We determine a new class of paracontact paracomplex Riemannian manifolds derived from certain cone construction, called para-Sasaki-like manifolds, and give explicit examples. define hyperbolic extension paraholomorphic manifold, which is local product two spaces equal dimension, show that it manifold. If the original manifold complete Einstein space negative scalar curvature, then its curvatur...

Symmetric connections that are compatible with semi-Riemannian metrics can be characterized using an existence result for an integral leaf of a (possibly non integrable) distribution. In this paper we give necessary and sufficient conditions for a left-invariant connection on a Lie group to be the Levi–Civita connection of some semi-Riemannian metric on the group. As a special case, we will con...