ON HYPERSURFACES SATISFYING A CERTAIN CONDITION ON THE CURVATURE TENSOR
ON HYPERSURFACES SATISFYING A CERTAIN CONDITION ON THE CURVATURE TENSOR
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满足曲率张量一定条件的超曲面
DOI：
10.2748/tmj/1178243217
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发表时间：
1968
影响因子：
0.5
通讯作者：
K. Nomizu
中科院分区：
文献类型：
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作者：
K. Nomizu
其中自同态 R(X, Y) 对 R 进行运算，作为 M 的每个点处的张量代数的导数。相反，曲率张量场 R 上的这个代数条件 (•*) 是否意味着 M 是局部对称的（即 Vi? = 0）？我们推测，在 M 不可约且完备且 d i m M ^ 3 的情况下，答案是肯定的。有关部分和相关结果，请参阅 [4]、p.ll、[9]、定理 8 和 [6]。本文的主要目的是在M是欧几里得空间中的完全超曲面的情况下给出肯定的答案。更准确地说，我们证明
where the endomorphism R(X, Y) operates on R as a derivation of the tensor algebra at each point of M. Conversely, does this algebraic condition (•*) on the curvature tensor field R imply that M is locally symmetric (i.e. Vi? = 0) ? We conjecture that the answer is affirmative in the case where M is irreducible and complete and d i m M ^ 3 . For partial and related results, see [4], p.ll, [9], Theorem 8, and [6]. The main purpose of the present paper is to give an affirmative answer in the case where M is a complete hypersurface in a Euclidean space. More precisely, we prove