arXiv research

A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…

We give some fundamental properties of the induced structures on submanifolds immersed in almost product or locally product Riemannian manifolds. We study the induced structure by the composition of two isometric immersions on submanifolds in an almost product Riemannian manifold. We give an effective construction for …

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimen…

A 4-dimensional Riemannian manifold equipped with a circulant structure, which is an isometry with respect to the metric and its fourth power is the identity, is considered. The almost product manifold associated with the considered manifold is studied. The relation between the covariant derivatives of the almost produ…

In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…

In this work, almost product and almost golden structures are studied. Conditions for those structures being Integrable and parallel are investigated. Also harmonicity of a map between almost pruduct or almost golden manifolds with pure or hyperbolic metric is discussed under certain conditions.

Study new warped product metric in contact geometry.

The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an…

The paper classifies degenerate almost complex surfaces in a nearly Kähler space.

The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that the statistical structure of a para-Kähler-like statistical manifold of constant curvature in the Kurose's sense is a Hessian structure. We also derive the main properties of statistical su…

Study of complex structures on product twistor spaces for 4D manifolds.

The paper studies affine connections on singular warped products and their curvature.

The class of the Riemannian almost product manifolds with nonintegrable structure is considered. Some identities for curvature tensor as certain invariant tensors and quantities are obtained.

A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belo…

In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold $M\times\mathbb{R}$, where $M$ is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.

We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham ope…

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

Let (M, g) be a pseudo Riemannian manifold. We consider four geometric structures on M compatible with g: two almost complex and two almost product structures satisfying additionally certain integrability conditions. For instance, if r is a product structure and symmetric with respect to g, then r induces a pseudo Riem…

We study almost contact metric structures induced by 2-fold vector cross products on manifolds with $G_2$ structures. We get some results on possible classes of almost contact metric structures. Finally we give examples.

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-He…

Non-trivial examples of Riemannian almost product structures are constructed on the product bundle of the positive and negative twistor spaces of an oriented Riemannian four-manifold. The Gil-Medrano and Naveira types of these structures are determined and a geometric interpretation of the corresponding classes is give…

Study on 3D manifolds with specific tensor structures and their properties.

Paper studies non-gradient almost Yamabe solitons and their properties.

The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.

In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined …

The notion of relativistic observer is confronted with Naveira's classification of (pseudo-)Riemannian almost-product structures on space-time manifolds. Some physical properties and their geometrical counterparts are shortly discussed.

The paper studies Einstein-Hilbert action on complex manifolds.

In a previous paper, the authors together with L. Vrancken initiated the study of $3$-dimensional CR submanifolds of the nearly K\" ahler homogeneous $\mathbb S^3\times \mathbb S^3$. As is shown by Butruille this is one of only four homogeneous $6$-dimensional nearly Kähler manifolds. Besides its almost complex structu…

The purpose of this paper is to give an effective construction for some induced structures on spheres or product of spheres of codimension 1, 2 or 3, respectively, in Euclidean space endowed with an almost product structure.

It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. It is proven that the introduced mani…

We consider higher dimensional generalisations of normal almost contact structures, the so called f.pk-structures where parallelism spans a Lie algebra g (f.pk-g-structures). Two types of these structures are discussed. In the first case, we construct an almost complex structure on a product manifold mirroring K-struct…

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonic…

Our aim in this paper is to give some examples of $(a, 1)f$ Riemannian structures (a generalization of an $r$-paracontact structure) induced on product of spheres of codimension $r$ ($r \in \{1,2\}$) in an $m$-dimensional Euclidean space ($m>2$), endowed with an almost product structure.

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost co…

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity …

The paper defines and studies almost complex structures on product manifolds and their integrability.

Researchers extend Godbillon-Vey functional to almost contact manifolds, finding critical structures.

Manifolds endowed with three foliations pairwise transversal are known as 3-webs. Equivalently, they can be algebraically defined as biparacomplex or complex product manifolds, i.e., manifolds endowed with three tensor fields of type $(1,1)$, $F$, $P$ and $J=F \circ P$, where the two first are product and the third one…

On a Riemannian almost product manifold $(M,P,g)$ we consider a linear connection preserving the almost product structure $P$ and the Riemannian metric $g$ and having a totally skew-symmetric torsion. We determine the class of the manifolds $(M,P,g)$ admitting such a connection and prove that this connection is unique …

Study coKähler structures on Lie algebras using Fino-Vezzoni correspondence.

Noting that the complete lift of a Rimannian metric defined on a differentiable manifold is not 0-homogeneous on the fibers of the tangent bundle . In this paper we introduce a new lift which is 0-homogeneous. It determines on slit tangent bundle a pseudo-Riemannian metric, which depends only on the metric . We study s…

Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of…

The study examines conditions for weak nearly cosymplectic manifolds to split into products.

In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the implications of this theorem with regard to the definition of generalized Sasakian and ge…

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 …