S" is an invariant vector field on r - ] ( p ) and equation (9) follows immediately. We have, from Proposition 2,

[D,, D,]n = ( D I D , - D,D,)n = [ D , , D?]N. Since [D,?D,] is a derivation on M ,from Proposition 2 we have

[ D , , D,ln = [Q, D,ln

7

and hence equation (10)

[ D , T"]n = ( D ( n ( T ) )- ( D n ) ( T ) )

0

?r =

(4DT))

r,, equation

which gives equation (1 1 ) . Since X " = of (11). Since R(X,, X,)E T ; ( M ) , we have

7r

= (DT)YN)

9

(12) is a special case

[x:,x:]= [L 0 ,=~[L,, r , =~Rcx,Z2j+ rr,,,l?l , from which follows immediately equation ( 1 3).

[Xh,A]a = C,(An)

- A(Fyn) =

which gives equation (14). Since X c = of (10).

(V uA)n = (m>n ,

p,., equation (15) is

a special case

5. Almost complex structures We now consider the main problem, that is, to determine a class of tensor bundles which admit almost complex structures. For this purpose it is sufficient to consider contravariant tensor bundles since a Riemannian metric tensor field induces a fibre preserving diffeomorphism of T;M TrtSMM. Also --t

318

361

ALMOST COMPLEX STRUCTURES

the tangent bundle TIM of a Riemannian space always admits an almost complex structure [l], [ 5 ] . Hence we shall restrict attention to T'M, r > 1. Lemma 4. Let V and g be, respectively, a symmetric connection and a Riemannian metric tensor field on M , and E E P - ' ( M ) be nowhere zero on M . Then T'M admits three distributions which are mutually orthogonal with respect to a Riemannian metric tensor field 2 induced on T'M by V and g . Proof. For each p E M a scalar product <, > is defined on the vector space rr-'(p) by < T I , T , > = t,(T,), where, for any tensor T with components T i ~ i z " ' itr , is the covariant tensor associated to T by g. Thus t has components

tili2 . . . i r

- Tjlir'"i,gi -

.g. .

i2.i2 '

.

'

gi,j,.

7

where each repeated suffix indicates summation over its range. If S E T'M, then a scalar product, denoted by the same symbol <, >, is defined on the vector space (T'M),* by the three equations

(17)

< T,",T,"> = < T I ,T 2> < T " ,Xh > = 0 ,

(18)

<x;,xa>= < X , , X ? >

(16)

0

n ,

o x ,

where X'l is the horizontal lift of X with respect to F . These equations are easily seen to determine on T'M with respect to which the horizontal distribution H , induced by r, is orthogonal to the fibres of T'M [3]. We now make use of E . For X E S I ( M ) , define the vertical lift X; of X with respect to E by

x;<= ( E 0X)l'

,

The map X X ; is then a monomorphism of P ( M ) J ' ( T ' M ) . Hence an n-dimensional C- vertical distribution V" is defined on T'M. Let V I be the distribution on T'M which is orthogonal to H and V s . Then H , L'/
4

~-&(p= ) { T :< T , E

0X >

= 0 for ull

X

Then L ' i = ( Y k ( p ) ) : . Let E l ( p ) be the sirbspuce of T r + ' ( p )defined by EL@)

Then 3 g ( p )

=

= {T:

< T , E > = O}

E L @ )0, F ' ( p ) .

319

,

E

S 1 ( p ) }.

362

A.

J. LEDGER 8: K. YANO

Proof. The first part of the lemma follows from the fact that the vertical lift preserves scalar products. To prove the second part it is sufficient to note that El-(p)0P ( p ) c J ; ( p ) , and dim ( E I ( p )0F ( p ) ) = n(n'-' - 1) = nr - n = dim F & ( p ) . Theorem. I f M admits an almost complex structure-and-a nowhere zero tensor field E E F T - l ( M ) ,then TrM admits an almost complex structure. Proof. Let F be an almost complex structure on M . We define a Cm tensor field J of type (1,l) on T'M by its action on the distributions H, T"$ and V L . Thus for X E Y ( M ) and T E P ( M ) define J by

J(X'l) = xi,J(X%) = - X " , J(T")= PJ,

(19)

where is obtained by contracting T 0F , and has components T*172. %iLF;r, where Tfi1?. and Ff are local components of T and F respectively. The restrictions of J to H f V L and V 1 are endomorphisms. and hence J is a tensor field on T'M. It is easily seen that J is C" and J? = - I , I being the unit tensor. Hence J is an almost complex structure on T'M. Corollary 12. Suppose a Riemannian manifold M admits an almost complex strirctirre. Then T'M admits an almost complex structure if (i) r is odd or (ii) r is even and M admits a nowhere zero vector field. Proof. ( i ) For r = 2s 1 choose E = (Og-')s, where g-' is the inverse of a metric tensor field g on M , and (Og-')sis the tensor product of g-I with itself s times. (ii) For r = 2s, s > 1, choose E = (Og-')"-'OX,where M is assumed to admit a nowhere zero vector field X . For Y = 2 choose E = X . 17

+

6. Integrability of the almost complex structure J

We now establish necessary and sufficient conditions for the integrability of J . Let e be the covariant tensor field of order r - 1 associated to E by g ; thus, with respect to local coordinates, e has components e ~ l i ~ , ,given . ~ ~ -by l eil~2...ir-l - gglj,gi2j

2...

.

~ i ~ - , . j ~ - , E ~ ~ ~ ~ " ' ~ r ~ ~

Proposition 3. Suppose M admits an almost complex structure F and a nowhere zero tensor field E E Y ' - ' ( M ) . Then the induced almost complex structure J is integrable if and only i f , f o r X , Y E F 1 ( M ) ,

R ( X , Y ) = 0 , TrE = 0 , P.rF = 0 , T.y--------- e <E,E>

3 20

=0

,

363

ALMOST COMPLEX STRUCTURES

Proof. Let N be the Nijenhuis 2-form on T'M with values in F ( T r M ) , defined by

N(W1, W,) =

[Wit

W2l

+ J [ J W , , + J [ W i ,JW2l W2l

-

[ J W , ,JW2l

for W,, W , E P ( T r M ) . Then J is integrable if and only if N = 0. Suppose N = 0. Then for X , Y E P ( M ) ,N(X;., YY3)= 0 . Hence, putting W , = Xi,W 2 = Y ; we have, from (9), (12). (13), and the definition of J ,

R T X T ) = J(Pl.(E 0X))" - J(T.y(E O Y))" - [ X , Y ] "

since r has zero torsion. Now since E O F ( M ) is a subspace of .Yi(M) there is a unique T E P ( M ) orthogonal to this subspace and a unique Z E Y ' ( M ) such that

(V1,E)OX - ( P , E ) @ Y = T

+EOZ.

Then from (19) and (20) _ _

R ( X , Y ) = TI

-

Z" .

-~

Since R ( X , Y ) is vertical, 2'1= 0 and hence Z = 0. It follows from Corollary 11 that

(21) (22)

0,

R(X,Y ) T=O.

We thus have for all X , Y E Y ' ( M ) ,

( T I E )0Y =

(r,E ) 0X .

Since M is assumed to admit an almost complex structure, dim M 2 2. Hence by choosing X , Y to be linearly independent it follows that (23)

r,E = 0 .

We next consider the case N ( X g , T I ) = 0, where X i Then from (9), (12) and the definition of 1 we have (24)

It follows that

J ( ~ , T )=I

(r,T ) ' E V 1 . Choose

E

V' and TI

E

VI.

(r,Q .

T = S 0Y where S E Y r - I ( M ) , Y EF ( M )

321

A . J. LEDGER S: K . YANO

364

and < S, E > = 0 (since M is paracoinpact such an S exists and can be chosen to be non-zero in a neighbourhood of a point). Then by Lemma 5, TI'E V l and (24) imply that

( r , ~o ) FY t s o Fr,Y

=

( r , qo FY + s o r,(FY) .

Hence sB(r,F)Y

0 ,

and it follows immediately that (25)

T,F

1

0

.

Finally, from Lemma 5 the condition ( r , T ) ' \ E V l implies that 0 = e(T,S) = - ( r , e ) S

(26)

But S is any tensor field which satisfies (27)

where

I ,

S,

E

.

> = 0.

Hence we deduce that

r,e = n(X)e , (Y

E

F 1 ( M ) . Then

N I$

determined by

Thus (28)

N

= tl log e ( E ) = cl log

:<

E, E

>.

r

is the Riemannian connection associated with g then (23) implies (27) e and cy = 0.) Hence, from (27) and (28), the tensor field has zero <E,E> covariant derivative, This proves the necessity of the conditions in Proposition 3 . To prove the sufficiency we note that (If

N ( X " , YV,)= N ( Y " , X h ) = JN(Y",, X " ) , N(X",, T") = JN(T", X " ) , N(Tp, T,")= 0

Thus N = 0 if N(X",, Y;) = N(X",, T") = 0.

,

Suppose TsE = 0 and

R ( X , Y ) = 0 for all X , Y E Y1(M). Then N ( X k , Y;) = - J [ X " , Y k ] - J [ X & , Y h ]- [X',, Y " ] = ( r y Y ) " - ( r , X ) h - [ X , Y]h = 0

322

.

365

ALMOST COMPLEX STRUCTURES

e = 0. The:i (27) follows and hence if T” E I/’ then <E,E> (r.yT)i’E V l . If we next assume r,F = 0 then we have

Suppose V.y-

~

N ( X g , T ” )=

(r.yT)‘ - J(r.yT)v= 0 ,

which proves the sufficiency.

7. Kahlerian structure on T ’ M We now determine necessary and sufficient conditions for the metric 2 on T I M , defined in € j 5 , to be Kahlerian with respect to J . Proposition 4. g is Hermitian with respect to J if and only if < E , E > = 1 and g is Hermitian with respect to F . Proof. Suppose g is Hermitian with respect to J . Then for X , Y EF ( M ) ,

:X , y >

r

Y l X ’ t , y’I > = -1 JXZ., JYZ. i =

--

= <[email protected]@Y;

( . -

# ’

>

X>,Y k

<E,E,”X,Y>~~T.

= 1 . Now let p E M and let S E Y-’ - ‘ ( p ) be non-zero such Hence ,E , E that < S , E > = 0. Then by Lemma 5 and the definition of J we have, for X , YE 7 l ( p ) ,

S,.S>>f X , Y ‘

x =

.\%X,S@Y

=

(S0X ) l , (S 8 Y ) ’

=

S O F X . S @ FY

’

J(S @

~

,T =

X)Ij.

.Y,S

r

J ( S @ Y ) ”>,

F X . FY

>

J

;r

.

Thus at p , < X , Y = F X , FY . . Since p is arbitrary, g is Hermitian with respect to F . The s u t k i e n c y of the above conditions is easily proved by the same method. Proposition 5. Slippose 2 is Hermitian with respect t o J . Then 2 is Kahlerian with respect t o J if und only if r is the Riemnrinian connection ussociatetl witli g , R = 0, T E = 0 unti T F : 0. Proof. Let N be the field of 2-forms on T r Mdefinedfor all W,,W, E 3 ’ ( T r M ) by n(W,. W,) = < W , , J W 2 >. Then 2 is Kiihlerian with respect to J if and only if N is closed and J is integrable [6, Chapter VII]. As usual it is sufficient to consider the action of N and d r y on the three distributions H , F‘>: and I/’ on T r M . Then for X , Y E F 1 ( M )and 7:, TI E V 1 we have

rU(XL, Y L ) = n(X’1, Y ” )= tu(T;‘,XYJ (29)

~(xl;, Y ” )= cu(Ty. Ty) =

*<

E@X,E @Y

< T , ,T , >

:z

,

323

= N(T:’,

X’t) = 0 ,

> -’ :: = < X , Y >

I

i~

,

366

A.

J. LEDGER & K. YANO

Suppose g is Kahlerian with respect to 1. Then by Propositions 3 and 4, R = 0, PsE = 0, and B,e = 0, for all X E Y ' ( M ) . Let p E M ,X E Y1(p), and choose T E F - l ( M ) such that < T , E > = 0 and < T , T > = 1 on some neighbourhood U of p . Since R = 0 parallel vector fields Y and Z exist on U with arbitrary initial values at p . Then using (9), (12) and Lemma 5 we have, on .-I@), 0 = &((T

0 Y ) " ,(T 0X ) " , X " )

< T @ Y , T @ F X > + < T 0FY, Bs(T 0Z ) > - < V.I(T 0Y ) , T 0FZ > = X < Y , F Z > + 2 < T,P.iT > < F Y , Z > + < F Y , r,z > - < B ~ YFZ , > = ( P , p ) ( Y ,FZ) - 2 < T , P \ T > < Y , FZ > . =X

Since F is non-singular it follows that Bsg = n(X)g

9

for some a E ,TI@), Then since P,E = 0 and Bse = 0 it follows easily that for all X E F ( p ) , 0 = Vse = ( r - l)cu(X)e

.

The tensor e is non-zero and so n = 0. Thus Vg = 0 at p and hence on M since p is arbitrary, It follows that P, having no torsion, is the Riemannian connection associated with g . We now prove the sufficiency of the above conditions by showing that the 2-form n is exact. Let X E Y1(M), and Ti' E V l , Define a 1-form?! , on TrM as follows: at each point S E T r M , P(X") = <S, E O X > ,

j ( X ; ) = 0, /?(T")= & < S , T >

.

Then using (29) it follows after some calculation that LY = d p . Hence d n = 0, and this together with Proposition 3 proves the sufficiency. 8. Integrability of H Proposition 6. H

+ V" and H + Y-L

+ V Eis integrable if and only i f R = 0 and for X Ep ( M ) ,

P x E = a ( X ) E , where n ( X ) = < E , V s E > <E,E> * Proof. It follows from (12) and (13) that H tion if and only if for X , , X , E Y1(M), (31)

f

(BXI(E0XJ)" E V's ,

3 24

V Eis an integrable distribu-

367

ALMOST COMPLEX STRUCTURES

R ( X , , X,)

E

VE

.

Let Y , and Y , be orthogonal vectors at p c M , and let < T , E > = 0 at p . Then from (16), (32) and Corollary 6, 0 =
0Y , >

= < T , T >

.

Hence R ( X , , X , ) Y , = cY, where c is some real number which depends on X , and X , . Since Y , is arbitrary -~ it follows that R ( X , , X , ) = cl at p . Then at any point S E n-’(p> we have R ( X , , X , ) = -crS”, and by choosing S” E V 1 it follows that R ( X , , X , ) = 0 at S; hence c = 0. Since p , X , and X , are arbitrary we have R = 0 on M . Using (30) and Lemma 5 we obtain FAYE = n ( X ) E and N is then uniquely determined by this equation. The proof of the sufficiency is immediate, Proposition 7. H V 1 is integrable if and only if R = 0 uncf for X EP ( M ) , < e , Pse > V,ye = a ( X ) e , where LY = _____

+

<e, e >

Proof. The proof is similar to that of Proposition 6 and we shall use the same notation. It follows from (12), (13) and Lemma 5 that H + V-L is an integrable distribution if and only if for S“E V l ,

(r,l(s

(33)

oX N

E v1

,

R(X,,X?) E V l .

(34)

then from (16), (34) and Corollary 6, 0 = < R ( X , , X,)(E 0 Y , ) , E

0Y , >

= < E , E > < R ( X , , X 2 ) Y , ,Y , >

.

Hence, as before, R = 0. From (33) we obtain

o = < x , ,Y > for Y E P ( p ) . Hence

o =

=

e(r,,s) = -(rlle)s.

It follows that r , , l e = n ( X , ) e at p . Since p and X , are arbitrary we obtain r , , e = cu(X,)e on M , and N is then uniquely determined, The proof of the sufficiency is immediate.

325

368

A. Y. LEDGER S. K. YANO

References P. Dombrowski, On the goenietry of tarigent brtridles, J. Reine Angew. Math. 210 (1962) 73-88. S. Helgason, Diflerential geometry arid symmetric sprrces, Academic Press, New York, 1962. S.Sasaki, On the differentialgeometry of tarigerzt brindles of Rierrimriirrrz r~ioiiifolds, TBhoku Math. J. 10 (1958) 338-354. A. J. Ledger & K. Yano, The tnrzgerit brrridle of a locally syrnnietric space, J. London Math. SOC.40 (1965) 487-492. S. Tachibana & M. Okumura, 011the rrlniost complex strrrctrrrc of tangent brrridles of Riemarirzicrri spaces, TBhoku Math. J . 14 (1962) 158-161. K. Yano, Diflereritirrl geometry on corriplex arid rrlmost complex spaces, Pergamon, New York, 1965. K. Yano & S . Ishihara, Almost complex strrrctrrrrs iritlrrced i r r torigeril brindles, K6dai Math. Sem. Rep. 19 (1967) 1-27. K. Yano & S . Kobayashi, Puolorigatiaris of terisor fields arid coririectiorrs to trrrrgerit brirrdles I, J. Math. SOC.Japan 18 (1966) 194-210. K. Yano & A. J. Ledger, Lirieor eoririectioris ori trrngerzt birridleJ, J. London Math. SOC.39 (1964) 495-500. K . Yano & E. M. Patterson, Vertical aritl corripletc lifts from ( I m~riifolr/ta its cutrrrigerit brrridle, J. Math. SOC.Japan 19 (1967) 91-1 13.

UNIVERSITY OF LIVERPOOL TOKYO INSTITUTE OF TECHNOLOGY

3 26

Reprinted from Transactionsof the American Mathematical Society, VoL 181, 1973. @ 1973 American Mathematical Society.

D I F F E R E N T I A L GEOMETRIC S T R U C T U R E S ON PRINCIPAL TOROIDAL B U N D L E S BY DAVID E. BLAIR, G E R A L D D. LUDDEN AND KENTARO YANO ABSTRACT. Under a n a s s u m p t i o n of regularity a manifold with an f-struc-

ture s a t i s f y i n g c e r t a i n c o n d i t i o n s a n a l o g o u s to t h o s e of a K i l l e r s t r u c t u r e a d m i t s a fibration a s a principal toroidal bu.idle over a K i l l e r manifold. In some natural s p e c i a l c a s e s , additional information about t h e bundle s p a c e i s obtained. F i n a l l y , curvature r e l a t i o n s between t h e bundle s p a c e and t h e b a s e s p a c e a r e studied.

L e t M Z n t s be a C"

manifold of dimension 2n

+ s.

If t h e structural group of

M Z n t s i s reducible to U ( n ) x O ( s ) , then M Z n t s i s s a i d t o h a v e an f-structure o/

rank 272. If there e x i s t s a set of 1-forms { q ' ,

- . , q"I

s a t i s f y i n g certain proper-

t i e s described in $ 1 , then M 2 n + s i s s a i d to have a n f-structure with complemented lrame.7. In [I1 it w a s shown that a principal toroidal bundle over a Kahler manifold with a certain connection h a s an /-structure with complemented frames and dv' = . . = dqs a s t h e fundamental 2-form. On the other hand, the following theorem i s proved in $ 2 of t h i s paper. T h e o r e m 1. L e t M 2 n t S be a compact connected mani/old w i t h a regular nor-

mal /-structure. T h e n M 2 n + s is the bundle space o/ a prmczpal toroidal bundle over a complex mani/old N2" (= MZntS/m). Moreover, i f M Z n + = I S a K-manifold, t h e n N 2 n I S a Kahler mani/old. After developing a theory of submersions in $ 3 , we d i s c u s s in $ 4 further properties of t h i s fibration in the cases where d v x = 0, x = 1, . , s and d q X = u X F , F being t h e fundamental 2-form of t h e /-structure.

- .

Finally in $ 5 we study t h e relation between the curvature of M 2 n + s a n d N2".

Since U ( n ) x O ( s ) C O ( 2 n + s ) , M Z n t S i s a new example of a space in the c l a s s provided by Chern in h i s generalization of Ka'hler geometry [4]. S. I. Goldberg's paper [ S ] a l s o s u g g e s t s t t c study of framed manifolds a s bundle s p a c e s over Ka'hler manifolds with parallelisable fibers.

1. Normal / - s t r u c t u r e s . Let M2"+s be a 2n + s-dimensional manifold with a n /-structure. Then there is a tensor field / of type (1, 1) on M L n t S that is of rank R e c e i v e d by t h e e d i t o r s January 10, 1 9 7 2 and. in r e v i s e d form, April 18, 1972. AMS (MOS) subject classifications (1969). Primary 5 3 7 2 ; Secondary 5 7 3 0 , 5 3 8 0 . K e y uiords and phrases. P r i n c i p a l toroidal bundles, f-structures, Kahler manifolds.

175

327

176

D. E. BLAIR. 6. D . LUDDEN AND KENTARO YANO

2n everywhere and s a t i s f i e s

(1)

/3

If there e x i s t vector f i e l d s

/tx= 0 ,

(2)

+/

tx,x = 1, -

)7X(CY)

=

a;,

*

=

*,

0.

s on M 2 n t s s u c h that

o/=

7f

0,

f2 = - I

t- 1 7 Y @ t Y .

we s a y M Z n t S h a s an /-Structure with complemented frames. Further w e s a y that the /-structure is normal if

(3)

[I,

f1 + d q x @ tx= 0 ,

where [f, / I i s the Nijenhuis torsion of f, It i s a consequence of normality that [tx, 5 1 = 0. Moreover it i s known that there e x i s t s a Riemannian metric g on Y

M

~ satisfying ~ + ~

(4) where X and Y a r e arbitrary vector fields on M 2 n t s . Define a 2-form F on M2n+s

f,

Y

( 51

F ( X , Y)

=

g(x, / Y L

A normal /-structure for which F i s c l o s e d will be called a K-structure and a K-structure for which there e x i s t functions a l , . , as s u c h that aXF= dqx for x = 1, . . , s will be called a n 5-structure.

.

Lemma 1. If M Z n t s , n

> 1, bus an S-structure, then the a x are a l l constant.

Proof. a X F= dqx so t h a t dux A F = 0 s i n c e dF = 0. However F f 0 s o dux = 0 and hence a x i s constant. T h e s p e c i a l case where the a x are a l l 0 or a l l 1 h a s been studied in [ I ] . Also, the following were proved.

Lemma 2. 11 M L n t s h a s a K-structure, the f x a r e Killing vector f i e l d s a n d % fb zs the Riemannian connection o/ g on M 2 n t s .

dqx(X, Y) = - 2 ( v y q X ) ( X ) , Here

*

From Lemma 2, we c a n see that in the c a s e of an S-structure a X / Y

=

- 2Vy4,. [,emma 3. I / M 2 n t s h a s a K-structure, then x

(\lxF)(Y, Z )

=

1 -

2

(qx(Y)dqx(/Z, S) + ~ " ( Z ) r / $ ( . Y ,/ Y ) ) . x

2. Proof of Theorem 1. In Chapter 1 of [ g ] R. S. P a l a i s d i s c u s s e s quotient manifolds defined by foliations, In particular, a cubical coordinate s y s t e m , u " ) ! on a n n-dimensional manifold is s a i d t o be regular with r e s p e c t {U, (u*,

..

328

177

PRINCIPAL TOROIDAL BUNDLES

t o a n involutive m-dimensional distribution if ld(m)/du"l, x = 1, b a s i s of

mm

for every m

E

U and if e a c h leaf of

.-

&

*.

- ,m ,

is a

i n t e r s e c t s U in a t most one

m

m-dimensional s l i c e of { U , ( u ' , , u " ) ] . We s a y i s regular if every leaf of i n t e r s e c t s the domain of a c u b i c a l coordinate s y s t e m which i s regular with re-

m

s p e c t t o 3n. In [$)I it i s proven that if i s regular on a compact connected manifold M , then every leaf of is compact and that the quotient M/% i s a compact differ-

m

m

entiable manifold. Moreover the l e a v e s of

m

are t h e fibers of a C" fibering of

M with b a s e manifold M / m and the l e a v e s a r e a l l C"

isomorphic.

m

W e now note that t h e distribution spanned by t h e vector f i e l d s of a normal f-structure i s involutive. In fact we have by normality

from which it e a s i l y follows that

t,

3n

i s involutive. If

m

- ,ts

tl,

i s regular and t h e vector

fields a r e regular we s a y that the normal /-structure i s regular. T h u s from t h e r e s u l t s of [9] we see that if M Z n t s i s compact and h a s a regular normal /-structure, then M2"ts admits a C" fibering over t h e (2n)-dimensional manifold N 2 " = M Z n t S / N with compact, C" S i n c e the distribution

isomorphic, fibers.

m of a regular normal /-structure

c o n s i s t s of s I-dimen-

s i o n a l regular distributions e a c h given by one of the t x ' s , if M 2 n + s i s compact, tx a r e c l o s e d and h e n c e homeomorphic to c i r c l e s S'. T h e

the integral c u r v e s of tX's

being independent and regular show that the fibers determined by t h e distri-

m

a r e homeomorphic to tori T S . Now define t h e period function A, of a regular c l o s e d vector field X by

bution

X x' (rn)

=

inf11 > O/(exp t X ) ( r n ) =

rnl.

For b r e v i t y we denote A by A x . W. M. Boothby and H. C. Wang [ 3 ] proved 5, that h,(rn) i s a differentiable function on M Z n t s . We now prove the following Lemma 4. T h e functions

Ax a r e constants.

T h e proof of t h e lemma makes u s e of the following theorem of A. Morimoto [7I. Theorem (Morimoto [ i ' ] ) .L e t M be a complex manifold with almost complex

structure tensor

1.

L e t k' be a n a n a l y t i c vector f i e l d on M such that ,l' a n d

a r e c l o s e d regular vector fields. Set p(m) morphic function on M .

=

Proof' of l e m m a . For s e v e n ,

329

A,(m) + P l A

IX

(m). Then p

IS a

1X

bolo-

D. E . BLAIR, G. D. LUDDEN AND KENTARO YANO

178

d e f i n e s a complex structure on M

=

M Z n t s (cf. [6]). It i s c l e a r from t h e normality

that 5,;s a holomorphic vector field. For s odd, a normal almost contract structure ( I , t o q,o ) i s defined where go and qo generically denote one of the [,Is and q r ' s respectively [6]. It i s well known that t h i s structure induces a complex structure J on M = M 2 n t s x S'. Moreover, by the normality, toconsidered a s a -2

vector field on M i s analytic. Then p ( m ) = A,(m) + d-=-lA * ( m ) or p ( ( m , q ) ) = h g o ( ( m , q ) ) + \ / - T A I E o ( ( m , q ) ) , q E S', for s odd, i s a htlomorphic function on M by the theorem of Morimoto. Since M i s compact,

p must be constant, T h u s

A; is constant on M and s i n c e A,((m, q ) ) = A,(m), Ax i s constant on M 2 n t s . L e t C x = A x ( m ) , then t h e circle group Sj of real numbers modulo Cn acts on M2nts by ( t , rn) ( e x p t f x ) ( r n ) , t E R . Now the only element in T S = S: x . x St with a fixed paint in M2"'.' i s the identity and s i n c e M Z n t s i s a fiber s p a c e over N 2 " , we need only show that M 2 n t s i s locally trivial [31. L e t ]Ua]b e a cover of N2" s u c h that e a c h U, i s the projection of a regular neighborhood on

-

-.

M 2 n ' s and let sa: [ l a --+ M Z n t s b e the s e c t i o n corresponding t o u' = constant,

- .', u s = constant.

T h e n t h e maps Y:,

Y a ( p ,t l ,

-,

ts) =

U , x T S + M Z n t s defined b y

(exph

t

* *

+ ts4,))(sa(p))

g i v e coordinate maps for M Z n t s . Finally (cf. [l]) we note that y = ( q ' , , qs) defines a L i e algebra valued connection form on M 2 n t s and we denote by p the horizontal lift with r e s p e c t to y. Define a tensor field J of type (1, 1) on by J X = n,/pX. T h e n , s i n c e the distribution 2 complementary to i s horizontal with r e s p e c t to y ,

..

330

PRINCIPAL TOROIDAL BUNDLES

179

Now define t h e fundamental 2-farm 52 by Q ( X , Y ) = G ( X , I Y ) . T h e n for vector * % ,

f i e l d s X , Y on M Z n t s we have *

z

W

x

%

Y

n*R(S, Y ) = R(n,.Y, n*Y) -= G(n,X',

n,Y)

Thus F = n*52. If now dF = 0, then 0 = dn*a = n*d52 and h e n c e d52 = 0 s i n c e n* i s injective. T h u s the manifold N2" is Ka'hlerian.

3 . Submersions. Let v denote the Riemannian connection of g on M 2 n t s . Since the (s;'s a r e Killing, g is projectable t o the metric G on N 2 " . Then follow* + Y where a s we s h a l l see v i s the ing [8] the horizontal part of v* n Y i s %,

;vx

nX Riemannian connection of G. Now for a n S-structure we have s e e n that

*

* a X / X for any vector field X on M Z n t 5 . By normality f is projectable and the a x ' s a r e constants; thus we c a n write

v+tx X

-4

t

( a c x /= 0)

where H x i s a tensor field of type ( 1 , 1) on N 2 " . W e c a n now find the vertical part of

- . . L

8-

i7X

n Y.

T h u s we c a n write

where e a c h b x is a tensor field of type ( 0 , 2 ) and G(//x.Y, Y ) = h x ( . Y , Y ) . Lemma

5. C X

(;X)

=

0 /or uny w c t o r /ield X on N 2 " , where

operator o/ L i e differentiution in the

tXdirection.

*

.

Proof. We have that g(tYy,n X ) = 0 for y = 1, are Killing, that i s g = 0. From the normality of C X have that g(+AW)=o,

4,

(;XI

y=l,..',s,

i s horizontal. However, Tr*v %+ >x

(;X)

and s o

G'Y)

Y

= 77*[tx, ns

l = [n, F

is vertical.

C X

33 1

i s the

- ,S. By Lemma 2 , the tX /, e x 6Y = 0. Hence, we

=X

and s o

e x

z

, n*n?(I = 0

180

-v-

D . E. BLAIR, G . D. LUDDEN AND KENTARO YANO %

%

Using t h e lemma we s e e that Vc nX Since

tx is Killing, z

0

=

=

X

nX

we have z

%

g ( L txn ,x)= nX -

tXfor any vector

field X on N 2 " .

- g ( [ x , v;xn,~)= - g ( t x , h Y ( X , ,W,) = - h X ( X , x) z

for all X . That i s to say h X ( X , Y ) = - h X ( Y , X ) for all X and Y . Now we have that Y

0=

(6)

'u

v-R X ( Z Y ) - 8%RY GX) - [GX,;;YI 'u

=

77(vxY -

v y x - [x,Y ] ) + ( h X ( X , Y ) - / J X ( Y ,x) f dqx(GX, ; Y ) ) f x

R,

=

n P x Y - V,X -

[x,Y l ) + ( 2 h X ( X , Y ) + dqX(;r?(, > Y ) ) t X ,

where we have used the following lemma. Lemma 6. [ P X , ;Y] = n [ X , Y ] - d$(;X, %

%

%

%

;Y)tx.

%

%

Proof. Since n,[nX, n Y ] = [n,nX, n,n Y ] = [ X , Y ] we s e e that n [ X , Y ] is . - b % the horizontal part of [ R X , n Y ] . By Lemma 2 , we have

z

2dqx(;X, ;Y)

z

'u

=

2 g ( t X ; ,V;rynX

-

V-

RX

;Y)

or

dqx(;x, ;Y)[,

=

C

g(tX,

C X ,;yIKX

=

vertical part of [;x,

;YI.

X

From (6) we s e e v x Y

- V,X - [ X , Y ] = 0 and h X ( X , Y ) = - g d q x ( ; X , ;Y).

Furthermore , Y

XG(Y, Z ) =;Xg(;Y,;Z) = g(;vxY,

=g(b- ;Y, RX

;Z) +&Y,

-

Vz ;Z) nX

'u

nZ)+ g G Y , ;vxZ)= G ( P x Y , Z ) + G(Y, V X Z ) .

Thus, we have t h e following proposition. Proposition.

v

is the Riemannian connection of G on N 2 n .

4. The 5-structure case. Let M 2 n + s , n > 1, be a manifold with a n 5-structure. Then, a s we have seen, there exist constants a", x = 1, . . . , s , such that a X F = dq". We will consider two c a s e s , namely c x ( ~ x= )02and f 0. In the first c a s e each dq, = 0 and by Lemma 2 each txi s Killing, hence the

xx(~.x)2

332

PRINCIPAL TOROIDAL BUNDLES

181

e5 a r e parallel on M Z n t 5 .

Moreover the complemen-

.

regular vector f i e l d s

b"

tary distribution distribution

?

,

(projection map i s

- f 2 = I - qX @ t x )i s

parallel. If now the

is a l s o regular, we have a s e c o n d fibration of M Z n t s with fibers

s

the integral submanifolds L 2 " of

and b a s e s p a c e an s-dimensional manifold

N 5 . T h u s by a result of A . G. Walker [lo] we see that although M 2 n t s i s not

n e c e s s a r i l y reducible (even though it is locally the product of N 2 " and T 5 ) i t

is a covering s p a c e of N 2 " x N S and i s covered by L 2 " x T 5 . In summary we have Theorem 2. I / M Z n t S

regular, then M 2 n t s space

of

IS

is

a s in Theorem 1 with dr]" = 0 , x = 1,

a covering s pace

the fibration determined by

of

N 2 " x N 5 , where N S

Now a s in Theorem 1, s i n c e t h e {,'s,

jectable to P2"".

the base

2. x = I,

. .. , s , are

fibrate by any s - t of them to obtain a fibration of M bundle over a manifold 17"".

. - ., s, and f

is

*"

By normality the remaining

regular, we could

a s a principal T 5 - '

"

I

vector f i e l d s a r e pro-

Moreover they a r e regular on P 2 " + ' ; for if not, their integral

curves would be d e n s e in a neighborhood U over which M Z n t 5 is trivial with compact fiber TS-' contradicting their regularity on M Z n t s . T h u s P2"+' i s a principal T' bundle over N 2 " . Theorem 3. I f M Z n t 5 ,

T X ( a x ) f' 0 , [ h e n

n

> I,

is

a s in Theorem 1 with dqX = a X F and

M Z n t 5 i s a principal T5-'

bundle over a principal circle bun-

dle P2"+' over N 2 " and the induced structure on P2"" is a normal rontaci m c tr i c ( S a s a k i an ) s t ru c t u re.

Proof. Without loss of generality we suppose a s f 0. Then fibrating a s ahove by

. . . , t5-

we have that M Z n t 5 is a principal T 5 - ' bundle over a

principal c i r c l e bundle P2"+' over N 2 " . L e t jection map. By normality P2"+' by

/, [,,

T ] ~are

%

$X=p*/ p x , where

p : M 2 n t 5 --+P2""

denote the pro-

projectable, s o we define

€=p*t,,

4 , [, 7

on

(T]1, . .

.,r]

7jJ(x)=7jJ5(;XY)

p d e n o t e s the horizontal lift with r e s p e c t t o the connection n4

5-

I)

considered a s a L i e algebra valued connection form a s in the proof of Theorem 1. Then by a straight-forward computation we have

7(&1,

Cg-0,

7jJo+o,

$2=-I+5@',

[4,41+&-0dT]=O,

that is, (+, [, T ] ) i s a normal almost contact structure on Ij 2 " + ' . Defining a m e t ric g by g ( X , Y ) = g(F?i', p " Y ) w e have i ( X , [) = T ] ( X ) and k(+X, 4 Y ) = d ( X , Y ) r ] ( X ) r ] ( Y ) . Moreover setting @(X, Y ) = g(X, 4 Y ) we obtain F = p*@. T h u s s i n c e

333

D. E. BLAIR, G. D. LUDDEN AND KENTARO YANO

182

s i n c e qs i s horizontal. T h u s we have that q A(dq)n = V,,(as@)" & 0 and hence regular. that P2"+' h a s a normal contact metric structure with Remark 1. While it i s already clear that P 2 n t 1 i s a principal circle bundle over N Z n , it now a l s o follows from the well-known Boothby-Wang and Morimoto

f ibrations. Remark 2. Under the hypotheses of Theorem 3 , i t is p o s s i b l e to assume without l o s s of generality that a" e q u a l s 0 or l/df where t i s the number of non-

-

zero a" and hence there e x i s t c o n s t a n t s p", q = 1, , s - 1, s u c h that 9 ~ x p q x qand x 5js = ~ x a x qare x 1-forms with d?jq = 0 and d?js = F . T h e n e

e

-

?j9

=

/, 7"

and the d u a l vector fields 6" again d e f i n e a K-structure on M Zn t s . If now - this K-structure i s regular, then, s i n c e the distribution spanned by tl, . . , and i t s complement a r e parallel, M Z n t s i s a covering of t h e product of P2"+' and a manifold P s - a s in the proof of Theorem 2. Remark 3. In [l] one of the authors g a v e the following example of an S-manifold a s a generalization of t h e Hopf-fibration of the odd-dimensional sphere over complex projective s p a c e , T ' : S2"+' 4 PCh. Let A denote the diagonal map and define a s p a c e H 2 n t s by t h e diagram

ts-l

'

that i s H 2 n t s = { ( P I ,.

+ .

, P,)

E

S 2 n t 1x

.. . x

SZnt1(n'(P

=

. . . = n'(P,)I

and

thus H Z n t S i s diffeomorphic to SZntlx Ts-'. Further properties of the s p a c e H ~ are~ given + in~ [ I ] , [21. If however the d q x ' s a r e independent then there c a n be no intermediate bund l e P2"+' over N 2 " s u c h that M Z n t s i s trivial over P 2 " " . Remark 4. If MZnts i s a s in Theorem 1 with the d q x ' s independent, then ther, i s no fibration by s - t of t h e 6"'s yielding a principal toroidal bundle P 2 n over N Z n s u c h that M Z n t s = P 2 n t t x Ts-'. For s u p p o s e P 2 n t f i s s u c h an inter-2 mediate bundle, then it i s n e c e s s a r y that 5, = 0 ( s e e e.g. 181) and thus the l7X qx ' s are parallel contradicting t h e independence of the dq"'s.

+'

v-2

CCI

, l ,

5. Curvature. L e t R a n d R denote t h e curvature t e n s o r s of spectively. T h e n

3 34

v

and Q re-

€8I

S 3 7 a N n f f 1ValOllO.L 1 V d I 3 N I l l d

PRINCIPAL TOROIDAL B U N D L E S

In [I], one of the present authors developed a theory of manifolds with an

183

/-

structure of constant /-sectional curvature. T h i s i s the analogue of a complex manifold of c o n s t a n t holomorphic curvature. A plane s e c t i o n of M Z n t s i s called an /-section if there i s a vector X orthogonal t o t h e distribution spanned by the t X ' s s u c h that

{X,/Xi i s

an orthonormal pair spanning the section. T h e s e c t i o n a l

curvature of t h i s section i s called a n / - s e c t i o n a l curvature and is of course given -.,

by g(R,,,X,

/XI.

Mints

i s s a i d to be of constant f-sectzonal curvature if the

/-sectional curvatures a r e constant for a l l /-sections.

T h i s i s a n a b s o l u t e con-

stant. We then have the following theorem.

Theorem 5 . I / M 2 * +' zs I compact, connected manilold w i t h a regular S-strucof constant / - s e c t l w a l curvutrcre c , then N2" zs a KZhler rnanzfold of constant holomorphic curvature. ture

I S KBhler

. . . , u s ,n e c e s s a r i i y

follows from Theorem 1. By definition there e x i s t

constant s u c h that a X F = d q X . If X is a unit vector on

SEE

Proof. T h a t N 2 "

a',

335

D. E. BLAIR, G. D. LUDDEN AND KENTARO YANO

184

Remark. T h i s a g r e e s with the r e s u l t s in [ I ] on H Z n t s . H Z n t s i s a principal

toroidal bundle over PC" and PC" i s of constant holornorphic curvature equal to 1. Also, a x = 1 for x = 1, . . . , s and H Z n t s w a s found to be of constant 1sectional curvature e q u a l to 1 - 3s/4. REFERENCES 1. D. E. Blair, Geometry of manifolds w i t h structural g r o u p 'U(n) x O(s), J . Differential Geometry 4 (1970), 155-167.

MR 42 #2403. 2. --, On a generalization of the Hopf fibration, An. Univ. "Al. 1. Cuza" l a s i 17 (1971), 171-177. 3. W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. ( 2 ) 68 (1958),

721-734. MR 22 #3015. 4. S. S. Chern, On a generalization of KChler geometry, Algebraic Geometry and Topology (A Sympos. in Honor of S. L e f s c h e t z ) , P r i n c e t o n Univ. Press, P r i n c e t o n , N. J., 1957, pp. 103-121. MH 19, 314. 5 . S. I. Goldberg, A genern[ization of K i i h k r geometry, J . Differential Geometry 6 (1972). 343-355. 6. S. I. Goldberg a n d K. Yano, On normal globally framed f-manifolds, TGhoku Math. J . 22 (1970). 362-370. 7. A. Morimoto, On rlormal almost contact structures with a regularity. T&oku Math. J,. ( 2 ) 16 (1964), 90-104. MR 29 #549. 8. B. O ' N e i l l , T h e fundamental equations of a submersion, Michigan Math. J . 13 (1966), 459-469. MR 34 #751. 9. R. S. P a l a i s , A global formulation of the L i e theory of transformation groups, Mem. Amer. Math. SOC. No, 22 (1957). MR 22 #12162. 10. A. G. Walker, T h e fibring of Riemannian m a n i f o l d s , Proc. London Math. SOC.(3) 3 (19531, 1-19. MR 15, 159. D E P A R T M E N T O F M A T H E M A T I C S , MICHIGAN S T A T E U N I V E R S I T Y , E A S T L A N S I N G , MICHIGAN 48823

336

Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes By Kentaro YANOand Shigeru ISHIHARA

8 1.

Introduction

Let ill be a Riemannian manifold of dimension n 2 3 and of class C". We cover A l by a system of coordinate neighborhoods { U ;P}, where here and in the sequel the indices 11, i, j , k, run over the range { I , &... , n), and denote by q J L ,V % KkjLh, , K j , and K the positive definite metric tensor, the operator of covariant differentiation with respect to the Levi-Civita of A1 connection, the curvature tensor, the Ricci tensor and the scalar curvature respectively. A conformally flat Riemannian manifold is characterized by the vanishing of the Weyl conformal curvature tensor

CXjlh = h - A j L h

+ d:

cji-8; CALf CLhgj&-cjn 9,s

and the tensor C k j = ~

PA cj&-P j C.LL>

where

C,"

=

c,,q f h.

Ryan [4] proved Let A4 be a cottipact conforttially flat Rieniamian manifold THEOREM with constant scalar curvatrwe. If the Ricci tensor is positive senii-dejinitc>, tlien the siniply comected Rienia?inian covering of A f is one of

P ( c ) , R x P 1 ( c )or E", the real space forms of curvature c being denoted by S"(C)or E" dependitig O H zuhether c is positive or zero. (See also Aubin [l]. Goldberg [3], Tani [6]). He first proves that, in a conformally flat Riemannian manifold with constant scalar curvature I;, we have

337

298

where

and then that, if we denote by &(i=1,%,..., n ) the eigenvalues of K j i , then we have

He then assumes that the Ricci tensor K j Lis positive semi-definite and shows that in this case we have P>=Oon M . Thus he obtains d ( K , , t K " ) ~ O ,from which

P=O and V,K,,=O. From these, he obtains the theorem quoted above. W e can easily see that the conclusion of the theorem also applies if the assumptions of compactness and constant scalar curvature are replaced by local homogeneity of M . T h e main purpose of the present paper is to prove the following theorem corresponding to that of Ryan, replacing the vanishing of the Weyl conformal curvature tensor in a Riemannian manifold by that of the Bochner curvature tensor in a Kaehlerian manifold.

THEOREM Let A4 he a Kuelilerian nianz'jold of real diriniemion 12 with constant sculur curvature whose Boch?iei- curvature tcmor vuiiishcs U J ~ C ? whose Kicci tetisor is positive semi-dejiiiite. If A4 is compuct, tlicn thc universal covering naanifold is a coniplex projective spact CP''/' or a cornplex space P I 2 . From the method of the proof we easily see that the conclusion of the theorem is also valid if the assumptions of compactness and constant scalar curvature are replaced by local homogeneity of A t

Q 2. Preliminaries Let M be a Kaehlerian manifold of real dimension n and ( 9 ,F ) its Kaehlerian structure. T h a t is, g is a Riemannian metric and F a complex structure in M such that

338

where g 3 , and FAhare local components of g and F respectively. known that FjL

FJ1

It is well

9,'

is skew-symmetric. As a complex analogue to the Weyl conformal curvature tensor, Bochner (see also, Yano and Bochner [9]) introduced the following curvature tensor in a Kaehlerian manifold :

[a]

Bhj:

(3. 1)

= k'hj,'+~~LJ,-6~LL,+L,'ggf~-Ljkg,r

+ FA

~ 1 1 5' F," AIA I

- 2 ( A I L JF,"

+ FA

j

+ AILt' FjL- A

fj,"

FA

AIth),

where

AlL'

= AlL, g',

Hj, = -hVJt F,"

,

Bochner introduced this curvature tensor using a complex local coordinate system. W e call this curvature tensor the Bochner curvature tensor. T h e form (2. 1) of the Bochner curvature tensor with respect to a real coordinate system has been given by Tachibana [5] (see also Yano [ 8 ] ) . By a straightforward computation, we can prove

( 2 . 3)

V , B,

= - 11

( V , Lj, - 0, LL,) .

When the Bochner curvature tensor vanishes, we have, from (2. l),

+ FinH,, - Fj, HA./+ HkhFj,- Hj,,

- 2 (Hk Fi, +- Fi., H,,)]

- Fjn F k i -

for the covariant components K , tensor.

= K k j jg t n

j

Fdk]

of the Riemannian curvature

$ 3 . Lemmas In this section, we prove some lemmas which will be used in the proof of the theorem.

339

K. Y U N OU I I JS . Ishihut-u

300

LEMMA1. I f the Bochner curvature tetisor vanishes and the scalar curvature is constant in a Kaehlerian manifold, thett w e have V,+Kjd-VjKAC = 0 ,

(3.1)

that is, VkKjt is u sywtietric tens0.r. PROOF This follows from (2.3) and the definition of Lj,. LEMMA 2. If the Bochner curvature tensor of a Kaehleriun nianijold vanishes and the nianifold is an Einstein nranifold, then the Kaehlerici~r rrianifold is of constant hoZoniorphic sectional curvature (see Tachibana [5]). PROOFIf the Bochner curvature tensor vanishes, we have, from (a.1). (3. 2)

h'kjp

Lj, f6: Lkl- I*!,"g Jl+ L," g , , -Fkhj l M + Fj" Mk6 - MA'' Fj, + MjhFkl t 2 ( M ,j F," + FA j Ai','') . = - 6;

If the manifold is an Einstein space, we have

K,6

=

K

Hj6

gj<,

K

=y Fj& 9

and consequently we have, from (2.2),

Thus, substituting these into (3. a),we find

which shows that the manifold is of constant holomorphic sectional curvature. LEMMA 3. In a Kaehlerian manifold of dimension n with consta~rt scular curvature whose Bochner curvature tensor vanishes, we have

+ (V,K$J(VjKe'"). PROOFWe first have 1

( V d(&hKdh)= gkJVk[(VjK6h)Kdh]g k J (vk v1K6,)Kdh+(VjK$,J

that is,

(3.5)

1

7d(K6,K"") = g ~ " v , v * K ~ , ) K " * + ( v , K(ViK"), ~,)

340

j P ) ,

where we have used (3. 1) and K i l L = K h l . On the other hand, applying the Ricci formula to K j L 7we have

V kv,,K j i = v/,v,K j L- K,/,jl K t i- K k , t LKjt f . Substituting this into (3.5), we obtain 1 a>

3

d ( Ki,cP') = y j (V,, v, K j L - K,,<jl K ,i - Kh./(K j t ) P" + (Cj Ki,,) (F' Kit'),

that is, (3. 6)

1

,,

A ( K d / ( P=)K,"/K,.f

-Kh',ji,,K""'i+(vjKi/() ( 8 j P " ) ,

3

where we have used (3.1) and K = const. (2. 4). we find We now compute K k j i h K k t L K JUsing d. (:j.

7)

K k j i h K k h Kji = ~. -~ )l

+ 4 [ga,L KjL-

g j,t

i

+ ~ t ,g ,j i -- Kj,c g k ,

- Fj,,H k i - H j , Fkj--2 (HA.jFild + F,,H,,)] Kk"Kii

On the other hand, we have the following equalities:

Thus substituting this equation into (3. 6), we have ( 3 . 4).

LEMMA 4. (Ryan [4])

171

a Rienumniutz manifold of diniciisioii

341

11,

for

302

(3.8)

K. Yano and S. Ishihara

P = ?1K:K:Kri--

2n-1 1 KK,,Kjt+--K3, n-1 n-1

w e have (3.9)

342

303

PROOFComputing Q-P, we have

$4.

Proof of the theorem

Assume that, i n a Kaehlerian manifold .\I of dimension 11, the scalar curvature is constant, the Bochner curvature tensor vanishes and the Ricci tensor is positive semi-definite. Then we have h'20, and consequently, by Lemmas 3, Fi and 6, we have

(4.1)

.t J(h',/,K

")

2

+(rjh-,,) wi-"') 20. is compact, then we have J ( K L , , P h ) I herefore, if we assume that =O, from which and (4.l'i, if K+0, we have I .

(4.2)

h'.= h JI

gJ,

3

that is, Allis an Einstein manifold. Thus, by Lemma 2, the Kaehlerian manifold Jl is of constant holomorphic sectional curvature. If K=O, then we have

343

1, = &

... =i.,,ZO and

because of

=

. . . = A,,= ()

consequently

h-,; =0 , from which

Lj, = 0 ,

,\Ijj

=0

.

r .

I hus. the Rochner curvature tensor being zero, the curvature tensor KA,," vanishes, and consequently the Kaehlerian manifold A1 is of zero curvature. Thus the universal covering manifold of ,I1 is a complex projective space CI"' or a complex space C" '. Thus the theorem stated at the end of the introduction has been completely proved. T c i L \ (1

In\titute of 'I e c l i n o l o p

Bibliography [ 11

'r, r

[2j

stante, C. I<. :\cad. Sci. Paris, 266 (1968), 422-423. S. H O C H K E K : C'urvature and I k t t i numbers, 11, r\nnals of Math., 50 (1949),

\

~

~: ~ S 3u r ~ les ~

vari&t&s kiihl6riennes compactrs ii courl,urc sciilnire coii-

77-93, [3]

s. I.

[5j

S. 'I'~\CHlBAK.4

(;OLDBEKG : O n conformally flat spaces

bvitI1 definite Iticci curvature, Iiac h i Math. Sem. Rep., 21 (1969), 226-23'2. [ 4 ] P, 1 . ]
t o appear. : O n the I3ochner curviiturc tensor, Natural Science I<epori, Ochanomizu I;iiiversity, 18 (l967), 15-19. [ 6 1 14. ' ~ ' . \ N I : O n ii conformally f l a t Riemannian space \villi positive Ricci c u r v a ture, 'I'Alioku Math. .I.,19 (1967), 227-231. K. Y A N O : 1)ifferenti;il geometry o n complex and almcist complex spices, Per[7] #anion Press, 1965. J i n K d a i Math. Sem. [ 8 ] I<. YANO: O n c o m p l e s cunforninl connections, ~ C appear I<ep [ 9 1 I<. Y A X O a n d S. I ~ O C H S E R : Curvature a n d Retii numbers, Annnls of Math. Studies, "I. 32, 1953. i

Krceivetl January 6, 19741

344

J. Math. SOC. Japan vol. 32, No. 1, lY80

Notes on infinitesimal variations of submanifolds By Kentaro YANO (Received April 1, 1978)

§ 0.

Introduction.

In a previous paper [ 5 ] , the present author studied variations of the metric tensor, the Christoffel symbols and the second fundamental tensors of submanifolds of a Riemannian manifold under infinitesimal variations of the submanifolds. In this paper, we assume that submanifolds under consideration are compact and orientable and we obtain, using integral formulas, some global results on infinitesimal isometric, affine and conformal variations of the submanifolds.

3 1. Preliminaries [l]. We consider an m-dimensional Riemannian manifold A I m covered by a system of coordinate neighborhoods { U ;x h } and denote by g,,, and V, the metric tensor, the Christoffel symbols formed with g,, and the operator of A i m respectively, where, of covariant differentiation with respect to here and in the sequel, the indices h , 1, j , k , ... run over the range {l',2', ... , m'}. We then consider an n-dimensional compact orientable Riemannian manifold A i " covered by a system of coordinate neighborhoods { V ;y"} and denote by g c b , v,, K d c b " and I ( c b the metric tensor, the Christoffel symbols formed with gcb, the operator of covariant differentiation with respect to c h , the curvature tensor and the Ricci tensor of Ad" respectively, where, here and in the sequel, the indices a, 6,c, run over the range 11, 2, ... , ? I } . We assume that hin is isometrically immersed in AIm by the immersion: hI"-Al" and represent the immersion by

r:,

r:,

Xh=X"(ya).

Since the immersion is isometric, we have

(1.1)

gcb=Bc'BbigJt9

where we have put BcJ=dcxJ (ac=a/ayc). We can assume that [ B b h ] gives the positive orientation of hi".

345

46

K. YANO

We choose m--n mutually orthogonal unit normals C V hto M", where, here and in the sequel, the indices x, y, z run over the range { n f l , ... , m } . The metric tensor of the normal bundle of M" is given by

(1.2)

gzy=CzjCyigji. Now, the equations of Gauss for hf" are written as

where vc Bb h=

acB bh

+r j l B

Bbt- r,",B a

is the van der Waerden-Bortolotti covariant derivative of Bbh and hcbZ are components of the second fundamental tensor with respect to the normal C Z h . On the other hand, the equations of Weingarten for Ail" are written as

(1.4) where

VcCyh=-hcayBah,

vcc,h=accyh+r;& B c~cyt -r;vc,h

r&

is the van der Waerden-Bortolotti covariant derivative of C,", being components of the linear connection induced in the normal bundle, that is,

rAJ=(a,cy+r:L B c ~ i)c= cy h

h

and Cxh=CyEgyagph,g y x being contravariant components of the metric tensor of the normal bundle of 121" and hcay=hcbzgbagzy, gba being contravariant components of the metric tensor of

M".

$ 2 . Infinitesimal variations C23 [ S ] [ 5 ] . We now consider an infinitesimal variation of hP given by

(2.1)

Xh=xh+E"y)&,

where Eh(y) is a vector field defined along M" and E is an infinitesimal. Under the infinitesimal variation (2.1) the vectors B b h tangent to h i'" are transformed into Bbh=abzh=Bbh+abEh& tangent to the deformed submanifold. Carrying B b h at ( f h ) back to ( x h ) parallelly, we obtain gbh=Bbh+r3i(

X+E&)E'Bbt&

that is,

346

,

Infinitesimal variations

47

neglecting terms of order higher than one with respect to

E,

where

In the sequel we always neglect the terms of order higher than one with respect to E . Thus putting o"Rbh=E b l l - Bbh, we have

aI!?","=(V,f")&.

(2.4)

If we decompose (2.5)

thas ;'"=:" B , +E"C,h ,

equation (2.4) can be written as

where Vb=gbaV, and libax=hedrgEbgda. When 6gc6=0, we say that the infinitesimal variation is isometric and when dgcb=2Rpcbs,R being a certain scalar, we say that the infinitesimal variation is conforiiial. If the variation is conformal and R is a constant, we say that the infinitesimal variation is Iionzothetic. From (2.7) we have THEOREM A. [5] h i order f o r a n infinitesimal variation (2.1) o f a submanifold to be isometric, it is necessary and sujficieizt that

(2.9)

+

V c b VbtC-2 h C b X [ " =0

.

THEOREM B. [5] 112 order for a n infinitesimal variation (2.1) o f a submanif o l d to be conformal, it i s necessary and suficient that

(2.10)

VcEbtVb:c-2}2cbxE3=2Rgcb

9

A being a certain scalar. Using (2.7) and (2.8), we calculate the infinitesimal variation of the

347

48

K. YANO

Christoffel symbols 1

r?b=2(acgbe+ abgce-aegcb)gea and obtain (2.12)

6 l ' ~ b =1, C v c ( a g b e ) + v b ( s g ~ , ) - ~ e ( ~ ~ c b* ) l ~ ~ "

which, using (2.7), we car, write as

(2.13)

6fibb=CVcVbEa+ K d ~ b ' [ ~ - V c (hbaz[")-Vb( hc"xE")+V"( hcbzez)]s

-

If 6r,4,,=0, we say that the infinitesimal variation is affine and if =(6:pb+8$pc)& for some 1-form PO, we say that the variation is projective. T h u s we have THEOREM C. [ 5 ] In order for an infiiiitesiinal variation (2.1) o f a subinanifold to he af/ine, it is necessary and suficient that

(2.14)

+

VcVbE" KdCb"td-Vc( h b a z E z ) - v b ( hc"z[")+

vQ(/lb x E x ) = O L

*

Now we have the following integral formula C41

which is valid for an arbitrary vector field E" in a compact orientable Riemannian manifold M", dV being the volume element of the manifold. From (2.15), we can easily derive

1

c-211CbZE3)

+ ( v c ~ b + v b ~ c - ~ ~ c b ~ ~ " ) h c b ~ ~ z

-(Vc;c-hccxE")(Vb~b)]dI/=O,

which is valid for arbitrary and Ex. Now suppose that an infinitesimal variation (2.1) of the submanifold is isometric. Then since it is affine, we have (2.14), from which, we have, by transvection with gcb,

(2.17)

gcbvcVbEaf Kd"Ed-2Vc( h,",E")+ V a (h C c , ~ " ) = O .

We also obtain from (2.9),

348

49

Injnitesimal variations

(2.18)

(VCE~+V~EC-~~C~~E')~~~"E"=~

and (2.19)

(VcEC-

h ccz[J)(vb[b)

=o .

Conversely if (2.17), (2.18) and (2.19) are satisfied, we have, from (2.16), VcEbfVb~~-2~~cby[~=~ t which shows that the infinitesimal variation is an isometry. Thus we have

THEOREM D. [ 5 ] I n order f o r a n infinitesimal variation o f a coinpact orientable submanifold o f a Riemannian manifold to be isometric, it is necessary and sziflcient that we have (2.17), (2.18) and (2.19). $ 3 . Infinitesimal isometries. Suppose that the infinitesimal variation (2.1) is an isometry. have (2.17). T h u s substituting (2.17) into

from which, by integration over

M",

or

Since (2.1) is an isometry, we have

Thus substituting these into the above equation, we find

From (3.2), we have

349

Then w e

K. YAso

50

THEOREM 3.1. I f an iizfiiiitesimal isometric variation o f a compact orientable submanifold M" of a Riemannian inanifold M" satisjies (3.3)

~ ~ c b E c ~ b + 2 ( ~ ~ c b " , c b x9 ) ~ y ~ x ~ ~

6" satisfies

then

VcEa=O and cosequently KcbEcEb=O and hcb.dx=O, that is, IW is geodesic in the direction E x . Moreover i f izil" is irreducible, then ["=O, that is, the variation is normal and the submanifold is geodesic i n the direction o f the variation.

8 4. Infinitesimal affine variations. For an infinitesimal affine variation (2,1), we have (2.14), from which, by transvection with gcb, we obtain

~cbVc~~~afK~a~d-~VC(hcaS~")+Va(hcc,~~)=O

(4.1)

and, by contraction with respect to a and b, we have

Vc(V,E" - h,",~")=O

(4.2) and consequently we obtain

VaEa- haa,Ex=constant.

(4.3) Thus (2.16) becomes ~

[1 ~

(

~

~

~

~

+

~

~

~

~

-

Vbfc-2 h c b y Ey=o

.

then we have, from the equation above, (4.5)

0cfb-t

The converse being evident, we have

3 50

~

h

c

b

y

~

~

)

(

~

~

~

~

~

~

~

Injnitesimal variations

51

THEOREM 4.1. I f an infinitesiinal aflne variation o f a compact orientable submanifold o f a Rieiizannian manifold satisfies (4.4)then the variation i s a n isornet ry.

Q 5. Infinitesimal conformal variations. If an infinitesimal variation (2.1) is conformal, we have (5.1)

vctb+v b t c -

2 h cbtt" =21gcb

for a certain function I , from which, transvecting with gcb, we find (5.2)

1 I =-(VaEQ

-

71

h .",~").

Thus we can write (5.1) a s

n-2 +--Vn(VeEen

heez[")=O ,

Now, we can transform (2.16) into

351

K. YANO

52

Thus if the infinitesimal variation is conformal, we have (5.3) and consequently (5.8) ' ( V c E b f v b ~ ~ - 2 h c b 2y ~ ~ - ~ ( /Iee,EV)gc*}/~",i"=o V e ~ ~ and also (5.6). Conversely if (5.6) and (5.8) are satisfied, we have from (5.7)

2 V e ~ b f V b ~ ~ - 2 ~ ~ c b y ~ ~ - - ( V e E ~ - h e ~ y E * ) g c tb = O 11

which shows that the infinitesimal variation is conformal. Thus we have THEOREM. 5.1. In order f o r an infinitesimal variation (2.1) to be conformal, it is necessary and szificient that (5.6) and (5.8) hold. Substituting (5.6) into (3.1), we find

1 -2 ~ ( $ E U ) = - ~ C ~ ~ ~ ~ ~ + ~ ~ Q V ~ ( ~ C " X E " ) - ~ U ~ ~ ( ~ C ~

_ _n -2 - EuVa(Ve4e-heexSx)+(VcEb)(VcSb) n

*

from which, integrating over M",

![

-Ir,*EcEb-(hcb",E")(vc~b+V,E,)f

hcCx4"(vaEu)

+~ ~ ( ~ ~ E ' - h ~ e , ~ " ) ( ~ Q E Q ) + ( ~ C ~ b ) ( ~ C ~ b ) ] d ~ = o Since the variation is conformal, we have

2 n

VrSb+vb~c=2hcbxEx+-(VeEe-

heedX)gcb

Substituting this into the above integral formula, we find

![ -KcbEESb-2hrbgE'

2 n

hcbxf"--hccy,EY(V,E"-Ilee~EZ)+

hc'&"(vaE")

+ "i (VeEe--hee~,E")(v~EQ)+(aP")(VcEb)]dV=O, "

352

Injnitesimal variations

from which (5.9)

Thus from (5.9), we have THEOREM 5.2. I f a n infinitesimal conformal variation of a coinpact orientable subiiianifold M" of a Rieiiiannian inaiiifold M" satisjes (5.10)

that is, M" is unibilical i n the direction i". Moreover if i\.I" is irreducible, then Ea=O, that is, the variation is normal a n d the szibrnanifold is unibilical in the direction o f t h e variation. Bibliography [ 1 ] Bang-yen Chen, Geometry of Submanifolds, Marcel Dekker, Inc., 1975. [ 2 ] J. A. Schouten, Ricci-Calculus, Springer-Verlag, 1954. [ 3 1 K. Yano, S u r la thkorie des dkformations infinitksimales, J. Fac. Sci. Univ. Tokyo, 6 (1919), 1-75. [4] K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, Inc., 1970. [ 5 1 K. Yano, Infinitesimal variations of submanifolds, Kodai Math. J., I (1978), 30-44.

Kentaro YANO Tokyo Institute of Technology Meguro-ku, Tokyo 152 Japan

353

This Page Intentionally Left Blank

J. DIFFERENTIAL GEOMETRY 16 (1981) 137-145

CR-SUBMANIFOLDS OF A COMPLEX SPACE FORM AUREL BEJANCU, MASAHIRO KON & KENTARO YANO Dedicated to Professor Buchin Su on his 80th birrhaby

0. Introduction The CR-submanifolds of a Kaehlerian manifold have been defined by one of the present authors and studied by him [2], [3] and by B. Y. Chen [4]. The purpose of the present paper is to continue the study of CR-submanifolds, and in particular of those of a complex space form. In $1 we first recall some fundamental formulas for submanifolds of a Kaehlerian manifold, and in particular for those of a complex space form, and then give the definitions of CR-submanifolds and generic submanifolds in our context. We also include Theorem 1 which seems to be fundamental in the study of CR-submanifolds. In $2 we study the f-structures which a CR-submanifold and its normal bundle admit. We then prove Theorem 2 which characterizes generic submanifolds with parallel f-structure of a complex space form. In $3 we derive an integral formula of Simons’ type and applying it to prove Theorems 3, 4 and 5.

1. Preliminaries Let 2 be a complex m-dimensional (real 2m-dimensional) Kaehlerian manifold with almost complex structure J , and M a real n-dimensional Riemannian manifold isometrically immersed in M.We denote by ( , ) the metric tensor field of M a s well as that induced on M . Let 7 (resp. V ) be the operator of covariant differentiation with respect to the Levi-Civita connection in (resp. M ) . Then the Gauss and Weingarten formulas for M are respectively written as

-

V,Y = V,Y

+ B(X, Y),

Communicated May 5, 1979.

355

V , N = -A,X

+ DxN

138

A W L BEIANCU, MASAHIRO KON & KENTARO YANO

for any vector fields X , Y tangent to M and any vector field N normal to M , where D denotes the operator of covariant differentiation with respect to the linear connection induced in the normal bundle T ( M ) I of M . Both A and B are called the second fundamental forms of M and are related by ( A , X , Y ) = (B(X, Y),N > . For any vector field X tangent to M we put

+

JX = PX FX, (1.1) where PX is the tangential part of J X , and FX the normal part of J X . Then P is an endomorphism of the tangent bundle T ( M ) of M, and F is a normal bundle valued 1-form on T ( M ) . For any vector field N normal to M we put JN=tN+fN, (14 where tN is the tangential part of JN, andfN the normal part of J N . If the ambient manifold is of constant holomorphic sectional curvature is called a complex space form, and will be denoted by G"'(c). c, then Thus the Riemannian curvature tensor of @"(c) is given by

z

R ( X , Y ) Z = i c [ ( Y , Z ) X - (X,Z ) Y -(JX, Z ) J Y

+ (JY, Z)JX

+ 2(X, J Y ) J Z ]

for any vector fields X , Y and Z of H"'(c).We denote by R the Riemannian curvature tensor of M. Then we have R ( X , Y ) Z = ~ c [ ( YZ, ) X - ( X , Z ) Y (1.3)

( 1.4)

+2(X, PY)PZ]

+ (PY, Z)PX

-

(PX, Z ) P Y

+ AB(Y,Z)X - AB(X,Z)Y,

(VXBN y , Z ) - ( V Y B ) ( X ,2) = f c [( P Y , Z ) F x - ( P X , 2 ) F Y

+ 2(X, P Y ) F Z ]

for any vector fields X , Y and Z tangent to M . If the second fundamental form B of M satisfies the classical C o w equation (VXB)(Y , 2 ) = (V,B)(X, Z ) , then (1.4) implies (cf., [l, p. 4341) Lemma 1. Let M be an n-dimensional submanifold of a conlplex space form @(c), c # 0. If the second fundamental form of M satisfies the classical Codazzi equation, then M is holomorphic or anti-invariant. Definition 1. A submanifold M of a Kaehlerian manifold M is called a CR-submanifold of if there exists a differentiable distribution 9: x + c T X ( M )on M satisfying the following conditions: (i) 9 is holomorphic, i.e., J q X = qxfor each x E M, and l:' x + 9;c T X ( M )is (ii) the complementary orthogonal distribution 9 anti-invariant, i.e., J 9 : c T X ( M ) lfor each x E M.

356

CR-SUBMANIFOLDS

139

If dim 9," = 0 (resp. dim qx= 0) for any x E M , then the CR-submanifold is a holomorphic submanifold (resp. anti-invariant submanifold) of M.If dim 9 ; = dim Tx(M)* for any x E M , then the CR-submanifold is a generic submanifold of M (see [9]). It is clear that every real hypersurface of a Kaehlerian manifold is automatically generic submanifold. A CR-submanifold is called a proper CR-submanifold if it is neither a holomorphic submanifold nor an anti-invariant submanifold. From Lemma 1 we have Proposition 1. Let M be a proper CR-submanifold of a conplex space form M"(c). If the second fundamental form of M satisfies the classical Codazzi equation, then c = 0. A submanifold M is said to be minimal if trace B = 0. If B = 0 identically, M is called a totally geodesic submanifold. Definition 2. A CR-submanifold M of a Kaehlerian manifold is said to be mixed totally geodesic if B ( X , Y ) = 0 for each X E 9 and Y E 9*. Lemma 2. Let M be a CR-submanifold of a Kaehlerian manifold Then M is mixed totally geodesic if and only if one of the following conditions is fulfilled : (i) A,X E 9for any X E 9and N E T ( M ) I , @ ) A N Y E 9L forany Y E 9*andN E T ( M ) * . The integrability of distributions 9 and 9I on a CR-submanifold M is characterized by Theorem 1. Let M be a CR-submanifold of a Kaehlerian manifold Z. Then we have (i) 9 '-is always involutiue, [4], (ii) 9 is involutiue i f and only i f the second fundamental form B s tisfes B(PX, Y ) = B ( X , P Y )for all X , Y E 9, [2]. Definition 3. A CR-submanifold M is said to be mixed foliate if it is mixed totally geodesic and B(PX, Y ) = B ( X , P Y ) for all X , Y E 9. Now, let M * be a leaf of anti-invariant distribution 9l on M . then we have Proposition 2. A necessary and sufficient condition for the submanifold M * to be totally geodesic in M is that

a.

L

B ( X , Y ) E f T ( M ) L f o r a I / X E q L and^ E 9.

Proof. For any vector fields X and Y tangent to M , (1.1) and Gauss and Weingarten formulas imply

( 1.5)

t B ( X , Y ) = (V,P)Y - A,X,

where we have put (V,P)Y = V,PY - PV,Y

357

140

AUREL BEIANCU, MASAHIRO KON & KENTARO YANO

Let X, Z E q L and Y E q.Then (1.5) implies that ( P V X Z , Y ) = -(A,X,

Y ) = -(B(X, Y),F Z ) ,

which proves our assertion. Corollary 1. Let M be a mixed totally geodesic CR-submanifold of a Kaehlerian manifold g. Then each leaf of anti-invariant distribution q L is totally geodesic in M . Corollary 2. A generic submanifold M of a Kaehlerian manifold M is mixed totally geodesic if and only i f each leaf of anti-invariant distribution is totally geodesic in M . Lemma 3. Let M be a mixed foliate CR-submanifold of a Kaehlerian manifold M. Then we have A,P PA, = 0

+

for any vector field N normal to M . Proof, From the assumption we have B ( X , P Y ) = B(PX, Y ) for all X , Y E 9, On the other hand, we obtain B ( X , Y ) = 0 for X E 9 and Y E 6DL. Moreover, we see that PX E $7 for any vector field X tangent to M. Consequently we can see that B ( X , P Y ) = B(PX, Y ) for any vector fields X , Y tangent to M , from which it follows that A,P PA, = 0. Proposition 3. If M is a mixed foliate proper CR-submanifold of a complex space form G m ( c ) ,then we have c < 0. Proof. Let X , Y E 9 and Z E 9l. Then we have

+

( V x B ) ( Y ,Z ) - (V,B)(X, Z ) = B ( X , V y Z ) - B( Y , V,Z). If we take a vector field U normal to M such that Z = J U = tU, we obtain that V ,Z = -PA, Y + t D y U.Thus Lemma 3 implies that (V,B)( Y , Z ) - ( V , B ) ( X , Z ) = B ( P Y , A J ) + B ( X , A,PY). Putting X = PY and using (1.4) we see that 2 B ( P Y , A,PY) 1 -?c(PY, P Y ) U . Therefore we have

( 1.6)

0

< 2(A,PY,

A,PY)

=

=

- $ c ( P Y , P Y ) ( U , U),

which proves our assertion. Corollary 3. Let M be a mixed foliate CR-submanifold of a complex space form E m ( c ) . If c > 0, then M is a holomorphic submanifold or an antiinvariant submanifold of Mm(c). 2. f-structure

Let M be an n-dimensional CR-submanifold of a complex rn-dimensional Kaehlerian manifold g. Applying J to both sides of (1.1) we have -X = P2X tFX,

+

358

141

CR-SUBMANIFOLDS

from which it follows that P3X

+ PX = 0 for any vector field X

tangent to

M. Thus P3+ P=O. On the other hand, the rank of P is equal to dim Consequently, P defines an f-structure on M (see [7]). Applying J to both sides of (1.2) we obtain that -N

=

FtN

9,everywhere on M.

+ fZN,

so that f 3 N + fN = 0 for any vector field N normal to M , and the rank of f is equal to dim T , ( M ) - dim qXeverywhere on M. Thus f defines an fstructure on the normal bundle of M. Definition 4. If V,P = 0 for any vector field X tangent to M , then the f-structure P is said to be parallel. Proposition 4. Let M be an n-dimensional generic submanifold of a complex m-dimensional Kaehlerian manifold G. If the f-structure P on M is parallel, then M is locally a Riemannian direct product M T X M I , where M T is a totally geodesic complex submanifold of G of complex dimension n - m, and M is an anti-invariant submanifold of Mof real dimension 2m - n. Proof. From the assumption and (1.5) we have JB(X, Y ) = tB(X, Y ) = -A,,X. Thus J B ( X , P Y ) = 0 and hence B ( X , P Y ) = 0. On the other hand, we see that (2.1)

fB(X, Y ) = B(X, PY)

+ (V,F)Y.

Sincef = 0, we have (V,F)Y = - B ( X , P Y ) = 0. Let Y E 9l. Then we have that P V , Y = V,PY - (V,P)Y = 0 for any vector field X tangent to M , so that the distribution oi)* is parallel. Similarly, the distribution 9 is also parallel. Consequently, M is locally a Riemannian direct product M T X M I , where M T and M I are leaves of oi) and D L respectively. From the constructions, M T is a complex submanifold of and M I is an anti-invariant submanifold of G. On the other hand, since B ( X , P Y ) = 0 for any vector fields X and Y tangent to M , M T is totally geodesic in M.Thus we have our assertion. Theorem 2. Let M be an n-dimensional complete generic submanifold of a complex m-dimensional, simply connected complete complex space form c). If the f-structure P on M is parallel, then M is an m-dimensional anti-invariant submanifold of Mm(c), or c = 0 and M is C"-" X M2"-" of C", where M2"-"is an anti-inuariant submanifold of C". Proof, First of all, we have

e,

a"(

(V,B)( Y , P Z ) = D,(B( Y , P Z ) ) - B ( V , Y , P Z ) - B( Y , P V , Z ) = 0,

359

142

AUREL BWANCU, MASAHIRO KON & KENTARO YANO

which together with (1.4) implies $ c [ ( P Y , P Y ) F X - ( P X , P Y ) F Y ] = 0.

Thus we have c = 0 or P = 0. If P = 0, then M is a real m-dimensional If c = 0, then the ambient manifold anti-invariant submanifold of Gm(c). Mm(c) is a complex number space C", and our assertion follows from Proposition 3. Proposition 5. Let M be an n-dimensional complex mixed foliate proper generic submanifold of a simply connected complete complex space form H"'(c). If c 2 0, then c = 0 and M is C"-" X M2"-' of C", where M2"-"is an anti-invariant submanifold of C". Proof. From Proposition 3 we see that c = 0 and hence M"(c) = C". Then (1.6) implies that A ,X = 0 for any X E 9. From this and (1.5) we see that P is parallel. Thus theorem 2 proves our assertion.

3. Anintegralformula First of all, we recall the formula of Simons' type for the second fundamental form [6]. Let M be an n-dimensional minimal submanifold of an m-dimensional fiemamian manifold $. Then the formula of Simons' type for the second fundamental form A of M is written as V2A = -A

(3.1)

0

/i -4

0

A

+ R ( A ) + 5,

where we have put A" ='A A and 4 = 2:;; adA,adA, for a normal frame { V,}, a = 1, * * , m - n, and A , = Avg. For a frame { E i } , i = 1, * , n of M , we put 0

-

-

for any vector fields X , Y tangent to M and any vector field N normal to M,

R being the Riemannian curvature tensor of g.Moreover, we put = i- 1

(3.3)

[ 2(R(Ei, Y ) B ( X , Ei), N ) + 2 ( R ( E i , X ) B ( Y , Ei), N )

-(A,X,

R ( E i , Y ) E i ) - ( A N Y , R( E, , . X) E; )

+ ( R ( E i , B ( X , Y ) ) E i ,N )

360

-

2(A,Ei, R ( E i , X ) Y ) ] .

CR-SUBMANIFOLDS

143

In the following, we assume that the ambient manifold is a complex space form G m ( c ) . Since G m ( c ) is locally symmetric, we have F = 0. A straightforward computation gives ( R ( A ) N ( X ) ,Y )

= acn(ANX, Y

) - fc(A,X,

+ c(fB(Y,P X ) , N

) + $ c ( P X , P A N Y ) + $ c ( P Y , PANX) - ~ c ( A N P X ,P Y )

+c(jB(X, PY), N )

(3.4)

t N ) - ;c(AFXY, t N )

n

-$c

2 [ (AF&Ei,X ) ( F Y , N ) + (AFE,Ei, Y ) ( F X , N )

i= I

+ i ( A F 4 X , Y ) < F E i ,N ) ] .

We now prepare some lemmas for later use. Lemma 4, [9]. Let M be a generic submanifold of a Kaehlerian manifold Then we have A,Y = A,X

a.

for any vectorfielcis X , Y . Lemma 5. Let M be a minimal CR-submanifold of a Kaehlerian manifold %with involutive distribution 9.n e n we have

x

B(E,, E,) = 0

for ajrame { E ~oj} oi)'-. Prooj. We take a frame { E,, E,} of M such that { E , } and { E,} are frames of oi) and D * respectively. Since % is involutive, we have that X B(E,, E,) = 0 so that Z B(E,, E,) = 0. We now define a vector field H tangent to M in the following way. Let { E , } be a fame of o i ) l , and put H = 2, A,E,. Then H = X i A,eitfei for any frame { ei} of M and H is independent of the choice of a frame of M . In the following we assume that M is a generic minimal submanifold of z m ( c ) , c > 0, with the second fundamental form B satisfying that B ( P X , Y ) = B ( X , P Y ) for all X , Y E 9, which implies that oi) is involutive, and H E 6DL. From (3.4), using Lemmas 4 and 5 we obtain

(3.5)

( R ( A ) ,A )

> f ( n + l)cllA112.

On the other hand, we have [6]

(3.6)

361

144

AUREL BETANCU, MASAHIRO KON & KENTARO YANO

where p denotes the codimension of M, and 1IA )I is the length of the second fundamental form A of M. Thus (3.1), (3.5) and (3.6) imply (3.7) If M is compact orientable, then ( v ~ A ,A ) = L¶M

-J (VA,

VA).

M

Therefore (3.7) implies the following. Theorem 3. Let M be an n-dimensional compact orientable generic minimal submanifold of a complex space form G'"(c), c > 0. If 9 is inoofutive and H E9 ' , then we have

<

[ (2

I.

- i ( n f l)c(lA)I2 P As the ambient manifold M"(c) we take a complex projective space CP"' with constant holomorphic sectional curvature 4. Then we have Theorem 4. Let M be an n-dimensional compact orientable generic minimal submanifold of CP" with involutive distribution 9, If H E 9 ' and IJA1(2< ( n 1)/(2 - l / p ) , then M is real projective space RP" and n = m = p . Proof. From (3.8) we see that M is totally geodesic in CP". Thus M is a complex or real projective space (see [l, Lemma 41). Since M is a generic submanifold, M is a real projective space and anti-invariant in CP". Thus we have n = m = p and dim 9 = 0. Theorem 5. Let M be an n-dimensional compact orientable generic minimal submanifold of CP". If is involutiue, H E 9 ' , and ((A(12= (n+1)/(2- l/p),thenMisS'xS'inCP2,andn=m=p=2. Proof. From the assumption we have VA = 0, and M is an anti-invariant submanifold of CP", and hence m = n = p . Thus our assertion follows from [5, Theorem 31. (3.8)

JM(VA, V A )

M

- L)l(A114

+

Bibliography [ 11 K. Abe, Applications of Riccari type differential equation to Riemannian manifoldv with totally geodesic distribution, T6hoku Math. I. 25 (1973) 425-444. [2] A. Bejancu, CR submanifo& of a Kaehler manifold. I, Roc. h e r . Math. Soc. 69 (1978) 135-142. [3] -, CR submanifold of a Kaehler manifold. 11, Trans. h e r . Math. Soc. UO (1979) 333-345. [4] B. Y. Chen, On CR-submamifol& of a Kaehler manifod. I, to appear in J. Differential

Geometry. [5] G . D. Ludden, M. Okumura & K. Yano, A totally real surface in C P z that is not totally geodesic, Proc. h e r . Math. Soc. 53 (1975) 186-190.

362

CR-SUBMANIFOLDS

145

[6] J. Simons, Minimal oarieties in riemannian manifold, Ann. of Math. 88 (1968)62-105. [7] K. Yano, On a structure defined & a tensor field f of Vpe (1, 1) satisfving f' + f = 0, Tensor, N.S.,14 (1963)99-109. [8] K. Yano & M. Kon, Anti-inoariant submanifoldr, Marcel DeWrer, New York, 1976. [9] , Generic submanifolds, Ann. Mat. Pura Appl. 123 (1980)59-92. [ 101 , C R - s m - w r i t h k d'un espcrce projectif conplexe, C . R Acad. Sci. Paris 288 (1979) 515-5 17.

IASI UNIVERSITY, RUMANIA HIROSAKI UNIVERSITY, JAPAN TOKYO INSTITUTE OF TECHNOLOGY, JAPAN

363

This Page Intentionally Left Blank