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4th Conference Geometry and Topology of Manifolds
April 28 - May 4, 2002
Technical University of Lodz; University Mining and Metallurgy, Cracow; Jagiellonian University, Cracow
Krynica, Poland

Organizers
Jan Kubarski (chairman), Lodz, Poland; Tomasz Rybicki, Cracow, Poland; Robert Wolak, Cracow, Poland

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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.