arXiv subjects

Vladimir Rovenski

Publications and source records attributed to Vladimir Rovenski.

71 records · Page 4Linked to original sources

arXiv2016

Prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold

The mixed scalar curvature is one of the simplest curvature invariants of a foliated Riemannian manifold. We explore the problem of prescribing the mixed scalar curvature of a foliated Riemann-Cartan manifold by conformal change of the structure in tangent and normal to the leaves directions. Under certain geometrical assumptions and in two special cases: along a compact leaf and for a closed fibred manifold, we reduce the problem to solution of a leafwise elliptic equation, which has three stable solutions -- only one of them corresponds to the case of a foliated Riemannian manifold.

math.DG↗

arXiv2016

Variations of the total mixed scalar curvature of a distribution

We examine the total mixed scalar curvature of a fixed distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g. on the distribution). For each of those cases we obtain the Euler-Lagrange equation and its several solutions. The examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows and codimension-one foliations of interesting geometric properties (e.g. totally umbilical, minimal).

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arXiv2016

Integral formulae for codimension-one foliated Randers spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don't depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure.

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arXiv2016

Integral formulae for codimension-one foliated Finsler manifolds

We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,α+β)$. Using a unit vector field $ν$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(α,β)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $β$ which simplify derivations, we express them in terms of corresponding invariants arising from $α$ and $β$. Using our approach (2012), we produce the integral formulae for ${\cal F}$ on $(M, F)$ and $(M, α+β)$, which relate integrals of mean curvatures with those involving algebraic invariants obtained from the shape operator of a foliation, and the Riemann curvature in the direction $ν$. They generalize the formulae by Brito, Langevin and Rosenberg, which show that total mean curvatures (of arbitrary order $k$) for codimension-one foliations on a closed $(m+1)$-dimensional manifold of constant curvature $K$ don't depend on a choice of ${\cal F}$.

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arXiv2016

The Einstein-Hilbert type action on foliated pseudo-Riemannian manifolds

We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

math.DG↗

arXiv2015

The mixed Einstein-Hilbert action and extrinsic geometry of foliated manifolds

We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature over all planes that contain vectors from both distributions of an almost-product structure and the variations we consider preserve orthogonality of the distributions. We derive the directional derivative $D J_{\,\rm mix}$ (of the total ${\rm S}_{\,\rm mix}$) for adapted variations of metrics on closed almost-product manifolds and foliations of arbitrary dimension. The obtained Euler-Lagrange equations are presented in two equiva\-lent forms: in terms of extrinsic geometry and intrinsically using the partial Ricci tensor. Certainly, these mixed field equations admit amount of solutions (e.g., twisted products).

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arXiv2015

The mixed Yamabe problem for harmonic foliations

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a harmonic foliation, we derive the leafwise elliptic equation and explore the corresponding nonlinear heat type equation. We assume that the leaves are compact submanifolds and the manifold is fibered instead of being foliated, and use spectral parameters of certain Schrödinger operator to find solution, which is attractor of the equation.

math.DG↗

arXiv2014

The mixed scalar curvature flow and harmonic foliations

We introduce and study the flow of metrics on a foliated Riemannian manifold $(M,g)$, whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, $\Sc_{\,\rm mix}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{\,\rm mix}$ (e.g., positive or negative)\/? We observe that the flow preserves harmonicity of foliations and yields the Burgers type equation along the leaves for the mean curvature vector $H$ of orthogonal distribution. If $H$ is leaf-wise conservative, then its potential obeys the non-linear heat equation $\dt u=nΔ_\calf\,u +(nβ_{\mathcal D}+Φ)\,u+Ψ^\calf_1 u^{-1}-Ψ^\calf_2 u^{-3}$ with a leaf-wise constant $Φ$ and known functions $β_{\mathcal D}\ge0$ and $Ψ^\calf_i\ge0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of leaf-wise Schrödinger operator $\mathcal{H}_\calf=-Δ_\calf -β_{\mathcal D}\id$) there exists a unique global solution $g_t$, whose $\Sc_{\rm mix}$ converges exponentially as $t\to\infty$ to a leaf-wise constant. The metrics are smooth on $M$ when all leaves are compact and have finite holonomy group. Hence, in certain cases, there exist ${\mathcal D}$-conformal to $g$ metrics, whose $\Sc_{\rm mix}$ is negative or positive.

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arXiv2013

The partial Ricci flow on one-dimensional foliations

A flow of metrics, $g_t$, on a manifold is a solution of a differential equation $\dt g = S(g)$, where a geometric functional $S(g)$ is a symmetric $(0,2)$-tensor usually related to some kind of curvature. The mixed sectional curvature of a foliated manifold regulates the deviation of leaves along the leaf geodesics. We introduce and study the flow of metrics on a foliation (called the 'Partial Ricci Flow'), where $S=-2 r$ and $r$ is the partial Ricci curvature of the foliation; in other words, the velocity for a unit vector $X$ orthogonal to the leaf, $-2 r(X,X)$, is the mean value of sectional curvatures over all mixed planes containing $X$. The flow preserves totally geodesic foliations and is used to examine the question: Which foliations admit a metric with a given property of mixed sectional curvature (e.g., point-wise constant)? This is related to Toponogov question about dimension of totally geodesic foliations with positive mixed sectional curvature. We first consider a one-dimensional foliation, since this case is easier. We prove local existence/uniqueness theorem, deduce the government equations for the curvature and conullity tensors (which are parabolic along the leaves), and show convergence of solution metrics for some classes of almost-product structures. For the warped product initial metric the global solution metrics converge to one with constant mixed sectional curvature.

math.DG↗

arXiv2013

On the partial Ricci curvature of foliations

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors $T$ of a simple form (defined on $M$) to admit metrics $\tilde g$, conformal to $g$, that solve the partial Ricci equations. The solutions are given explicitly. Using above solutions, we also give examples to the problem of prescribing the mixed scalar curvature related to $D_i$. In aim to find "optimally placed" distributions, we calculate the variations of the total mixed scalar curvature (where again the partial Ricci curvature plays a key role), and give examples concerning minimization of a total energy and bending of a distribution.

math.DG↗

arXiv2012

The mixed scalar curvature flow on a fiber bundle

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case of integrable orthogonal distribution the flow velocity is proportional to the mixed scalar curvature. We observe that the mean curvature vector $H$ of $D$ satisfies along the leaves the forced Burgers equation, this reduces to the linear Schrödinger equation, whose potential function is a certain "non-umbilicity" measure of $D$. On order to show convergence of the solution metrics $g_t$ as $t\to\infty$, we normalize the flow, and instead of a foliation consider a fiber bundle $π: M\to B$ of a Riemannian manifold $(M, g_0)$. In this case, if the "non-umbilicity" of $D$ is smaller in a sense then the "non-integrability", then the limit mixed scalar curvature function is positive. For integrable $D$, we give examples with foliated surfaces and twisted products.

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arXiv2012

Deforming metrics of foliations

Our results concern geometry of a manifold endowed with a pair of complementary orthogonal distributions (plane fields) and a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies conformally along one of the distributions. Then we introduce the Extrinsic Geometric Flow depending on the mean curvature vector field of the distribution, and show existence/uniquenes and convergence of a solution as $t\to\infty$, when the complementary distribution is integrable with compact leaves. We apply the method to the problem of prescribing mean curvature vector field of a foliation, and give examples for harmonic and umbilical foliations and for the double-twisted product metrics, including the codimension-one case.

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arXiv2011

Extrinsic geometric flows on foliated manifolds, III

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae concerning deformations of geometric quantities as the Riemannian metric varies along the leaves of the foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second order parabolic PDEs, for which the existence/uniquenes and in some cases convergence of a solution are shown. Applications to the problem of prescribing mean curvature function of a codimension-one foliation, and examples with harmonic and umbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given.

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arXiv2011

Extrinsic geometric flows on foliated manifolds, I

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and estimate the existence time of solutions for some particular cases. The key step of the solution procedure is to find (from a system of quasilinear PDE's) the principal curvatures of the foliation. Examples for extrinsic Newton transformation flow, extrinsic Ricci flow, and applications to foliations on surfaces are given.

math.DG↗

arXiv2010

Extrinsic geometric flows on foliated manifolds, II

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliations, foliations on surfaces, and when the EGF is produced by the extrinsic Ricci tensor.

math.DG↗

arXiv2010

On surfaces with a prescribed curvilinear projection of one field of principal directions

A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $Π$ is considered. A problem of determination of such surfaces when both principal curvatures are given over a line in $Π$ is formulated and studied. The geometric problem is reduced to the Cauchy problem for quasilinear PDE's which, under certain conditions for data, are hyperbolic and admit a unique solution. It is shown that the parallel curved (PC) surfaces in space forms provide a special class of global solutions to the geometrical problem with weaker regularity assumptions. Such solutions may be found by an iteration function sequence.

math.DG↗

arXiv2009

Some extensions of the class of convex bodies

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of $\eps$-convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer.

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