ρ-共形平坦なリーマン空間について
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(2) Journal of Hokkaido University of Education (Section HA) Vol. 30, No. 2 March, 1980. ^m^H±Wm (^ 2 ^A) Hi 30 ^ ^ 2 -?- Bgffi 55 ^ 3 ^. Remark on the ^-conformally Flat Riemannian Manifold. Izumi HASEGAWA Mathematics Laboratory, Sapporo College, Hokkaido University of Education,. Sapporo 064. :^jii%n: p-^mw^s:)) —7>^^^^^-c. umw^m^^^^ Abstract Let M be an % - dimensional Riemannian manifold. As the generalization of the conformal curvature tensor, R. S. Kulkarni [2] defined the p-th confomal curvature tensor. The. Riemannian manifold M is called the />-conformally flat one if the p -th conformal curvature tensor vanishes and nt^^PIn the previous paper[l], we gave one characterization of the ^-conformally flat Riemannian. manifold. In this paper, we shall give another characterization of the p -conformally flat Riemannian manifold.. 1. Preliminaries For terminology and notation, we follow the previous paper [1]. The p-th conformal curvature. tensor Cp is given by ZP (— 1 \k. (1. D Cp=Ri3+^ ,,n.-i^_L;^o_^ 9k/\ckRP. k=i /?!ii'j=o {n—. Of course, Ci is the Weyl conformal curvature tensor.. T. Nasu and M. Kojima[3] proved the following Lemma. A Riem.annian manifold M of dimension n{ ^ 4:p) ?'s p-conformally flat if and only if Rp satisfies (1. 2) ^e{a)RP{X^Xa,,-,Xa^Xa,,X^,--,Xa^=0 a. for every Ap-tuple of orthonormal vectors {Xi, Xz, •", X^} at each point of M. Here, the sum extends over all 2^-tuples a=(o'i, 02, ••', a2p)with cn=i or i+2p and £(o')= 1. (91).
(3) Izumi HASEGAWA. or — 1 according to the number of cii such that a; =i-}-2p is even or odd.. 2. Theorem. Theorem. In order that a Riemannian manifold M of dimension n ( ^ 4:p ) ^s p-con formally flat, it is necessary and sufficient that Rp satisfies (2. 1) RP(Xz,X2,-,X2p){X2P+l,X2P+2,-,X4p)=0 for every ^p-tuple of orthonormal vectors {Xi, Xz, •••, X^p} at each point of M. Proof. If M is /)-conformally flat and {^i, ^2, "•, ^4/>} is any 4^-tuple of orthonormal. vectors, from (1. 1), it is clear that Rp satisfies (2. 1). Conversely, we assume that Rp satisfies (2. 1) for every 4^-tuple of orthonormal vectors i, Xs,'", X^p}. is clear that \X\JrX2p+-\.,X2~^~X2p+2,'",X2p~\~X^p,X\ —Xzp+i, Xz —Xzp+2, "', Xzp —-X-tp. is the 4^-tuple of orthogonal vectors.. From this fact and simple computations, we have (1. 2). Using the lemma, we conclude the theorem.. Q.. E.. D.. References [1] Hasega\va, I. (1976), Characterization of the p-conformally flat Riemannian manifold. Hokkaido Math. J., Vol. 5, 139-144.. [2] Kulkarni, R. S. (1972), On the Bianchi identities. Math. Ann., Vol. 199, 175-204. [3] Nasu, T. and Kojima, M., Characterizations of Riemannian manifolds with vanishing conformal curvature tensor of higher order, (to appear).. (92).
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