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In this paper we consider a manifold $(M,\nabla )$ with a symmetric linearconnection $\nabla $ which induces on the cotangent bundle $T^*M$ of $M$ asemi-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 byE. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitianstructures on $(T^*M,\overline g)$ which are almost para-K\"ahler orpara-K\"ahler and prove that the defined almost para-complex structures areharmonic. On certain hypersurfaces of $T^*M$ we construct almost paracontactmetric structures, induced by the obtained almost para-Hermitian structures. Wedetermine the classes of the corresponding almost paracontact metric manifoldsaccording to the classification given by S. Zamkovoy and G. Nakova (2018). Weobtain a necessary and sufficient condition the considered manifolds to beparacontact metric, K-paracontact metric or para-Sasakian.

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