Einstein like $(\varepsilon)$-para Sasakian manifolds

TL;DR

This paper introduces Einstein-like $(\\varepsilon)$-para Sasakian manifolds, providing conditions for their curvature properties, scalar curvature behavior, and characterizations of hypersurfaces within indefinite Riemannian product manifolds.

Contribution

It establishes necessary and sufficient curvature conditions for Einstein-like $(\varepsilon)$-para Sasakian manifolds and characterizes their scalar curvature and hypersurface properties.

Findings

Derived conditions for Einstein-like $(\varepsilon)$-para Sasakian manifolds.

Obtained the scalar curvature and its differential equation.

Proved that certain hypersurfaces are always Einstein-like.

Abstract

Einstein like $(\varepsilon)$-para Sasakian manifolds are introduced. For an $(\varepsilon) $-para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Einstein like $(\varepsilon) $-para Sasakian manifold is obtained and it is shown that the scalar curvature in this case must satisfy certain differential equation. A necessary and sufficient condition for an $(\varepsilon) $-almost paracontact metric hypersurface of an indefinite locally Riemannian product manifold to be $(\varepsilon) $-para Sasakian is obtained and it is proved that the $(\varepsilon) $-para Sasakian hypersurface of an indefinite locally Riemannian product manifold of almost constant curvature is always Einstein like.