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

Abstract

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla$ which induces on the cotangent bundle $T^{\ast }M$ 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 $(T^{\ast }M,\overline{g})$ which are almost para-Kähler or para-Kähler and prove that the defined almost para-complex structures are harmonic. On certain hypersurfaces of $T^{\ast }M$ 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.

17 pages