Almost Para-Hermitian and Almost Paracontact Metric Structures Induced by Natural Riemann Extensions

Abstract

The context of our work is 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 (due to Sekizawa and Kowalski) of the Riemann extension, introduced by Patterson and Walker (Q J Math Oxford Ser 2(3):19–28, 1952). We construct almost para-Hermitian structures on $$(T^*M,{\overline{g}})$$ which are almost para-Kahler or para-Kahler 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 Zamkovoy and Nakova (J Geom 109(1):18, 2018. https://doi.org/10.1007/s00022-018-0423-5 ). We obtain a necessary and sufficient condition for the considered manifolds to be paracontact metric, K-paracontact metric or para-Sasakian.