International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 4, Pages 787-790

Second order parallel tensor in real and complex space forms

Ramesh Sharma

Department of Mathematics, Michigan State University, East Lansing 48824, Michigan, USA

Received 9 February 1988

Copyright © 1989 Ramesh Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Levy's theorem ‘A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor’ has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form.

It has been shown that an affine Killing vector field in a non-flat complex space form is Killing and analytic.