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Second order parallel tensor in real and complex space forms. (English) Zbl 0696.53012
The following generalization of Levy’s theorem [cf. Symmetric tensors of the second order whose covariant derivatives vanish, Ann. Math., II. Ser. 27, 91-98 (1926)] is proved: A second order parallel tensor in a non-flat real (resp. complex) space form is proportional (resp. a linear combination with constant coefficients) to the metric tensor (resp. of the underlying Kählerian metric and Kählerian 2-form). Note that in the real case the dimension must be greater than 2. It is also proved that an affine Killing vector field in a non-flat complex space form is Killing and analytic.
Reviewer: E.Vassiliou

MSC:
53B20Local Riemannian geometry
53B35Hermitian and Kählerian structures (local differential geometry)