Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds

Details Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds Natural Connection with Totally Skew-Symmetric Torsion on Riemannian Almost Product Manifolds

2010-01-27

arxiv.org/abs/1001.4946v2

International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 1 (2012) 1250003

Mathematics

Differential Geometry

14 pages, a revised edition, an example is added

Scientific paper

10.1142/S021988781250003X

On a Riemannian almost product manifold $(M,P,g)$ we consider a linear connection preserving the almost product structure $P$ and the Riemannian metric $g$ and having a totally skew-symmetric torsion. We determine the class of the manifolds $(M,P,g)$ admitting such a connection and prove that this connection is unique in terms of the covariant derivative of $P$ with respect to the Levi-Civita connection. We find a necessary and sufficient condition the curvature tensor of the considered connection to have similar properties like the ones of the K\"ahler tensor in Hermitian geometry. We pay attention to the case when the torsion of the connection is parallel. We consider this connection on a Riemannian almost product manifold $(G,P,g)$ constructed by a Lie group $G$.

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