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Vladimir Rovenski

Publications and source records attributed to Vladimir Rovenski.

At least 55 records · Page 3Linked to original sources
arXiv2020

Deforming convex bodies in Minkowski geometry

We introduce and study deformation $T_{{\bf b},ϕ}$ of Minkowski norms in $\mathbb{R}^n$, determined by a set ${\bf b}=(β_1,\ldots,β_p)$ of linearly independent 1-forms and a smooth positive function $ϕ$ of $p$ variables. In particular, the $T_{{\bf b},ϕ}$-image of a Euclidean norm $α$ is a Minkowski norm, whose indicatrix is a rotation hypersurface with a $p$-dimensional axis passing through the origin. For $p=1$, our deformation generalizes construction of $(α,β)$-norm; the last ones form a rich class of "computable" Minkowski norms and play an important role in Finsler geometry. We use compositions of $T_{{\bf b},ϕ}$-deformations with ${\bf b}$'s of length $p$ to define an equivalence relation $\overset{p}\sim$ on the set of all Minkowski norms in $\mathbb{R}^n$. We apply M. Matsumoto result to characterize the cases when the Cartan torsions of a norm and its $T_{{\bf b},ϕ}$-image either coincide or differ by a $C$-reducible term.

math.DG↗
arXiv2020

On Ricci curvature of metric structures on $\mathfrak{g}$-manifolds

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the ${\mathfrak{g}}$-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian ${\mathfrak{g}}$-manifolds.

math.DS↗
arXiv2020

On non-gradient $(m,ρ)$-quasi-Einstein contact metric manifolds

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,ρ)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,ρ)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,μ)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,ρ)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $η$-Einstein manifold.

math.DG↗
arXiv2020

Integral formulas for a Riemannian manifold with several orthogonal complementary distributions

In the paper we prove integral formulae for a Riemannian manifold endowed with $k>2$ orthogonal complementary distributions, which generalize well-known formula for $k=2$ and give applications to splitting and isometric immersions of Riemannian manifolds, in particular, multiply warped products, and to hypersurfaces with $k>2$ distinct principal curvatures of constant multiplicities.

math.DG↗
arXiv2020

Almost $η$-Ricci solitons on Kenmotsu manifolds

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $η$-Ricci solitons and $η$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $η$-Ricci soliton is Einstein metric if either it is $η$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $η$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $η$-Ricci solitons and gradient $η$-Ricci solitons, which illustrate our results.

math.DG↗
arXiv2020

On the Bochner technique for singular distributions

In this paper we continue our recent study of a manifold endowed with a singular or regular distribution, determined as the image of the tangent bundle under a smooth endomorphism, and generalize Bochner's technique to the case of a distribution with a statistical type structure. Following the theory of statistical structures on Riemannian manifolds and construction of an almost Lie algebroid on a vector bundle, we define the modified statistical connection and exterior derivative on tensors. Then we introduce the Weitzenbock type curvature operator on tensors and derive the Bochner-Weitzenbock type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian on a distribution.

math.DG↗
arXiv2020

The Einstein-Hilbert type action on metric-affine almost-product manifolds

We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric~connections.

math.DG↗
arXiv2020

The curve shortening flow in the metric-affine plane

We investigate for the first time the curve shortening flow in the metric-affine plane and prove that under simple geometric condition it shrinks a closed convex curve to a "round point" in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in Euclidean plane.

math.DG↗
arXiv2019

An integral formula for a pair of singular distributions

The paper is devoted to differential geometry of singular distributions (i.e., of varying dimension) on a Riemannian manifold. Such distributions are defined as images of the tangent bundle under smooth endomorphisms. We prove the novel divergence theorem with the divergence type operator and deduce the Codazzi equation for a pair of singular distributions. Tracing our Codazzi equation yields expression of the mixed scalar curvature through invariants of distributions, which provides some splitting results. Applying our divergence theorem, we get the integral formula, generalizing the known one, with the mixed scalar curvature of a pair of transverse singular distributions.

math.DG↗
arXiv2019

Variations of the Godbillon--Vey invariant of transversely parallelizable foliations

We consider a $(2q+1)$-dimensional smooth manifold $M$ equipped with a $(q+1)$-dimensional, a priori non-integrable, distribution ${\cal D}$ and a $q$-vector field ${\bf T}=T_1\wedge\ldots\wedge T_q$, where $\{T_i\}$ are linearly independent vector fields transverse to~${\cal D}$. Using a $q$-form $ω$ such that ${\cal D} = \ker\,ω$ and $ω({\bf T})=1$, we construct a $(2q+1)$-form analogous to that defining the Godbillon--Vey class of a $(q+1)$-dimensional foliation, and show how does this form depend on $ω$ and ${\bf T}$. For a compatible Riemannian metric $g$ on $M$, we express this $(2q+1)$-form in terms of ${\bf T}$ and extrinsic geometry of~${\cal D}$ and normal distribution ${\cal D}^\bot$. We find Euler-Lagrange equations of associated functionals: for variable $(ω,{\bf T})$ on $(M,g)$, and for variable metric on $(M,{\cal D})$, when distributions/foliations and forms are defined outside a "singularity set" under additional assumption of convergence of certain integrals. We show that for a harmonic distribution ${\cal D}^\bot$ such $(ω,{\bf T})$ is critical, characterize critical pairs when ${\cal D}$ is integrable and find sufficient conditions for critical pairs when variations are among foliations, calculate the index form and consider examples of critical foliations among twisted products, Reeb foliations and transversely holomorphic flows.

math.DG↗
arXiv2019

On the Betti and Tachibana numbers of compact Einstein manifolds

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein manifold $(M,g)$ with Einstein constant $α>0$ is a homo-logical sphere when the minimum of its sectional curvatures $> α/(n+ 2)$; in particular, $(M,g)$ is a spherical space form when the minimum of its sectional curvatures $> α/ n$. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold $(M,g)$ with $α< 0$.

math.DG↗
arXiv2019

The Bourguignon Laplacian and harmonic symmetric bilinear forms

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In this case, there are the exterior differential and codifferential defined on the vector space of these differential one-forms. Then a symmetric bilinear form is said to be harmonic if it is closed and coclosed as a one-form with values in the cotangent bundle of a Riemannian manifold. In the present paper we prove that the kernel of the little known Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold. We also prove that every harmonic symmetric bilinear form on a compact Riemannian manifold with non-negative sectional curvature is invariant under parallel translations. In addition, we investigate the spectral properties of the little studied Bourguignon Laplacian.

math.DG↗
arXiv2019

Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications

There are very few general theorems on the kernel of the well-known Lichnerowicz Laplacian. In the present article we consider the geometry of the kernel of this operator restricted to covariant (not necessarily symmetric or skew-symmetric) tensors. Our approach is based on the analytical method, due to Bochner, of proving vanishing theorems for the null space of Laplace operator. In particular, we pay special attention to the kernel of the Lichnerowicz Laplacian on Riemannian symmetric spaces of compact and noncompact types. In conclusion, we give some applications to the theories of infinitesimal Einstein deformations and the stability of Einstein manifolds.

math.DG↗
arXiv2018

The weighted mixed curvature of a foliated manifold

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The novel concepts of the "mixed curvature-dimension" condition and "synthetic dimension of a distribution" allow us to update the estimate of the diameter of a compact Riemannian foliation and to prove new splitting theorems for almost-product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. In the case of positive (and nonnegative) weighted mixed sectional curvature we explore the weighted generalization of Toponogov's conjecture on totally geodesic foliations.

math.DG↗
arXiv2018

A Godbillon-Vey type invariant for a 3-dimensional manifold with a plane field

We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary, \textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a vector field $T$ transverse to ${\cal D}$. Using a 1-form $ω$ such that ${\cal D} = \ker\,ω$ and $ω(T)=1$ we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on $ω$ and~$T$. For a compatible Riemannian metric on $M$, we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of ${\cal D}$. We deduce Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$ on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable, and prove that these critical pairs are not extrema.

math.DG↗
arXiv2017

Integral formula for codimension-one foliated $(α,β)$-spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. Recently, we associated a new Riemannian metric to a codimension-one foliated Finsler space and proved integral formulae for general and for Randers spaces. In the paper, we study this metric for a wider class of codimension-one foliated $(α,β)$-spaces and embody it in a set of metrics that can be viewed as a perturbed metric associated with $α$. For such metrics we calculate the Weingarten operator of the leaves and deduce the Reeb type integral formula, which can be used for $(α,β)$-spaces, e.g. Randers and Kropina spaces.

math.DG↗
arXiv2017

Integral formulas for a metric-affine manifold with two complementary orthogonal distributions

We obtain integral formulas for a metric-affine space equipped with two complementary orthogonal distributions. The integrand depends on the Ricci and mixed scalar curvatures and invariants of the second fundamental forms and integrability tensors of the distributions. The formulas under some conditions yield splitting of manifolds (including submersions and twisted products) and provide geometrical obstructions for existence of distributions and foliations (or compact leaves of them).

math.DG↗