Foundations of differential geometry vol 1 kobayashi, nomi

-, -., _*Y A '&-*: , ;\y 1*:., I; ,: + "2 i Number, 15, vol j e: >&?a~ : DIFFAER$NTI& GE&ETRY T By Shoshichi Kobayashi 6ad Kabumi~No&zn ,.a - ‘ _ FOUNDATIONS OF FOUN@ATIOkS OF DIFlf’ERENl%.AL GEOMETRY ’ ’ r , VOLUME I SHOSHICHI KOBAYASHI University of California, Berkeley, California and KATSUMI NOMIZU Brown University, Providence, Rhode Island INTER8CIENCE PUBLISHEqS a dl~ision of i& Wiley & Sons, New York +* - 1963 t : ?I, INTERSCIENCE PUBLISHERS n., i“ - London a di~is&n zf John Wilby i So&, New York s PREFACE Differential g&etry has a long l&dry as a &ld of mathematics and yet its rigorous foundaticvn in”&e realm of contemporary mathematics is:nlatively new We h& written this book, the first of the twe;ivolumes of the Fdatiwdf D@nntial Geometry, with the intition of provid& a”l v&&tic introduction to differential ‘&ometry w&h @ also, &ve’ ‘a~ a reference book Our primary conce~ was to make it ,+l&&ta.@ed as much as possible and to give compl& ~MO& of ali’&d&d results in the foundation We hope that; && pw has been $chieved with the following arrangtments In chapter I we have given a brief survq ;of differentiablei m&nKMs, ,I$& mtips Ad ,fibre bundles The readers who-areu&amil& G&h &din.,qay learn thi subjects from t+e books +f Chevotlley, ppin, Pontrjagin, and Steenrod, listed in the Bilk aie our standard references in Chapter I We a coficise account of tensor- algti &d MU& fiek& th&“&n&l theme of which is the notiib;i ef8~h’aFtihc ‘k&ebr& of tensOr fields In the Appdiccs, we have @Gn some ‘r&t&s from topology, Lie grbup theory and othirs which %ve n’eed & the iain text With these preparations, the main t&t of thi book is &l&ontained Chapter II -c&ntains&e conilection theory of Ehresmann and its lateq development Results in this chapter are applied to linear and affifie cdnnections in Chapter III atid to Riemannian connections in Chapter IV Mat~y basic results on normal coordinates, convex neighborhoods, dis!ance, cotipleteness and Monomy groups are’ p&&l here completely, including the di: Rham decomposition thearem for Ri&mannian m&lds; ‘I In Chhpter v, -we ‘int&&ce the sectional curvature of’ a Rieme lM%&Id.‘a&i tbt Spaces of constant curvattire A more complete treMment!&“&$ert& of Riemanni& manifolds involving Hctional ‘&&+attrS depends on “talculus of iariations and will tie given II We’ discuss flat’ affine and a* IV Riemannian connec In Chapter VI; trqnsfornihtions and infinitesimal transformations which p&cx+e a given linear connection or A Eemambn metric We include here various results concerning R$ci tensor, holoztomy and infmitesimal isometries We then’ ’ V Vl PREFACE treat the wtension of local, transformations and the so-called equivaleqC Q problem for a&e and Riemannian connections The resuh in this chap ter are closely related to differential geometry ,c$ qhomogeaeous spaces (in particula*$~ symmetric sPaces) wt\i& ‘are planocd for Volume&I In al!, t& chapters, we have tried& familiarize $b readers with various techniques of computatio&phich are curmtly in use in di&rential geometry These T:,- (1) classical tensor calculus with indices ; (2) exterior differew calculus of E Cartan and (3) formalism of covariant differentiatiqn a,Y, which is thd newest among the the’& -We have also jllu,sfr;tsed as wk see fit bundle or &o&&g directly in) the methods of using ,? suitable , the base space /.,, :.’ The Notes include B&F ,histprical facts and supplementary results Pertinent to thi &i;&fent of the present, v&me The Bibliogr@!Y at the en&&+* only tboge booti and papers * which we quote thro~$$&uZ~~,~~k ,+ -_ I ‘ ‘.$ f Theoqems, prop~i~~n~~~~l~o~~~~~ ;IKe”&mbered for each section For example, in e+& ,&@@qs qayr, &apter II, Theorem 3.1 is in S,sMon IrJ.the re&&,+~~~ r; it will be referred tQ simply as Theorem 3.l,:~~~iR~ot~~~.~IFUbsequent chapters, it is referred to as TheA)rem $.I cQf.(&pta&* ‘.Ve originally phmg$ toAte ~@&&&,which would include the content o< he present ,vc&@ ~5 weU.+s &e following topics: submanifolds; variat+ns of’ t& leng# ‘hgral; differential geometry of complex and K4&r;unanifblds;,~fferential geometry of homogeneo?rs spaces; symme& spaceg;.cbaracteristic classes The considerations of tjple,;an$ spase,h3ve made it desirable to divide the book in two,voluqes The:top& mentihned above will ’ therefore be included in.V@ume II In concluding the preface, we sb+d like, to thank ,Professor L Bers, who invited US to undertgke this project, a@ Interscience Publishers, a~‘di$sidn of Jc&n Wiley a,nd ‘Sons, for their patience and kind coaparat$n We ‘se greqtly indebted tp, Dr A J Lohwater, Dr ,H C)sreki,Mess+!A Howard,and E Ruh for their kind help which resulted in maqy improvements of both the content and the prescnt+#on .We al& acknowledger;- th.e grants of the National Science Fo&&&n which support&p+rt oift:he work included in this book., ;,,, j r: -;f : ’ * StiOSHICFiI fbi3AYASHI KATSUMI NOMI zu i Contents xi Interdependence of the Chapters and the Sections C HAP T I R Manifolds Diffrentiable E Differentiable manifolds Tensor algebras Tensor fields Lie groups -.:, Fibre bundles 17 26 38 50 ,;‘.L II C HAPTER Tbeory of Connections Connections in a principal fibre bundle Existence and extension of connections i I Parallelism Holonomy groups Curvature form Bnd structure equation Mappings of connections Reduction theorem , Holonomy theorem Flat connections : 10 Local and infinitesimal holonomy groups 11 Invariant ‘connections 63 67 68 71 75 79 83 ’ 89 92 94 103 1, CHAPTER III Linear and Affine Connections Linear connections Affine connection 7; ‘+ Developments Curvature and torsion tensors Connections in a vector bundle Geodesics Expressions in local coordinate systems ? vii i3 118 125 130 132 138 140 +-tr* !a CONTENTS Normal coordinates *?‘ * f Linear infinitesimal holonomy’groups 146 : : izl k%P~R Iv Rie-ashdan conneclions Riemannian metrics 2, Kemannian connections 3, Normal coordinates and ~&I neighborhoods : : 4, Completeness ’ Holonomy groups 6.The decomposition theorem of de I&& : : : : Affine holonomy groups 154 158 162 172 179 167 193 &APTRRV Curvature and space Forms Algebraic preliminaries ; ‘ -193 Sectional curvature 201 SpaCeS of constant +rvature 204 Flat a&e and Ri &ar&nconrLkt&i 209 ,‘i ; CHAPTER 1 vz;y T~iWZWblU ~emapp;dgr~*,~tioas Infinitesimal affuw tansfbrmatwns hometries aryl infinitesimal isom& ‘ .g * Holonotny anh &finitesimp!iaometries ‘.‘I 1 * ’ Ricci tensor and infinitesimal iaomet& Extension of local &morphisms - Equivalence problem -225 229 236 ’ 4 ’ : : : 248 * ‘: * * * * ‘252 256 IAPPRUDICES Ordinary F differential equati0n.s 267 A connected, locally compact metric space,.is separable 269 Partition af unity , 272 On an arcwise connected subgroup of a Lie group , 275 Irreducible subgroups Of o(n) : 277 Green’s theorem : 281 Factorization lemma , 284 ix CONT&NTS N&s Connections and holonomy,groups i?: Complete affine and Riemannian connections Ricci tensor and scalar curvature 292 Spaces of constant positive curvature 2% Flat Riemannian manifolds 297 Parallel displacement of curvature - - : - 366 Symmetric spaces 366 L i n e a r c o n n e c t i o n s with r~urrent ~urv~ture i: The automorphiim group of a geometric structure Groups of &met&s and affine transformations with 308 maximum dimeiisions 309 Conformal transformationd of a Riemannian manifol& Summary of Basic Notations Bibliography ; 313 315 325 of the Chapters and the Sections Interdeprndence ui’ I Ii-1 to r-7 r .I, + VI-1 : I III-l,2 I II-8 ‘$-5,6 - I ’ I-10 , LJ ? ,.: M-7,8 I I I n-11 VI-3 -p-4 cl VI-5 I Exceptions Chapter I1 : ‘Theorem 11.8 requires Section II-IO Chapter lil , Proposition 6.2 requires Section 111-4 Chapter I\‘: Corollby 2.4 requiws Proposition i.4 in Chapter III Chap1t.r IV: ‘I’hcorcm I, (4) requires Section III-4 and Proposition in C!lnptcr JIJ Chapter 1.: P r o p o s i t i o n wquirrs Section 111-7 Chapter 1’1: ‘1’hcon.m 3.3 requires Section \;-2 Chapter \ I : Corollary 5.6 requires Example 4.1 in Chapter V Chapter \‘l : Corollary 6.4 requires Proposition 2.6 in Chapter I\‘ C h a p t e r \ ‘ I : ‘I‘hcorrm IO rcq:zilcs Section V-2 6.2 A @d&&~~of %@.$&h& on’s &I;&&& ‘space S is a set r of transformations sai$y&g the foli~wG3~ a&ms: (1) Each J-e F is-a hotie@merphism ef an open set (called the domain ,off) of S 0.~0 another *open set (called the range off) of s ; it -,- ” j x (2) Iffc T’, their the restriction off to an arbitrary open subset of the domain off is i.n F; ~(3) &et Us= uUj where, each Ui.is.an open set of S A homeomkphisn;ijcof, c’ontb a‘n?open s&df S belongs to I’ if the’restric“ i tion off t0 Ul’is in I‘ fcir Lverv i:- :, (4) -For every cfpen set U iFS;‘tlie identity transfbrmation of Lr , , ‘,, : is in r; (5) Iffc l’, thenf-’ f I’; : ; 1* 446) :Iff* 11 is aq bomeamorphism~of U on&o VTadi,*f-“ I: is a homeomorphism of IJ’ onto V’ and if% V n U’.~ is hen-empty, then the homeomorphism f’ df ,of j-1( K n U’)bofitb f’( P n Li’) is in 1‘ Ii kkgi?#S:qfe~ examples of psixdogroupk~~l&h ai: used in this b% -@ Jj# bsrhe pwofu- t.upla~ of re.4: numbers (xl, x2 , x”) ~~th’th~~~~~t~l~g~~~.A fitdppingJ of an open set oikni into R”! js sai&+l,be of ~~~~-,.C$ r =+1.,:-Z, , aa, if f is continuously r tirnq &@&qentiable By -class CO l&z mea.2 thatf:$ continuous By cla&CX&yz mean tkt Lf is real anal& :Tbe @dagroup 6’(W) ~&#@g#iv7natitilPs of.class CT of R” -is &he :f*et of! homeomocphisms J d,?,n:&peaiset +&R” onto: an own; set of*w such that both f an&f? iare pf cl&O OsW;ously i”fiR”) & ca pseudogroup of transformations ‘of RI If r < s, then P(‘“) is a I r AxawmJ vr IJlPP~RaNTIAL ~qyp=, I subpseudogroup of l?(R”) If we consider only those f~ P(R”) whose Jacobiang ,are positive everywhere, we obtain a subpseudogroup of ,P(R”) T h i s subpseudogroup, denoted by QR”), is calied~ thy of doSs the pseudogroup of o$ntatt of R” Let C” be”tl&#&e numbers with the usual topology The ps (i.e., complex analytic) yqf~iions of C* cm be defined and will be den& ii by%+@&) We shall iden R*“, when necessary, by mapping (21, , P) E Cn into (x!, ; x”, y’, , y”) e R2”, where ,$ = xi + &j ‘8 Under this id&iiication, l?(O) is a, of rq(RsR) for,any Y F An atlas of a ti Mcom$Q.iile %v.i” 4&*&ogroup r is a fa~~t~&g~ gUj, vi>, @Ied charts, s~!y~;.~ (a) Each Ut.$,bz+n,ov$pt of $ and CJU, = M; (b) Each qi isra~hor&&norphism of &i onto an open set of S; j’ (c) Whenever U;, ti, U, is non-empty, the mapping qi $’ of gf(Ui A U,) onto qj( Ui n U,) is an element of r .:-:A complete atlas of Mcompatible with r is an atlas of&f comther atlas of M patible with r which is not contained in e with I’ is concompatible with r Every atlas of M eo tained in a unique complete fact, given an atlas A = Qj p-l: Sp(U n Vi) + yi(U”i Wi) is an element of I’ whenever U A Vi is non&rpty Then ;kis the ‘ I complete atlas containing A If l” is a subpseudogroup of I’, then an SatEas of Mucompatible with l” k compatible with l? A dt@xntiable manifold of class c’ is a Hausdorff space with a fixed complete *atlas compatible with P(R”) The integer I is’ called the dimension of the manifold Any atlas ofJa RMdorff space compatible with P(R”), enlarged to al compkte ad& defines a dii$erentiable structure of class’O.,Since p(R”) T’(R”) for t < s, a diffekentiable structure of~4ass (?fldefinh uniquely a di&ren&ble structure of class C k differentiable mamk~ld of c&s @” is “also called a rcal’ktj& manifold (TW&W! *he bIc we shall hostly consider differentiable manifolds ofi t&s cm %’ ii ‘,’ l I DIFFERENTIABLE MANIFOLDS a dl&entiable manifold or, s’ ly, manifold, we shall mean a ‘differentiable manifold of class T y.) A complex (anabtic) manifold of complex dimension n is a Ha rff space with a fixed complete atlas comp.a$>le with I’(CP)“’ ented differentiable manifold of class C’ 1s a Hausdorff spa ,a fixed complete atlas compatible with.~~(R”) An orien&j differentiable structure of class Cr gives rise to a differenti#e structnre ‘of class $7 uniquely Not every difherentiable str&&e of&l& Cr is thus dbtained; if it is obtained from an &kntied Ont$&“is .called orientable &I orientable man&Id of class $&&s &.actly two orientations if it is connect&L I&$&g the pro$‘o&ris fact to t,he reader, we shal15pn~y i$$.$ ,&w h’ r&ii,& &&e of an oriented +anifold If af2ariZly ;(afc&&j(&, ~st) d&es anfo$ented manifold, then the, ,f{mjly of &2&s ;( p6, vi) &#nes the manifold with the reverse$ o+ntdt$ori where tyi is the’composition of i, with the transformation (xl,xs~ ; , x”) + (-&, a?, , x”) of R” Since r(v) c r;(w), every complex man&d is oriented as a ma&-‘ fold of class C ; For any structure under consider%io~ (e.g., differentiable structure of class Cr), an allowable chart is a chart which belongs to the fixed complete atlas de&&g the structure From now on, by a chart ,WF shall mean an allowable chart Given an allowable chart (Vi, vi) of an n-dimensional manifold M of class CI, the system of.functions x1 qi, , X” vi defined on Ui is called a local coordinate q&m in -Ui We say then that U, is a coordinate neighborhood For every point p of M;it is possible to find a chart ( Ui, yi) such that &) i the origin of R” and v; is a homeomorphism of Vi onto kn?op& set of R” defined by I.&l < a, , , Ix”] < a for some‘“” “* number a Vi is then called a cub+ tuighbor&od of p yhve ’ IIl,&‘~aturd’ manner R” is an’oriented manifold of class Cr for auy “Y; ‘a~cliart ‘con&s of an elementfof I’i(R*) and the domain off Si$@rly?,, q.!‘q a conipIex mar$bld Any open sr&et N of a manifold M of&&~ is a manifiold of class C in a natural manner a chart of N is &ve&‘bi (oi, n A!‘, -$I,) where (Vi, &) is a chart 0; M and y$ is the re&&@ tif ++ t& ‘VI&p X SimilaAy, f&i complex manifolds ’ ‘ ” ‘ ” ,,;,‘:;:-’ Given two man&l& &” *‘MI of class 0; a’ mapping f: M *‘MI is said to be dif&&iable of class clc, k r, if, for every chart (Vi, vi) of M and every chart (V,, yj) of M’ such that I D I F F E R E N T I A B L E MANIPOLDS l~O~‘NDATIONS OF DIP&&;~TIAL GEOMETRY the n$pping ylj Lf vi I of q),( U,) into wj( V,j ‘is = I;, differentiable of c$ass 12”; If ul, , U” is a local co$inate syst& in IT, and ,vl., :‘; , v r’s is a local coordinate systefn:;\i yj, lhenf may be” YxprcsSed by a set of diffei&tiable functions, oft~~~s~‘~~~: r: ‘: II ‘r ,,tal Efl(d, ,TP), ;.v,‘” ep(d, , , li”) i!” ;, By a’dQ@rentiable VM/$& or ,simp&, a ‘$.@ing, we shall &$,n a mapping of class C” ,A &Gxentiable f&&ion 0~ cl+ Ck, on # is zi inapping of class C” ?f A4 into R Thet&fin,it& ‘1 of QT a h&&@& (or complex anal$k) *ppifig b; function i?si&lar By a dferenti$le &e of cl& cn: in M;;ck~~a$~$$$ c@&&$able map@ng,g,f _.“clasd Ck of a closkd intq@3”fgl.b] 6% R.‘into 111, namely, tEe’ rcst?iction of a differentiable mapping of claps CLi$f an open inteival containing [q-h] into A4 We shall riow define’s tangent vector (or simply’ a @for) a point p of ‘M Let s(p) be tke glgebra of differentiab!e‘functions ofcla’ss C’ defined in a neighbdigood ofp Let x(t) %e P c&% ‘& &Gs’ Cl, n t S,;fi, such that x(t,,) = p The vector tansent to_the cufve x(t&$‘p 1s a mapping X: iQ) ,*.R de3ined by+“‘: , ‘,‘I .I fir’,, , /’ ’ Xf = ~,d&(t))/a)~o ‘$ In other words, dyf is the ‘&i\i”&%e off in‘ the~directidn ‘of the ‘ cllrve x(t)% t’== to Tlie~~~rror,;~~atisfrdsthe fdllowinq conditiond: L )I “: (,l) X is a linear mapl+&r#$(p) into R; the dimension’of M Let ul, , f , U” be a local coordinate syqqn in aLcoordinate neighb&hood G pf p For ea& j, ($/,@j), is a mapping of 5Cp) &to, &,, which satisfies condit;ons lQl+vd (2) ah:ove We shall s&v that‘?t.he set -of vyct@$ at b is ‘thq.,vqctor space with basis (i318~‘)~, :, , (a/a~‘~)~,.‘ c;iy$n any &r% x(t) with h ‘k x(t,);‘l&‘uj - x’(t)j.,j 1, :.:‘n, ibc its equAtio1~!5 in terms of the local coordinate sy2tt:m $, a $!, $ThqqV, ).,” > (nf(x(t)~~po,” ;I; xj (@i&q, (&j(d)~dk\jo~:*, ‘.s I c 5,: ,,,,!,,, +‘yf11.14 ’ T-o::,!~~,,; * !“lll I?,,: \Illnnr:l!icirl Ill),,,lirri:, -(‘” which proves tll,kt cvcry \‘cctor at p is a linear ccimbination (apfl) ),, (a/&) i, C:on\,crsely, gi\.cn a lincnr combination c p(aptfl) ,,, ronsidcr the curv~‘&finecl by I$ -.-= IIJ( p) i- pkp I j = 1, , 72 Then the vector tnn,g:cllt to 21?iii &rve at t -~ is x S’(a/&‘),, T o prove the l i n e a r ind&~~ence tJf (aj+‘) xj, , (a/au”),, assur11c ?; p(a/allq,, :: : ~~~*~,::; Trig’:’ j : ; Tz x p(auqj3+j; ,;.+k f o r k - l, ,?l This complctcs thd *probf of’&r as$crtion ‘I’hc set of tangent vectors at p, denoted by 7’;(%) or 7’,,, ’IS callrcl the tcqenl space 01 M at p: The n-tuple of not&&s El, , :‘I \vill be called the com@one@ of the.vector p(apj, with respect to the local coordinate system d, , Y” Remark It is known that if a manifold AI is of class C,‘“, then T,,(M) coincides with the space of X: 3(p) -+ R satisfking conditions (1) and (1) above, where z(p) now denotes the algcbrn of all C” fimctions around p From now on we shall consider mainly manifolds of class c,“‘: and mappirqs of class C*‘ ’ A&ctorjeld X on a mafiifold 111 is an assignment of n ve:tor XP to cd point of A/ Iffis a $iffcrentiablc function on IZ~, then XJis a functioh on li defined by (.yf) (pi -:: LY,,.J A vcsctor field X is rallcd djfhvztiobfe if x/-is diffcrcntiablc for cbverv differentiable function J In terms of a 10~21 coordinatr svstcn; uI, , ~‘1, ii vector field ,Y may be cxl~r~sscd by Y x IJ(~/&J), whcrc ij arc functions defined in the coordinatnrlloc~d, called the components of Y with rcspcct to $, , 11” Y is differentiable I if and only if its componc~rrts E1 arc diff~~l-c:ntiablc Let F(M) br th e s,$ of‘ all diffirrntinblc \.cctor fields on \I It’ is” a’ real vector spacr undet tllc* ri;ltural atl~lition and scalar multiphcatioh If X and I’ arc ill X( \f), dcfinc the bracket [X, IrT as’s mapping from the riii of’ fiitic:tions on \I into itsctf bY [X, Yjf == X( I:/‘; “l’(A4’f’) We shall show that [KI’J is a \.ciLor ficlu In terlns of a local coordinate system ul;.* , II”, \vc \Vrite FOUNDATIONS OF DIFFERENTIkL GEOhfETR+ I \ ‘_ Then DIFFERENTIABLE MANIFOLDS A l-form w can be defined also as an g(M)-linear mapping of the S(M)-module S(M) into S(M) The two definitions are related by (cf Proposition 3.1) , (y(X)), = (wp, X,>> ,& WW, P E M [X q”f = Zj,E(SP?‘w)~ - ?www>(aflau’) This means &t IX, yl is a veeto’$ield whose components 4th respect to G, ‘1 , ZP iffe given &(~~(a$/&~) - $(w/&.P)), j= l, , n; With respect to this’ b@ket operation, Z@f) is ,a (of infinite dimcnsioris~ Lie algebra over the real number In particular, we have Jacobi’s id Let AT,(M) be the exterior algebra over T,*(M) An r-form OJ ’ is an assignment of an element of degree r in A T,*(M) to each point p ‘Of M In terms of a local coordinate system al, , un, ti can be expressed uniquely as 0= Is d1 implies VR = (This was known by Lichnerowicz [3, p 41 when M is compact.) They remarked later that the assumption of completeness is not necessary Note Linear connections with recurrent curvature zero on P(uo) such that for u Q P(u& f(u) = v(u) -f (uo) As a special case, K is parallel if and only if f (u) is constant on Wo) * Using this result and the holonomy theorem (Theorem 8.1 of Chapter II), Wong obtained THEOREM Let r be a linear ‘connection on M with recurrent curvature tensor R Then the Lie algebra of its linear holonomy group Y (u,,) is spanned by all elements of the form a,,( X, Y) , where is the curvature form and X and Y are horizontal vectors at u,, In particular, we have Proof The first assertion follows from Corollary 7.9 of Chapter VI Let a, be the local symmetry at a point o of M By Corollary 6.2, a, can be extended to an affine transformation of M onto itself which is involutive Define an involutive automorphism of G by a” = o0 a o IT, Then H lies between the subgroup of all fixed QED elements of (T and its identity component Let M be an n-dimensional manifold with a linear connection I’ A non-zero tensor field K of type (r, s) on M is said to be recurrent’ if there exists a l-form a such that VK = K a The following result is due to Wong [ 11 In the notation of $5 of Chapter III, letf: L(llif) * T HEOREM T;(R”) be the mapping which corresponds to a given tensorjeld K of OPe (r, s) Then K is recurrent if and only if, for the holonomy bundle P(uo) through any u E L(M), there exists a dzjkentiable function q(u) Wzth no 305 _ Ricci tensor field, then fS is also parallel The irreducibility of M implies that 1s = c - g, where c is a,constant and g is the metric tensor (cf Theorem of Appendix 5) If dim M and if the Ricci tensor S is non&trivial, then ;il is a constant function by Theorem of Note Since JR is parallel and since is a constant, R is parallel Next we shall consider the case where the Ricci tensor 5’ vanishes identically Let V R = R @ a’and let Rj,, and a,, be the components of R and a with respect to a local coordinate system xl ,*a*, By Bianchi’s second identity (Theorem 5.3 of Chapter III; see also Note 3), we _ have Multiply by gJm and sum with respect to j and m Since the Ricci tensor vanishes identically, we have Cj,~,~gj”‘Rj~~,~ = Cj,,,gjnlRj,,t = Hence, EjRjktaj = 0, where aj = Zn,gjmam :3r;ti FOI!NDATIONS OF DIFFERENTIAL NOTES GEOI\lETRY ‘I’llis ccluation has the following geometric implication Let x be an arl)itr;\rily fisrcl point of hl and let X and _I’ be any vectors at x If \VI’ clc*notc by I’ the vector at x with components &(x), then the linc’ar translbrmation R(X, E’) : 7’,(A4) -+ 7’,(.14) maps I’ into the zero \‘cctor On the other hand, by the Holonomy Theorem e also (‘l’hcorcm 8.1 of Chapter II) and Theorem of this Note \\‘ong [ l]), the Lie algebra of the linear holonomy grou Y(x) is spanned by the set of all endomorphisms of 7’,(A4) given by fER WAERDEN B L [I] Uber metrisch homogencn Raume, Abh ;2lath Sem Hamburg (19281, - DIEUDONN~, J [I] Sur lcs espaces uniformes 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connection, 81 of a Lie algebra, 40 of a Lie group, 40 , Bianchi’s identities, 78, 121, 135 Bundle associated, 55 holonomy, 85 homomorphism of, 53 induced, 60 of affine frames, 126 of linear frames, 56 of orthonormal frames, 60 principal fibre, 50 reduced, 53 sub-, 53 tangent, 56 tensor, 56 trivial, 50 vector, 113 L Canonical decomposition ( = dc Rham decompdsition), 185, 192 flat connection, 92 form on L(.M), 118 invariant connection, 110, 301 invariant Riemannian metric, 155 linear connection, 302 metric, 155 l-form on a group, 41 parameter of a grodcsic, 162 Chart, Christoffel’s symbols (I’:.,), 141 Compact-open topalogy, 46 Complete, * linear connection, 139 Riemannian manifold, 172 Riemannian metric, 172 vector field, 13 Components of a linear connection, 141 of a l-form, of a tensor (field), 21, 26 of a vector (field), Conformal transformation, 309 infinitesimal, 309 Connection, 63 affine, 129 canonical, 110, 301 canonical flat, 92 canonical linear, 302 flat, 92 flat affine, 209 form, 64 generalized affine, 127 induced, 82 invariant, 81, 103 by parallelism, 262 Levi-Civita, 158 325 326 Connection linear, 119 metric, 117, 158 Rkmannian, t 58 Uniwtd, 230 Constant curvature, 202 space of, 202, 204 Contraction, 22 Contravariant tensor (space), 20 Convex neighborhood, 140, 166 Coordinate neighborhood, Covariant derivative, 114, I1 5, 122 differential, 124 djfferentiation, 115, 116, 123 tensor (space), 20 Covector, Covering space, 61 Cross section, 57 adapted to a normal coordinate system, 257 Cubic neighborhood, Curvature, 132 c constant, 202 form, 77 recurrent, 305 scalar, 294 sectional, 202 tensor (field), 132, 145 Rkmannian, 201 transformation, 133 Cylinder, 223 Euclidean, 10 twisted, 223 Derivation of Z\(M), 33 of T(M), 30 of the tensor algebra, 25 Devclopmtnt, 131 DifTcomorphism, Differential covariant, 124 of rl function, of a mapping, Discontinuous group, 44 properly, 43 Distanw function, 157 Distribution, 10 involutive, 10 Divergencr, 281 Mectivc action of a group, 42 kinstein manifold, 294 Elliptic, 209 Iklidran cylindw, 21 O locally, 197, 209, 210 metric, 154 motion, 15 subspace, 218 tangent space, 193 torus, 210 Exponential mapping, 39, 140, 147 Exterior covariant derivative, 77 covariant differentiation, 77 derivative, 7, 36 differentiation, 7, 36 327 INDEX INDEX c Fibre, 55 bundle, principal, 50 metric, 116 transitive, 106 Flat affine connection, 209 connection, 92 canonical, 92 linear connectibn, 210 Riemannian manifold’, 209, 210 Form curvature, 77 l-form, r-form, tcnsorial, 75 pseudo-, 75 torsion, 120 Frame affinr, 126 linear, 55 orthonormal, 60 Free action of a group, 42 Frobenius, theorem of, 10 Fundamental vector field, 51 Geodesic, 138, 146 minim&ng, 166 totally, 180 Green’s theorem, 281 G-structure, 288 Holomorphic, Holonomy bundle, 85 Holonomy group, 71, 72 affine, 130 homogeneous, 130 infinitesimal, 96, 151 linear, 130 local, 94, 151 restricted, 71, 72 Holonomy theorem, 89 Homogeneous Riemannian manifold, 155, 176 space (qubtient space), 43 symmetric, 301 Homomorphism of fibre bundles, 55 Homothetic transformation, 242, 309 infinitesimal, 309 Homotopic, 284 C’-, 284 Horizontal component, 63 curve, 68, 68 lift, 64, 68, 88 subspace, 63, 87 vector, 63 Hyperbolic, 209 Hypersurface, Imbedding, 3, 53 isometric, 161 Immersion, isometric, 161 Indefinite Ricmannian metric, 135 Induced bundle, 60 connection, 82 Ricmannian metric, 154 Inner product, 24 Integral curve, 12 C Integral manifold, O Interior product, 35 Invariant by parallelism, 262 connection, 81, 103 Riemannian metric, 154 Involutive distribution, 10 Irreducible group of Euclidean motions, 218 Riemannian manifold, 179 Isometric, 161 imbedding, 161 immersion, 161 Isometry, 46, 161, 236 infinitesimal, 237 Isotropv ’ group, linear, 154 subgroup, 49 Killing-Cartan form, 155 Killing vector field, 237 Klein bottle, 223 Lasso, 73, 184, 284 I,eibniz’s formula, 11 Levi-Civita connection, 158 Lit algebra, 38 dcrivativc, 29 difftwntiation, 29 group, 38 subgroup, 39 transformntion group,‘41 I.ift, 64 OR, 88 horizontal, 64, 68, 88 naturA, 23O I,inr:tr connection 11’) frame, i5 holrinomy group, 130 isotropy group, 154 Local I&s of a distribution, 10 coordinate system, 328 INDEX ‘ ‘\ 329 INDEX Locally affine, 210 Euclidean, 197, 209, 210 symmetric, 303 Lorentz manifold (metric), 292, 297 Manifold, 2, complex analytic, differentiable, 2, oriented, orientable, real analytic, sub-, Maurrr-Cartan, equations of, 41 Metric connection, 117, 158 Mobius band, 223 Natural lift of a vector field, 230 Non-prolongable, 178 Normal coordinate system, 148, 162 -parameter group of transformations, subgroup, 39 Orbit, 12 Orientation, Orthonormal frame, 60 12 Paracompact, 58 Parallel cross section, 88 displacement, 70, 87, 88 affine, 130 tensor field, 124 Partition of unity, 272 Point field, 131 Projection, covering, 50 Properly discontinuous, 43 Pseudogroup of transformations, 1, Pseudotensorial form, 75 Quotient space, 43, 44 Rank of a mapping, Real projective space, 52 Recurrent curvature, 305 Recurrent tensor, 304 Reduced bundle, 53 R e d u c i b l e connection, 81, 83 Riemannian manifold, 179 structure group, 53 Reduction of connection, 81, 83 of structure group, 5.3 Reduction theorem, 83 de Rham decomposition, 185, 192 Ricci tensor (held), 248, 292 Riemarmian connection, 158 curvature tensor, 201 homogeneous space, 155 manifold, 60, 154 ’ metric, 27, 154, 155 canonical invariant, 155 indefinite, 155 induced, 154 invariant, 154 Scalar curvature, 294 Schur, theorem of, 202 Sectional curvature, 202 Segment, 168 Simple covering, 168 Skew-derivation, 33 Space form, 209 Standard horizontal vector held, 119 Structure ‘constants, 41 equations, 77, 78, 118, 120, 129 group, 50 Subbundle, 53 Submanifold, Symmetric homogeneous space, locally, 303 Riernannian, 302 Symmetrization, 28 Symmetry, 301 Tangent affine space, 125 301 +-gent &ndle, space, vect*; Tensor algebra, &!, 24 56 contrairariant,bun%, 20 q _ covariant , 20 ‘f%d, 26 product, 17 space, 20,21 Tcnsorial form, 75 pseudo-, 75 Torsion form, 120 of two tensor fields of type (l,l), 38 tensor (field), 132, 145 translation, 132 Torus, 62 Euclidean, 210 twisted, 223 Total differential, Totally geodesic, 180 Transformation, Transition functions, 51 Trivial fibre bundle, 51 Twisted cylinder, 223 torus, 223 Type ad G, 77 of tensor, 21 Universal factorization Vector, bundle, 113 field, Vertical component, 63 subspace, 63, 87 vector, 63 Volume element, 281 property, 17 ... transpose of 11 Since T is generated by F, II and I’*, 11 is uniquely dctcrminc,tl its restriction to F, V a n d V* It follows that 11 *Ii is iujccti.c Conversely, given an endomorphism 11 of 1, ‘... induced by a tensor field of type (I,? In other words, the set of Ieruor,fie/dr qf Qpe (1, 1) is an ideal of th&Lie algebra of derivations oj’ 1( .11 1) On the other hand, the ret of Lie d{fffrentiations... follows that the set of all tensor fields S of type (1, 1) forms a subalgebra of the Lit algebra of derivations of Z(M) In the proof of Proposition 3.3,’ we showed that a derivation of l(.Zlj is induted