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Abstract: Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection is introduced such that the structure of these manifolds is parallel with respect to . 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 is -parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized. [ABSTRACT FROM AUTHOR]

Abstract: This paper is a survey of results obtained by the authors on the geometry of connections with totally skew-symmetric torsion on the following manifolds: almost complex manifolds with Norden metric, almost contact manifolds with B-metric and almost hypercomplex manifolds with Hermitian and anti-Hermitian metric. [Copyright &y& Elsevier]