Almost para-Hermitian and almost paracontact metric structures induced by natural Riemann extensions

TL;DR

This paper explores the geometric structures induced by natural Riemann extensions on cotangent bundles, constructing almost para-Hermitian and paracontact structures, and classifying their properties in the context of differential geometry.

Contribution

It introduces new almost para-Hermitian and paracontact structures on cotangent bundles with natural Riemann extensions and characterizes their geometric classes.

Findings

Constructed almost para-Hermitian structures that are almost para-Kähler or para-Kähler.

Proved the harmonicity of the almost para-complex structures.

Derived conditions for manifolds to be paracontact, K-paracontact, or para-Sasakian.

Abstract

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla $ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is called natural Riemann extension and it is a generalization (made by M. Sekizawa and O. Kowalski) of the Riemann extension, introduced by E. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitian structures on $(T^*M,\overline g)$ which are almost para-K\"ahler or para-K\"ahler and prove that the defined almost para-complex structures are harmonic. On certain hypersurfaces of $T^*M$ we construct almost paracontact metric structures, induced by the obtained almost para-Hermitian structures. We determine the classes of the corresponding almost paracontact metric manifolds according to the classification given by S. Zamkovoy and G. Nakova (2018). We obtain a necessary and sufficient condition the considered manifolds to be paracontact metric, K-paracontact metric or para-Sasakian.