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Holomorphically projective curvature tensors. (English) Zbl 1119.53010

The author examines the geodesic mapping of an almost Hermitian (almost para-Hermitian) manifold using the holomorphic curvature tensor. She obtains the corresponding invariant tensors. As a consequence, she gets that for the Kähler (para-Kähler) manifolds the holomorphically projective curvature tensor can be obtained either as the invariant tensor with respect to the holomorphically projective mapping using the curvature tensor, or as the invariant tensor with respect to the geodesic mapping using the holomorphic curvature tensor. An almost Hermitian (almost para-Hermitian) manifold satisfying the Rizza’s Bianchi-type identity is also considered.

MSC:

53B35 Local differential geometry of Hermitian and Kählerian structures
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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