arXiv research

We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendr…

The canonical paracontact connection is defined and it is shown that its torsion is the obstruction the paracontact manifold to be paraSasakian. A $\mathcal{D}$-homothetic transformation is determined as a special gauge transformation. The $η$-Einstein manifold are defined, it is prove that their scalar curvature is a …

We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,η)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures …

The study of *-Ricci solitons on para-Sasakian manifolds.

The paper classifies hypersurfaces with parallel almost paracontact structures.

Study on hypersurfaces with specific geometric structures.

We prove that any contact metric $(κ,μ)$-space $(M,ξ,φ,η,g)$ admits a canonical paracontact metric structure which is compatible with the contact form $η$. We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold $(M,η)$ a sequence of com…

In monograph of D. E. Blair "Riemannian geometry of contact and symplectic manifolds" and in the paper of S. Zamkovoy "Canonical connections on paracontact manifolds", the curvature identities respectively for contact and paracontact metric manifold are proved. We obtain the curvature identity in the wider class of man…

New connections defined for a specific geometric structure.

The paper defines $η$-normality for contact and paracontact manifolds and explores their properties.

Study connects Lie groups to specific Riemannian manifolds.

An example of a three dimensional flat paracontact metric manifold with respect to Levi-Civita connection is constructed. It is shown that no such manifold exists for odd dimensions greater than or equal to five.

The paper classifies geometric structures of δ-almost Yamabe solitons on paracontact metric manifolds.

The paper explores new structures on cotangent bundles induced by natural Riemann extensions.

The notion of generalized almost paracontact structure on the generalized tangent bundle $TM\oplus T^*M$ is introduced and its properties are investigated. The case when the manifold $M$ carries an almost paracontact metric structure is also discussed. Conditions for its transformed under a $β$- or a $B$-field transfor…

We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $ξ$ is harmonic. We prove that they are characterized by the condition that $ξ$ is a Ricci eigenvector. We then investigate how harmonicity of the Reeb vector field $ξ$ of a paracontact metric manifold…

Study revises results on a specific type of 3D paracontact metric manifolds.

In this paper, we define almost paracontact and normal almost paracontact Finsler structures on a vector bundle and find some conditions for integrability of these structures. We define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracon…

The paper reclassifies almost paracontact metric manifolds into 12 classes.

We study the Schouten-van Kampen connection associated to an almost contact or paracontact metric structure. With the help of such a connection, some classes of almost (para) contact metric manifolds are characterized. Certain curvature properties of this connection are found.

Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-homothetic…

Study of $η$-Ricci solitons on $(\varepsilon)$-almost paracontact metric manifolds.

Study classifies 3D paracontact manifolds based on curvature properties.

Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and suffcient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact …

We completely describe paracontact metric three-manifolds whose Reeb vector field satisfies the Ricci soliton equation. While contact Riemannian (or Lorentz\-ian) Ricci solitons are necessarily trivial, that is, $K$-contact and Einstein, the paracontact metric case allows nontrivial examples. Both homogeneous and inhom…

Study on specific types of Riemannian manifolds with detailed tensor analysis.

The goal of this paper, using lifting theory it is to produce almost paracomplex struc- tures on the tangent bundle of almost Lorentzian r-paracontact manifold endowed with almost Lorentzian r-paracontact structure. Finally, we discuss the effect over dynamics systems of the produced geometrical structures.

In this paper eleven basic classes of almost paracontact manifolds are introduced and some examples are constructed.

We study paracontact metric $(κ,μ)$-spaces with $κ=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We w…

This paper is a study of three-dimensional paracontact metric (\k{appa},μ,ν)-manifolds. Three dimensional paracontact metric manifolds whose Reeb vector field ξ is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric (\k{appa},μ,ν)-manifolds under a condition …

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. I…

The study explores δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds.

In this paper we define and study the slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold. Some necessary and sufficient conditions for a submanifold of a Lorentzian almost paracontact manifold to be slant are obtained. We give some examples for the slant and semi-slant submanifolds of a Loren…

We study a remarkable class of paracontact metric manifolds which have no contact metric counterpart: the paracontact metric $(-1,\widetildeμ)$-spaces which are not paraSasakian (i.e. have $\widetilde h\neq0$). We present explicit examples with $\widetilde h$ of every possible constant rank and some with non-constant r…

The paper classifies 3D paracontact and almost paracosymplectic spaces.

Study of 3D Lie group structures with special Riemannian properties.

Study on a specific type of Riemannian manifolds constructed from 2D space-forms.

Study on hyperspheres in 4-spaces as special Riemannian manifolds.

The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.

We give a local classification of generalized (kappa,mu)-Paracontact Metric Manifold which satisfies the condition xi(mu)=0. An example of such manifolds is presented.

The paper constructs and classifies 3D Walker manifolds with specific structures.

The purpose of this paper is to classify paracontact metric $3$-manifolds $M^3$ such that the Ricci operator $S$ commutes with the endomorhism $φ$ of its tangent bundle $Γ(TM^3)$.

In this study, taking into considering lifting theory, we shall obtain both almost complex and paracomplex structures on the tangent bun- dle, based on almost Lorentzian r-contact and r-paracontact manifold.

The presented paper is devoted to study the curvature and torsion of slant Frenet curves in 3-dimensional normal almost paracontact metric manifolds. Moreover, in this class of manifolds, properties of non- Frenet slant curves (with null tangents or null normals) are obtained. The achieved results are illustrated by ex…

We study non-paraSasakian paracontact metric $(κ,μ)$-spaces with $κ=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$.

We prove that any non-Sasakian contact metric (κ,μ)-space admits a canonical η-Einstein Sasakian or η-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of κand μfor which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Convers…

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $$ and $\tildeμ$). This class of pseudo-Riemannian manifolds, which includes para-Sasak…

Connections with (skew-symmetric) torsion on non-symmetric Riemannian manifold satisfying the Einstein metricity condition (NGT with torsion) are considered. It is shown that an almost Hermitian manifold is an NGT with torsion if and only if it is a Nearly Kähler manifold. In the case of an almost contact metric manifo…