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Zbl 0696.53012
Sharma, Ramesh
Second order parallel tensor in real and complex space forms. (English)
[J] Int. J. Math. Math. Sci. 12, No.4, 787-790 (1989). ISSN 0161-1712; ISSN 1687-0425/e

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.
[E.Vassiliou]

MSC 2000:: *53B20 Local Riemannian geometry
53B35 Complex differential geometry (local)

Keywords: second order parallel tensor; space form; metric tensor; affine Killing vector field

Cited in: Zbl 1179.53042 Zbl 1135.53317 Zbl 0872.53031 Zbl 0782.53025

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