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

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

2009-02-05

arxiv.org/abs/0902.0868v2

C. R. Acad. Bulgare Sci., 2010, 63, p.19-28

Mathematics

Differential Geometry

15 pages

Scientific paper

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.

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