On the geometry of the connection with totally skew-symmetric torsion on almost complex manifolds with Norden metric

Abstract

We consider an almost complex manifold with Norden metric (i. e. a metric with respect to which the almost complex structure is an anti-isometry). On such a manifold we study a linear connection preserving the almost complex structure and the metric and having a totally skew symmetric torsion tensor (i. e. a 3-form). We prove that if a non-Kaehler almost complex manifold with Norden metric admits such connection then the manifold is quasi-Kaehlerian (i. e. has non-integrable almost complex structure). We prove that this connection is unique, determine its form, and construct an example of it on a Lie group. We consider the case when the manifold admits a connection with parallel totally skew-symmetric torsion and the case when such connection has a Kaehler curvature tensor. We get necessary and sufficient conditions for an isotropic Kaehler manifold with Norden metric.

15 pages

References in corpus (2)

• On the geometry of quasi-Kaehler manifolds with Norden metric
• Canonical connection on quasi-Kaehler manifolds with Norden metric

Cited by in corpus (5)

• Natural connection with totally skew-symmetric torsion on almost contact manifolds with B-metric
• Distinguished connections on $(J^2=\pm 1)$-metric manifolds
• A connection with parallel totally skew-symmetric torsion on a class of almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics
• On Geometry of Manifolds with Some Tensor Structures and Metrics of Norden Type
• On some Lie groups as 5-dimensional almost contact B-metric manifolds with three natural connections