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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
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