arXiv research

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…

Develops a theorem for a 6D manifold with boundary.

Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.

The paper proves formulas and theorems for J-Witten deformation on specific manifolds.

The paper proves a theorem for a twisted Dirac operator on specific manifolds.

The paper proves a theorem for a twisted Dirac operator on specific manifolds.

Study connects spectral and algebraic torsion in geometric contexts.

It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a p…

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.

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…

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 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…

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…

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

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 …

Maps are shown to be Riemannian products with Ricci-flat fibers.

Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.

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…

Study on properties of tangential hypersurfaces in product-like manifolds.

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.

In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.

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…

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…

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

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

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

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fund…

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.

We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin$^{c, r}$ structure carrying a partially pure spinor field. We study various integrability conditions of the alm…

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its Kähler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to…

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when …

We present the necessary and sufficient conditions for constructing gradient Ricci almost solitons that are realized as warped products. This will be done by means of Bishop-O'Neill's formulas and a particular study of Riemannian manifolds satisfying a Ricci-Hessian type equation. We prove existence results and give an…

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

The main aim of the present work is to obtain some curvature properties of the manifolds from two classes of Riemannian product manifolds. These classes are two basic classes from Naveira classification of Riemannian almost product manifolds.

The paper explores warped-like product metrics with exceptional holonomy groups.

We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows $M$ carrying non-trivial solutions to that equation in case $M$ is a local Riemannian product, a Sasakian manifold or 3-dimensional.

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …

The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of an almost quaternion Kaehler product manifold.

The paper studies Einstein-Hilbert action on complex manifolds.

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.

Proves almost flat spin^c manifolds bound compact manifolds.

This paper classifies 4D spin manifolds with skew Killing spinors.

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 show that any closed spin manifold not diffeomorphic to the two-sphere admits a sequence of volume-one-Riemannian metrics for which the smallest non-zero Dirac eigenvalue tends to zero. As an application, we compare the Dirac spectrum with the conformal volume.

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold…

The class W_1 of conformal Riemannian P-manifolds is the largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric. This class is an analogue of the class of conformal Kaehler manifolds in almost Hermitian geometry. In the p…