On the holomorphic pseudosymmetry of Kähler manifolds
by
Zbigniew Olszak
Institute of Mathematics, Wroclaw University of Technology

The holomorphic pseudosymmetry is a natural generalization of the semisymmetry, in particular, the local symmetry.

A Kähler manifold M(J, g) is said to be holomorphically pseudosymmetric if its Riemann curvature R satisfies the condition

R·R=f  ~ R   ·R,

where f is a certain function on M and [R\tilde] is the derivation of the tensor algebra generated by the curvature type endomorphism

~ R   (X, Y)=X /\ Y+(JX) /\ (JY)-2g(JX, Y)J.

Basing on the Lichnerowicz's integral formulas, we can investigate the holomorphic pseudosymmetry of compact Kähler manifolds. The main result is the following:

Let M be a compact Kähler manifold with constant scalar curvature. If M is holomorphically pseudosymmetric with f >= 0, then it is locally symmetric.

Date received: April 18, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajc-14.