Second order parallel tensor in real and complex space forms (Q910727)

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.