Sharma, Ramesh Second order parallel tensor in real and complex space forms. (English) Zbl 0696.53012 Int. J. Math. Math. Sci. 12, No. 4, 787-790 (1989). 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 Cited in 5 ReviewsCited in 32 Documents MSC: 53B20 Local Riemannian geometry 53B35 Local differential geometry of Hermitian and Kählerian structures Keywords:second order parallel tensor; space form; metric tensor; affine Killing vector field PDFBibTeX XMLCite \textit{R. Sharma}, Int. J. Math. Math. Sci. 12, No. 4, 787--790 (1989; Zbl 0696.53012) Full Text: DOI EuDML