Para-Sasaki-like Riemannian manifolds and new Einstein metrics

TL;DR

This paper introduces a new class of para-Sasaki-like Riemannian manifolds derived from cone constructions, providing explicit examples and establishing their Einstein properties, thereby expanding the landscape of Einstein manifolds with negative scalar curvature.

Contribution

It defines para-Sasaki-like Riemannian manifolds via cone constructions and demonstrates their Einstein properties, including explicit examples and hyperbolic extensions.

Findings

New class of para-Sasaki-like Riemannian manifolds introduced

Hyperbolic extension preserves Einstein property with negative scalar curvature

Explicit examples of complete Einstein para-Sasaki-like manifolds provided

Abstract

We extract a new class of paracontact paracomplex Riemannian manifolds arising from certain cone construction, call it para-Sasaki-like Riemannian manifold and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces with equal dimensions, showing that it is a para-Sasaki-like Riemannian manifold. If the starting paraholomorphic paracomplex Riemannian manifold is complete Einstein with negative scalar curvature then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian manifold with negative scalar curvature thus producing new examples of complete Einstein Riemannian manifold with negative scalar curvature.