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Curvature properties of para-Kähler manifolds. (English) Zbl 0843.53029
Tamássy, L. (ed.) et al., New developments in differential geometry. Proceedings of the colloquium on differential geometry, Debrecen, Hungary, July 26-30, 1994. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 350, 193-200 (1996).
The authors prove that for a para-Kähler manifold $$(M, g, J)$$, the paraholomorphic (resp. antiparaholomorphic) sectional curvature $$H$$ (resp. $$K$$) can be extended to degenerate planes only if $$H$$ (resp. $$K$$) is constant. Thereafter they introduce the important class of isotropic para-Kähler manifolds for which they obtain the following local classification result. If ($$M, g, J)$$ is an isotropic para-Kähler manifold with constant scalar curvature and diagonalizable Ricci operator, then $$M$$ is a space of constant paraholomorphic sectional curvature, or $$M$$ is locally isometric to a product $$M_1 \times M_2$$ of para-Kähler manifolds of constant paraholomorphic sectional curvature $$c$$ and $$-c$$, respectively.
For the entire collection see [Zbl 0833.00034].
Reviewer: V.Cruceanu (Iaşi)

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C35 Differential geometry of symmetric spaces 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics