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On pseudosymmetric para-Kähler manifolds. (English) Zbl 0903.53025

By a para-Kähler manifold the authors mean an almost Hermitian manifold \((M,J,g)\) whose curvature tensor satisfies the additional condition \(R(JX,JY)=R(X,Y)\). The metric \(g\) does not have to be positive definite. The authors study various pseudosymmetry type conditions for the curvature. For a detailed explanation of the pseudosymmetry notion, see the paper by R. Deszcz [Bull. Soc. Math. Belg., Sér. A 44, 1-34 (1992; Zbl 0808.53012)].
The main results of the paper under review state that, for para-Kähler manifolds of dimension \(\geq 4\), pseudosymmetry, Ricci-pseudosymmetry and Weyl-pseudosymmetry reduce, respectively, to semisymmetry, Ricci-semisymmetry and Weyl-semisymmetry. The results generalize certain already known theorems for Kähler manifolds.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citations:

Zbl 0808.53012
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