A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Abstract

The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied.

18 pages

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Cited by in corpus (8)

• Natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric
• Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian-Norden Type
• Holomorphic submanifolds of some hypercomplex manifolds with Hermitian and Norden metrics
• On Geometry of Manifolds with Some Tensor Structures and Metrics of Norden Type
• Quaternionic Kaehler manifolds with Hermitian and Norden metrics
• Natural Connections with Totally Skew-Symmetric Torsion on Manifolds with Almost Contact 3-Structure and Metrics of Hermitian-Norden Type
• On the Geometry of Connections with Totally Skew-Symmetric Torsion on Manifolds with Additional Tensor Structures and Indefinite Metrics
• Lie groups of dimension 4 and almost hypercomplex manifolds with Hermitian-Norden metrics