Homogeneous almost para-Hermitian structures (Q1891442)

Let \((M,g)\) be a simply connected pseudo-Riemannian manifolds and \(J\) an almost product structure behaving as an anti-isometry. A homogeneous almost para-Hermitian structure is a (1,2)-tensor field \(S\) such that \(g\), \(R\), \(S\) and \(J\) are parallel under the action of the connection \(\widetilde {\nabla} = \nabla - S\) (we denote by \(\nabla\) the Levi-Civita connection of \(g\) and by \(R\) the curvature tensor of \(\nabla)\). In this paper, the classification of homogeneous almost para-Hermitian manifolds is given, by using previous results of the authors concerning the decomposition of some tensor spaces under the action of the pseudo- orthogonal group. Finally, some examples for the respective structures are given.