A natural connection on a basic class of Riemannian product manifolds

TL;DR

This paper explores a natural connection on a special class of Riemannian product manifolds, analyzing its curvature, torsion, and conformal invariance, and providing explicit examples involving Lie groups.

Contribution

It introduces and studies a natural connection on a broad class of Riemannian product manifolds, extending concepts analogous to Kähler geometry and examining curvature and torsion properties.

Findings

Weyl tensors for the connection D and Levi-Civita coincide

Curvature tensor of D is invariant under conformal transformations

Constructed explicit example with a Lie group where D has non-parallel torsion

Abstract

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M; P; g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kahler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M; P; g) (i.e. DP = Dg = 0). We find necessary and suffcient conditions the curvature tensor of D to have properties similar to the Kahler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion.We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a at connection. We construct an example of the considered manifold by a Lie group where D is a at connection with non-parallel torsion.