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

Abstract

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 Kähler 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 $G$.

arXiv comment14 pages, a revised edition, an example is added

References in corpus (1)

• Vanishing Theorems and String Backgrounds

Cited by in corpus (2)

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• Invariant tensors under the twin interchange of the pairs of the associated metrics on almost paracomplex pseudo-Riemannian manifolds