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

Abstract

In this paper we consider a manifold (M, ∇) with a symmetric linear connection ∇ which induces on the cotangent bundle T * M of M a semi-Riemannian metric g with a neutral signature. The metric 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, 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 * 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.Date: 9th November 2018.

Magnetic curves on cotangent bundles endowed with the Riemann extension

“…In [1] it is shown that, for any t ∈ R * , the restriction of the natural Riemann extension g to H t , denoted by g, gives rise to an almost paracontact structure (ϕ, ξ, η, g) on H t with (6.6)…”

“…More precisely, the second aim here is to study geodesics and magnetic curves on a family of level surfaces of the phase space, whose construction is recalled from [2]. In [1], these hypersurfaces are endowed with almost paracontact structures. Paracontact geometry appears in [8] as a counterpart of contact geometry.…”

“…(1) Relation(4.1) shows that every integral curve of X is a geodesic of (M, ∇) up to reparametrization. (2) In Corollary 4.3, the converse of (i), (ii), (iii) and (iv) holds if we add respectively the assumption (4.4), (4.5), (4.6) and (4.7) appearing below:C((∇X)(∇Y )) + C(R(•, X)X) = 0, (4.4) C((∇X)(∇Y )) + C(R(•, X)X) = (∇X) v , (4.5) (∇ X α) v − (i α (∇X)) v + 2C((∇X)(∇X)) + 2C(R(•, X)X) = 0, (4.6) (∇ X α) v − (i α (∇X)) v + 2C((∇X)(∇X)) + 2C(R(•, X)X) (4.7) = 2(∇X) v − α v .…”

Generalized para-Kähler spaces in Eisenhart’s sense admitting a holomorphically projective mapping

“…C.-L. Bejan [1,2] classified almost para-Hermitian spaces and found some examples of spaces with hyperbolic structures. Recently, C.-L. Bejan and G. Nakova [3] studied almost para-Hermitian and almost paracontact metric structures induced by natural Riemann extensions. Some interesting results concerning para-Kähler-like statistical submersions were obtained by G. E. Vîlcu [4].…”

Notes on some properties of the natural Riemann extension

“…As a particular situation, when a = 1 and b = 0, we get the Riemannian extension. For further references relation to the natural Riemann extension, see[5][6][7]13].…”