A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

Abstract

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly constant almost complex structures and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.

20 pages, This paper is dedicated to Professor Kostadin Gribachev, the author's teacher in the theory of manifolds, on the occasion of his 70th birthday

References in corpus (2)

• Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure
• On hypercomplex pseudo-Hermitian manifolds

Cited by in corpus (13)

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• The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor
• Holomorphic submanifolds of some hypercomplex manifolds with Hermitian and Norden metrics
• On Geometry of Manifolds with Some Tensor Structures and Metrics of Norden Type
• Moduli spaces of oriented Type A manifolds of dimension at least 3
• 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
• Hypercomplex structures with Hermitian-Norden metrics on four-dimensional Lie algebras
• Lie groups of dimension 4 and almost hypercomplex manifolds with Hermitian-Norden metrics
• Moduli spaces of Type~$\mathcal{B}$ surfaces with torsion