Second order parallel tensor in real and complex space forms

Ramesh Sharma

doi:10.1155/s0161171289000967

Abstract

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.

Highlights

In 1923, Eisenhart [l] proved that if a positive definite Riemannian manifold admits a second order- parallel symmetric tensor other than a constant multiple of the metric tensor, it is reducible. ]n 1926, Levy [2] proved that a second order parallel symmetric non-singular tensor in a space of constant curvature is proportional to the metric tensor
In Theorem 3 we have proved that a Killing vector field in a non-flat complex space form is analytic vector field of J
Our result assumes the vector field to be just affine Killing and proves it to be Killing and analytic in a complex space form, whereas Yano’s result proves a Killing vector to be analytic if the space is compact Kaehler

Summary

Introduction

In 1923, Eisenhart [l] proved that if a positive definite Riemannian manifold admits a second order- parallel symmetric tensor other than a constant multiple of the metric tensor, it is reducible. ]n 1926, Levy [2] proved that a second order parallel symmetric non-singular (with non-vanishing determinant) tensor in a space of constant curvature is proportional to the metric tensor. In 1923, Eisenhart [l] proved that if a positive definite Riemannian manifold admits a second order- parallel symmetric tensor other than a constant multiple of the metric tensor, it is reducible. ]n 1926, Levy [2] proved that a second order parallel symmetric non-singular (with non-vanishing determinant) tensor in a space of constant curvature is proportional to the metric tensor. Using_Theorem__2_ it has been proved in Theorem 3 t_h_a_t___.an affine Killing vector field in a non-flat co,nplex space form is Killing and analytic.

Results

Conclusion