高次アインシュタイン空間について

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(1)Title. 高次アインシュタイン空間について. Author(s). 長谷川, 和泉. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 26(2) : 73-78. Issue Date. 1976-02. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5993. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section H A) Vol. 26, No. 2 February 1976. »ig»ir:*:<N.e® (is 2 $]; A ) ^5 26 ^ % 2 -f Hgffi 51 ^ 2 fl. On Higher Order Einstein Spaces Izumi HASEGAWA Laboratory of Mathematics, Sapporo College, Hokkaido University of Education. 064 Sapporo. WTA >^^^ A vW^^^r :^J11 W 7^ -lfc®itf!(1;*-^tL?^K&'^»H. Abstract We shall consider an n( ^4/»—2)-dimensional Riemannian manifold M satisfying the curvature condition L=fg2t>~1, where L denotes a certain curvature structure of order 2p—l. For n^4p—l,c2t'(R11) is constant in this space, where c is the contraction and R denotes the Riemannian curvature tensor. The basic tool of proof is the generalized Bianchi identity. We see that this space has properties analogous to the usual Einstein space.. Introduction Recently, the study of higher order curvature structures has been developed by J. A.. Thorpe [6], A. Gray [I], R. S. Kulkarni [3], T. Nasu [4] and other authors. In this paper, we shall define a higher order Einstein space and investigate the properties of this space.. Throughout this paper, we shall assume that all manifolds and all objects are of differenliability class C" 1. Preliminaries In this section, we shall recall a brief summary on the culculas of double forms for later use. (for the details, see [I], [3] and [4]). Let M be an n -dimensional Riemannian manifold with Riemannian metric g, let <5'(M) be the algebra of smooth functions on M and let 3£(M) be the Lie algebra of vector fields on M- For p an integer between 0 and n, let AP(M) be the bundle of ^-vectors on M and let A*"(M) denote the bundle of^-forms on M. Here we put A°(M) = A*°(M} =5(M). We put. ®M=A*p(M)(g)A*'?(M), (Q^p,g^n), where the tensor products are taken over 'S-(M). We call an element w of ® •'' a double form. of type (p, q) on M. It is an 3(M)-multilinear map. (11).

(3) 74 Izumi HASEGAWA. ^: ^(M)l'X^(M)tl —— ^(M), which is skew-symmetric in the first ^-variables and also in the last 17-variables. We shall use. the notation (1.1) w(Xi,...,Xp)(Yi,..,, ¥„) to denote the value of 01 in the vector fields X\, ..., Xp, Yi, ... Yq e 3£(M). Then. ^X,,...,Xp)-MMV —^(M) is the g-(AO-multilinear map whose value in the vector fields Yi,..,, Yq is given by (1.1).. If p = q and (1.2) w(Xi,...,Xp)(Y,,..., Yp)=co(Y,,..., Yi,)(X,,...,Xp). for any X\,..., Xp, Yi, ..., Yp^^(M), we call a> a curvature structure of order p and denote the set of curvature structures of order p by (£p. We can see that Riemannian metric g is an element of <£1 and Riemannian curvature tensor R is an element of K2 For (ye(£/>, we regard &i as a quadratic form on A (M). And so, for ueS'', we can define a function tr(.&)) on M as the trace of co on A (M), i. e. (1.3) tr(a})=^w(E^,...,Ej,)(E^,...,Ej,), l&j i <•••<] ftn. where {Ei,..., En} is locally defined orthonormal frame field. Next, the exterior product co/\0 of two double forms <y e'3)A<? and Qe'Sr's is defined by. the formula (co^9)(X^...,Xi,+r)(Y,,..., Y^s) (1.4) ='Z£a£T(v(Xsw, ..., Xa(p))( Yrw, ..., y<r(?)) o-6S/!(p,r) r eS/i(<?,s). 6(Xa(t>+i), ..., Xs(p+r))( Yr(q+i), ..., Y^q+s)},. (X., r,e3EW). Here, Sh{p, r) denotes the set of all (p, r) -shuffles ; specifically Sh(p, r)=(a<= Sp+r\ff(l)<---<ff(p) and a(p+l) <••• <o(p+r)} where Sp+r is the symmetric group of degree p+r.. It is easy to show that A is an associative multiplication and (1.5) wA0=(-l)pr+9s0Au. Let &)A denote the A-th exterior power of a>- For k=0, we put u ==1.. We can define the inner product on A (M) by (1.6) <Xi^-/\Xt,, Yi/\-/\Yp>=-^gl'(Xi,-,Xt,)(Y^-, Yp~). We introduce four basic operations on 'S1''11. Let <y be an element of 3)/''1' and let V denote. (12).

(4) On Higher Order Einstein Spaces 75 the Rimannian connection on M(I) Contraction c maps 'SV'9 into 'St'-1-"-1, li p=Q or g=0, set c(&>)=0.. If both p and q^l, then we put d.?) c(oj)(x,,-,Xi,-i)(Y,.-, y,-i) =T.Ê,.,X,.-,Xp-,)(Ei,, Y,,-, V,-i), *=1. (X,y,e3£(M)), where [Ei, •••, En] is locally defined orthonormal frame field. (II) The first Bianchi sum b maps 'S"'1' into CSP+l."-1 and is defined as follows. If ^=0,. set b(ai)=Q. If q>l, set b(u)(Xi,-,Xp^)(Y,,'-, y,-i) d.8) =^(-i}k+^(x,,-,x,,-,x^)(x,, Vi,-, y,-i), !^)(X, A=l. (X.; V,e3£(M)), where V denotes omission as usual. (III) The covariant differentation Vx maps '3)p'17 into 'S"'1' and is defined by. (1.9) FxM?,-,^>)= ^(^(î,-,^))-2.^(î,-, Fx(^*),-,^),. (x,x^W)). (IV) The second Bianchi sum D maps 'S^9 into c£)p+l''?. We set f+i. V. (1.10) D(w}(Xi, -, ^+i)=2.(-l)A+l F^(a))(^-i, -, X,., •-, Xp^), A=l. (^,e3£(M)). We shall call w e ®"''7 the double form satisfying the first or the second Bianchi identity, according to b{oj)=0 or Z)(&))=0. It is well known that g and 7? satisfy the first and the second Bianchi identities. 2. Lemmas In this section, we shall give the lemmas for later use. LEMMA 1. Let (D and 0 be the elements o/®/''9 and S)r's respectively. Then we have the. following formulas (e. g. see [3] and [5]): (2.1) c(g/\w)=g/\c(&>)+(n-p-q)&>;. (2.2) b(uê)==b((o)/\e+(-l)tl+''u^b(0); (2.3) be = cd;. (2.4) f7x(w/\e)=7x(w)/\Q+a>/\^x(8); (2.5) c7x=VxC;. (2.6) D(w/\6)=D((o)/\9+(-l)t'u/\D(9);. (13).

(5) 76 Izumi HASEGAWA. (2.. 7) 8). c(g"-l^c''(Rt'))=g'i-l/\c"+l(Rt'}+(k-l)(n-^+2+k)g't-2/\c"(R"};. (2.. 9). tr(gk)=Tnn^-. (2.. 10). (2.. 11). (2.. c(g'')=k(n-k+l)g"-1;. .ii\ n\. c21'Rt'=tr(c21'-l(RI'))=-=(2p-k)[tr(c"(Rt'))=-=(2p)ltr(Rt');. ."-'A r"( R")} = {^Z^±kM^R'^. 'A C"(K v)) = tM^fl)T(2^)T.. When n^ 4^—2, let L be a curvature structure of order 2^—1 defined by. (2.12) L=^^,^J^^-^-^gk-l^ck(RI')+c(R1'). ^2 k\n'/^(n-'. Using Lemma 1, we have the following: LEMMA 2.. (2.13) c(L)=^^mn^4p+2) ^gk-2^c"(Rt'); ^2 k\W=l{n-4p+2+j)':' ''" '" ". .14) tr(L}={n^p+2\c2w^-. l=^n^2p+T)(2p)T.. Next lemma follows from D(Rt>)==0. LEMMA 3. (2.15) D(c'l(Rll))(Xi, •-, X^-,.^)( Y,, •••, Y2p-,,) =k^7E,{c"-l(Rt'))(X,,--,X2t,-^)(Ej, ¥,,-••, Y^-k),. (Z,, Y^K(M)). 3. An p-th Einstein space and it's properties In this section, we shall consider an n(^^p~2)-dimenswnal Riemannian manifold M satis-. fying the following curvature condition : (3.1) L^fg2"-1 where L denotes the curvature structure of order 2p—l defined by (2.12) and / is a smooth function on M. Using Lemma 2, we obtain. (n-^p+2)(n-2p)\c2t'(R1'). (3,2) ^^_^^^_^_ DEFINITION. A Riemannian manifold M of dimension n-^4p—2 is called an p-f.h Einstein space, if M. satisfies the follo'wing cu.rvature condition: (3.3) L^fg2"-1,. ,^,,, ^(n^P±2){n^-2p)\c2tl(Rt')_. -(2^lT^.. REMARK. In the case f=\,M is an usual Einstein space.. THEOREM 1. If an p-th Einstein space is of dimension n^^p—1, then c2 (R ) is constant. PROOF. Let X be any vector field and let [X, Xi, •••, ^2^-2} be any orthononnal system. (14).

(6) On Higher Order Einstein Spaces 77 of locally defined vector fields containing X. Using Lemmas 1, 2 and 3, we obtain. (n-^p+2)(n-2p)\ ^^ p^. ^^_^_^(. =2 ((W)^/'-'(X, X,, -, X2t,-2)(E,, X,, -, X2P-2) J'=l. =2 VEW(X, Zl, •••, X2ft-2}(E,, Xl, •-, Xzp-z). ^ n -4/»+2. _-s-l. (34) ~^k\ Tl'^(n-^p+2+j) (g'!-2^D(ck(Rll)))(X, Zi, -, X^){X,, -, X^p-2) =D(c(L)}(X, Zi, •-, X^){X,, •-, X^-2). (n-^p+2)(n-2p)\ ^^ pp^r^p-r ^2p)Vn\ " 7-AC"AK' ;cl , Xl, •", X2P-2)(Xl, •", X^P-2. (n-'ip+2)(n-2p+2){n-2p)l ^ /^pfo^. "2p~n\'~"" ~'" V^C^K"^-. For n^4/)—l, we have. (3.5) Fx(c2W ))=(), whence we obtain that c2t>(Rt>) is constant. Q. E. D. The 2p -th sectional curvature 7-2;, of Thorpe [6] is given by (—2)t> Rtl(Xi, •••, X2t>)(Xi, •••, Xip). (3'6) râ}=^^2P^x[^^'". for any 2^-plane o-e Gzp(M), where {^"i, •••, ^2?} is a base of CT and G2p(M) denotes a Grassmann bundle of tangent 2p -planes of MIn the case p=l, 7-2 is the usual sectional curvature. A Riemannian manifold M of dimension n ^ 2p is called the space of p -constant curvature /c, if the 2p -th sectional curvature 7-2^ is the constant K on Gzp(Af).. LEMMA 4. A Riemannian manifold M is the space of p-constant curvature K if and only if 7?'' satisfies the following cwvat.u.re condit.i.on: K. .2p (3,7) Rtl=-^y,g2. THEOREM 2. A s^zce of p-constant curvatuf'e K of dimension n^4rp—2 is an p-th Einstein. space which, satisfies (3.8) c2"(Rt')=, (o^).'n'o^-. (-2r(n-2fi)T-. PROOF. It is easy to show this theorem from Lemma 4. Q. E. D.. COROLLARY (Generalized Schur's Theorem). Let M be an n(^4p—l)-dimenswnal Riemannian manifold. If the 2p-th sectional curvature f •if is constant on each fiber of Gii>{M),. (15).

(7) 78 Izumi HASEGAWA then M is of p-constant cnrvatnre. REMARK. More generally, this corollary is satisfied for n>2p. Next, as the generalization of the conformal curvature tensor C, we define the p -th conformal curvature structure C/> by the formula [3]:. (3.9) ^=R^-^^l)^^g^c'W, (n^p-1). LEMMA 5. (3.10) Ci,=0 for n =4^-1. PROOF. See Proposition 2. 6. in [3]. Q. E. D.. Using this lemma, we get the following: THEOREM 3. An (4:p—l}-d.imensionalp-th Einstein space M is a space of p-constant curvature.. Furthermore we obtain the following :. THEOREM 4. In order that an p-th Einstein space M of dimension n^4-p is of p-constant curvatw'e, it is necessary and sufficient that M is p-conformally flat. We omit the proof, which is a straightforward calculation.. Addendum : Let M be an n(^2p}-dimensmna\ Riemannian manifold and let Qp,^ be the curvature structure of order 2p-k defined by. Q^=cW)-(n^j^^^-k (0^2,-l). We shall consider the space M with Qi,,,,=Q. 1). In the case Qi,i=0,M is an usual Einstein space. 2). Qp,o=0=>Q/>,i=0=>"-=>Op,2p-i=0.. 3). If n>2p, then c2tl(Rt'} is constant. 4). If n^4:p—2 and ^=1, then M is the p-th Einstein space. 5). In the case n>2p and k=0,M is of ^-constant curvature. References [1] Gray, A. (1970), Some relations between curvature and characteristic classes. Math. Ann. Vol. 184, p. 257 - 267.. [2] Hasegawa, I. (1975), Characterization of the ^-conformally flat Riemannian manifold. Hokkaido Math. J. Vol. 5, (in press).. [3] Kulkarni, R. S. (1972), On the Bianchi identities. Math. Ann. Vol. 199, p. 175-204. [4] Nasu, T. (1975), On conformal invariants of higher order. Hiroshima Math. J. Vol. 5, p. 43 — 60. [5] Tachibana, S. (1972), On a generalization of Schur's theorem in Riemannian and Kahlerian spaces. Differential Geometry, in honor of K. Yano. Kinokuniya, Tokyo, p. 459 — 468. [6] Thorpe, J. A. (1964), Sectional curvatures and characteristic classes. Ann. of Math. Vol. 80, p. 429 - 443.. (16).

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