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On the holomorphic sectional curvature on an indefinite Kähler manifold. (English. Abridged French version) Zbl 0765.53051

Summary: It is shown that for a general indefinite almost Hermitian manifold boundedness on the holomorphic sectional curvature leads to spaces of pointwise constant holomorphic sectional curvature. A family of null holomorphically flat indefinite Kähler manifolds but not of constant holomorphic sectional curvature is exhibited. Finally, we provide two criteria for the constancy of the holomorphic sectional curvature.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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