arXiv subjects

Vladimir Rovenski

Publications and source records attributed to Vladimir Rovenski.

At least 37 records · Page 2Linked to original sources

arXiv2024

Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds

A weak metric $f$-structure $(f,Q,ξ_i,η^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak $f$-K-contact structure, which is a weak $f$-contact structure, whose characteristic vector fields are Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the property known for $f$-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak $f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci flat, then find sufficient conditions for a weak $f$-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak $f$-K-contact structure to an $f$-K-contact structure. We define an $η$-Ricci soliton and $η$-Einstein structures on a weak metric $f$-manifold (which for $s=1$, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak $f$-K-contact manifold with an $η$-Ricci soliton structure of constant scalar curvature to be $η$-Einstein.

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arXiv2024

Willmore-type variational problem for foliated hypersurfaces

We study new Willmore-type variational problem for a hypersurface $M$ in $\mathbb{R}^{n+1}$ equipped with an $s$-dimensional foliation ${\cal F}$. Its general version is the Reilly-type functional $WF_{n,s}=\int_M F(σ^{\cal F}_1,\ldots,σ^{\cal F}_s)\,{\rm d}V$, where $σ^{\cal F}_i$ are elementary symmetric functions of the eigenvalues of the second fundamental form restricted on the leaves of $\cal F$. The first and second variations of such functionals are calculated, conformal invariance of some of $WF_{n,s}$ is also shown. The Euler-Lagrange equation for a critical hypersurface with a transversally harmonic (e.g., Riemannian) foliation $\cal F$ is found and examples with $s\le2$ and $s=n$ are considered. Critical hypersurfaces of revolution are found, and it is shown that they are a local minimum for special variations.

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arXiv2024

Foliated structure of weak nearly Sasakian manifolds

Weak contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed a new look at the theory of contact manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are automatically satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and the weak nearly Sasakian structure foliates into two types of totally geodesic foliations. Our main result generalizes the theorem by B. Cappelletti-Montano and G. Dileo (2016) about the foliated structure of nearly Sasakian manifolds.

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arXiv2024

Characterization of Sasakian manifolds

Weak contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact manifolds. In this paper, we continue our study, see arXiv:2312.11411, of a structure of this type, called a weak nearly Sasakian structure, and prove two theorems characterizing Sasakian manifolds. Our main result generalizes the theorem by A. Nicola - G. Dileo - I. Yudin (2018) and provides a new criterion for a weak almost contact metric manifold to be Sasakian.

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arXiv2023

Geometry of a weak para-$f$-structure

We study geometry of the weak almost para-$f$-structure and its subclasses. This allow us to produce totally geodesic foliations and also to take a fresh look at the para-$f$-structure introduced by A.\,Bucki and A.\,Miernowski. We demonstrate this by generalizing several known results on almost para-$f$-manifolds. First, we express the covariant derivative of $f$ using a new tensor on a metric weak para-$f$-structure, then we prove that on a weak para-${\cal K}$-manifold the characteristic vector fields are Killing and $\ker f$ defines a totally geodesic foliation. Next, we show that a para-${\cal S}$-structure is rigid (i.e., a weak para-${\cal S}$-structure is a para-${\cal S}$-structure), and that a metric weak para-$f$-structure with parallel tensor $f$ reduces to a weak para-${\cal C}$-structure. We obtain corollaries for $p=1$, i.e., for a weak almost paracontact structure.

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arXiv2023

Back to almost Ricci solitons

In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we examine compact almost Ricci solitons using the orthogonal expansion of the Ricci tensor, this allows us to substantiate the concept of almost Ricci solitons.

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arXiv2023

Generalized Ricci solitons and Einstein metrics on weak $K$-contact manifolds

We study metric structures on a smooth manifold (introduced in our recent works and called a weak contact metric structure and a weak K-structure) which generalize the metric contact and K-contact structures, and allow a new look at the classical theory. First, we characterize weak K-contact manifolds among all weak contact metric manifolds by the property well known for K-contact manifolds, and find when a Riemannian manifold endowed with a unit Killing vector field forms a weak K-contact structure. Second, we find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.

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arXiv2023

Variations of the mutual curvature of two orthogonal non-complementary distributions

On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having trivial intersection, we consider the integral of their mutual curvature, as a functional of Riemannian metrics that make the distributions orthogonal. The mutual curvature is defined as the sum of sectional curvatures of planes spanned by all pairs of vectors from an orthonormal basis, such that one vector of the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\cal D}_2$. As such, it interpolates between the sectional curvature of a plane field (if both distributions are one-dimensional), and the mixed scalar curvature of a Riemannian almost product structure (if both distributions together span the tangent bundle). We derive Euler-Lagrange equations for the functional, formulated in terms of extrinsic geometry of distributions, i.e., their second fundamental forms and integrability tensors. We give examples of critical metrics for distributions defined on domains of Riemannian submersions, twisted products and $f$-$K$-contact manifolds.

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arXiv2023

Variational problem on a metric-affine almost product manifold

We study a variational problem on a smooth manifold with a decomposition of the tangent bundle into $k>2$ subbundles (distributions), namely, we consider the integrated sum of their mixed scalar curvatures as a functional of adapted pseudo-Riemannian metric (keeping the pairwise orthogonality of the distributions) and contorsion tensor, defining a linear connection. This functional allows us to generalize the class of Einstein metrics in the following sense: if all of the distributions are one-dimensional, then it coincides with the geometrical part of the Einstein-Hilbert action restricted to adapted metrics. We prove that metrics in pairs metric-contorsion critical for our functional make all of the distributions totally umbilical. We obtain examples and obstructions to existence of those critical pairs in some special cases: twisted products with statistical connections; semi-symmetric connections and 3-Sasaki manifolds with metric-compatible connections.

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arXiv2022

On isometric immersions of sub-Riemannian manifolds

We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.

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arXiv2022

On isometric immersions of almost $k$-product manifolds

A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called the mixed curvature is also related to the extrinsic geometry of a Riemannian almost $k$-product manifold. Our inequality also contains mixed scalar curvature type invariants related to B.-Y Chen's $δ$-invariants. Applications are given for isometric immersions of multiply twisted and warped products (we improve some known optimal inequalities by replacing the sectional curvature with our invariant) and to problems of non-immersion and non-existence of compact leaves of foliated submanifolds.

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arXiv2022

On the geometry of a weakened $f$-structure

An $f$-structure, introduced by K. Yano in 1963 and subsequently studied by a number of geometers, is a higher dimensional analog of almost complex and almost contact structures, defined by a (1,1)-tensor field $f$ on a $(2n+p)$-dimensional manifold, which satisfies $f^3 + f = 0$ and has constant rank $2n$. We recently introduced the weakened (globally framed) $f$-structure (i.e., the complex structure on $f(TM)$ is replaced by a nonsingular skew-symmetric tensor) and its subclasses of weak $K$-, ${\cal S}$-, and ${\cal C}$- structures on Riemannian manifolds with totally geodesic foliations, which allow us to take a fresh look at the classical theory. We demonstrate this by generalizing several known results on globally framed $f$-manifolds. First, we express the covariant derivative of $f$ using a new tensor on a metric weak $f$-structure, then we prove that on a weak $K$-manifold the characteristic vector fields are Killing and $\ker f$ defines a totally geodesic foliation, an ${\cal S}$-structure is rigid, i.e., our weak ${\cal S}$-structure is an ${\cal S}$-structure, and a metric weak $f$-structure with parallel tensor $f$ reduces to a weak ${\cal C}$-structure. For $p=1$ we obtain the corresponding corollaries for weak almost contact, weak cosymplectic, and weak Sasakian structures.

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arXiv2022

Integral formulas for a foliated sub-Riemannian manifold

In this article, we deduce a series of integral formulas for a foliated sub-Riemannian manifold, which is a new geometric concept denoting a Riemannian manifold equipped with a distribution ${\mathcal D}$ and a foliation ${\mathcal F}$, whose tangent bundle is a subbundle of ${\mathcal D}$. Our integral formulas generalize some results for foliated Riemannian manifolds and involve the shape operators of ${\mathcal F}$ with respect to normals in ${\mathcal D}$ and the curvature tensor of induced connection on ${\mathcal D}$. The formulas also include arbitrary functions $f_j\ (0\le j<\dim{\mathcal F})$ depending on scalar invariants of the shape operators, and for a special choice of $f_j$ reduce to integral formulas with the Newton transformations of the shape operators. We apply our formulas to foliated sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of ${\mathcal F}$ and to codimension-one foliations.

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arXiv2022

On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

We introduce new metric structures on a smooth manifold (called "weak" structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures $(φ,ξ,η,g)$ and allow us to take a fresh look at the classical theory. We demonstrate this statement by generalizing several well-known results. We prove that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. We show that a weak almost contact structure with parallel tensor $φ$ is a weak cosymplectic structure and give an example of such a structure on the product of manifolds. We find conditions for a vector field to be a weak contact infinitesimal transformation.

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arXiv2021

A series of integral formulas for a foliated sub-Riemannian manifold

In this article, we prove a series of integral formulae for a codimension-one foliated sub-Riemannian manifold, i.e., a Riemannian manifold $(M,g)$ equipped with a distribution ${\mathcal D}=T{\mathcal F}\oplus\,{\rm span}(N)$, where ${\mathcal F}$ is a foliation of $M$ and $N$ a unit vector field $g$-orthogonal to ${\mathcal F}$. Our integral formulas involve $r$th mean curvatures of ${\mathcal F}$, Newton transformations of the shape operator of ${\mathcal F}$ with respect to $N$ and the curvature tensor of induced connection on ${\mathcal D}$ and generalize some known integral formulas (due to Brito-Langevin-Rosenberg, Andrzejewski-Walczak and the author) for codimension-one foliations. We apply our formulas to sub-Riemannian manifolds with restrictions on the curvature and extrinsic geometry of a foliation.

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arXiv2021

The Partial Ricci Flow on $\mathfrak{g}$-foliations

In the paper we introduce new metric structures on $\mathfrak{g}$-foliations that are less rigid than the well-known structures: almost contact and 3-quasi-Sasakian structures as well as $f$-structures with parallelizable kernel and almost para-$ϕ$-structures with complemented frames. We discuss the properties of the new structures in order to demonstrate similarities with the corresponding classical structures. Then using the flow of metrics on a $\mathfrak{g}$-foliation, we build deformation retraction of our structures with positive partial Ricci curvature onto the subspace of the aforementioned classical structures.

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arXiv2021

The Einstein-Hilbert type action on almost $k$-product manifolds

A Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called here a Riemannian almost $k$-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, and proper Dupin hypersurfaces of real space-forms. In the paper, we consider the mixed scalar curvature of such structure for $k>2$, derive Euler-Lagrange equations for the Einstein-Hilbert type action with respect to adapted variations of metric, and present them in a nice form of Einstein equation.

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arXiv2020

On the mixed scalar curvature of almost multi-product manifolds

A pseudo-Riemannian manifold endowed with $k>2$ orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in stu\-dies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbert action, and we generalize certain results on the mixed scalar curvature of pseudo-Riemannian almost product manifolds.

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