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
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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;+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/
-
= *IIj/ 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 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
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 0,
when
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 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<mijkl.
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
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)
: 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\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 = - 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.
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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'
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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,
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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
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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
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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
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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 ' .
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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
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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 $(A{), then
+,
[+V, w V ]
PROOF. If Z E T ; ( M ) , then
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100
[+", W " ] Z " = +"(wv(zv))-w"(+"(Z")) = +v(w(z))"-wv(+(z>>v by (7.3). Since
~ ( z )+(Z) , E %(M) and
+T',
uv are vertical, we get
[$", 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
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. 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; 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
:
.
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) =
*
: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 =
= < 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
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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 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
