Vladimir Rovenski

Deforming metrics of foliations

Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D⊥-closed. Assuming that D⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.

On solutions to equations with partial Ricci curvature

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold (Mn, g) endowed with the complementary orthogonal distributions D1 and D2. We provide conditions for symmetric (0, 2)-tensors T of a simple form (defined on M) to admit metrics g̃, 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 Di. 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.

Mixed gravitational field equations on globally hyperbolic spacetimes

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting T (M) = D ⊕ R∇T of M, as exhibited by Bernal and S´ anchez (2005 Commun. Math. Phys. 257 43–50). We give sufficient geometric conditions (e.g. T is isoparametric and D is totally umbilical) for the existence of exact solutions −β dT ⊗ dT + g to mixed field equations in free space. We linearize and solve the mixed field equations RicD(g)μν − ρD(g) gμν = 0 for empty space, where ρD(g) is the mixed scalar curvature of foliated spacetime (M, D) (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If g� = g0 + �γ is a solution to the linearized field equations, then each leaf of D is totally geodesic in (R 4 \ R, g� ) to order O(�) . We derive the equations of motion of a material particle in the gravitational field gμν governed by the mixed field equations RicD(g)μν − ρD(g )ω μων − � gμν = 2 πκ c −2 Tμν − 1 Tg μν . In the weak field (� � 1) and low velocity (� v� /c � 1) limit, the motion equations are d 2 r/dt 2 =∇ φ + F, where φ = (�/ 2)c 2 γ00.

Integral formulae for a Riemannian manifold with two orthogonal distributions

We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.

VARIATIONAL FORMULAE FOR THE TOTAL MEAN CURVATURES OF A CODIMENSION-ONE DISTRIBUTION

where (M, g) is a fixed compact oriented n-dimensional Riemannian manifold with the curvature tensor R, D ranges over the space of all codimension-one distributions (plane fields) on M , m = 1, 2, . . . n−1 (especially, m even), N the unit normal of D, and σm(D) is the m-symmetric function of the shape operator AN : D → D. For example, the minimal value of I2(D) can be used for estimation from below of the energy E(N) of N , because [1], E(N) ≥ 1 2n−2 ∫

Prescribing the Mixed Scalar Curvature of a Foliation

We introduce the flow of metrics on a foliated Riemannian manifold (M g), whose velocity along the orthogonal (to the foliation \(\mathcal{F}\)) distribution \(\mathcal{D}\) is proportional to the mixed scalar curvature, Scalmix. The flow preserves harmonicity of foliations and is used to examine the question: When does a foliation admit a metric with a given property of Scalmix (e.g., positive/negative or constant)? If the mean curvature vector of \(\mathcal{D}\) is leaf-wise conservative, then its potential function obeys the nonlinear heat equation \((1/n)\partial _{t}u = \Delta _{\mathcal{F}}\,u + (\beta _{\mathcal{D}} + \Phi /n)u + (\Psi _{1}^{\mathcal{F}}/n){u}^{-1} - (\Psi _{2}^{\mathcal{F}}/n){u}^{-3}\) with a leaf-wise constant \(\Phi \) and known functions \(\beta _{\mathcal{D}}\geq 0\) and \(\Psi _{i}^{\mathcal{F}}\geq 0\). We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of Schrodinger operator \(\mathcal{H}_{\mathcal{F}} = -\Delta _{\mathcal{F}}-\beta _{\mathcal{D}}\mathrm{id\,}\)) the flow of metrics admits a unique global solution, whose Scalmix converges exponentially to a leaf-wise constant. Hence, in certain cases, there exists a \(\mathcal{D}\)-conformal to g metric, whose Scalmix is negative, positive, or negative constant.

Variations of the total mixed scalar curvature of a distribution

We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result 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. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).

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.

The mixed Yamabe problem for 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 leafwise 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 on a closed manifold (leaf). Then we assume that a foliation is defined by an orientable fiber bundle, and use spectral parameters of certain Schrödinger operator to find solution, which is an attractor of the equation.

The Partial Ricci Flow for Foliations

We study the flow of metrics on a foliation (called the Partial Ricci Flow), ∂ t g = −2 r(g), where r is the partial Ricci curvature; in other words, for a unit vector X orthogonal to the leaf, r(X, X) is the mean value of sectional curvatures over all mixed planes containing X. The flow preserves total umbilicity, total geodesy, and harmonicity of foliations. It is used to examine the question: Which foliations admit a metric with a given property of mixed sectional curvature (e.g., constant)? We prove local existence/uniqueness theorem and deduce the evolution equations (that are leaf-wise parabolic) for the curvature tensor. We discuss the case of (co)dimension-one foliations and show that for the warped product initial metric the solution for the normalized flow converges, as t → ∞, to the metric with \(r = \Phi \,\hat{g}\), where \(\Phi \) is a leaf-wise constant.