On the holomorphic pseudosymmetry of Kähler manifolds

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 R̃ · R,

where f is a certain function on M and R̃ is the derivation of the tensor algebra generated by the curvature type endomorphism

R̃(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.