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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…
The Riemannian geometry of elastica in one and two dimensions is considered. An example is given of the deflexion or Frenet curvature of the elastic filament rod where the Riemannian curvature vanishes, since the curve is one dimensional.…
We produce examples of codimension one foliations of the Euclidean and hyperbolic planes with bounded geometry which are topologically products, but for which leaves are non-recursively distorted. That is, the function which compares…
The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the…
The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at…
We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow.…
We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian manifold whose Pfaffian form is a positive multiple of the volume form, and $y\in C^{1,\alpha}(M;\mathbb{R}^{n+1})$ is an isometric immersion with $n/(n+1)<…
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…
In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group…
We examine the solution of the constraints in spherically symmetric general relativity when spacetime has a flat spatial hypersurface. We demonstrate explicitly that given one flat slice, a foliation by flat slices can be consistently…
In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in [8]. In this part we mainly focus on evolution equations involving fourth order derivatives. The global existence and exponential…
We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and…
We use the transverse K\"ahler-Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse K\"ahler-Einstein structure. We also study the main features of…
In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…
The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as…
We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these…
We consider foliations of the whole three dimensional hyperbolic space H^3 by oriented geodesics. Let L be the space of all the oriented geodesics of H^3, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics…
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…
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…
We develop a gradient-flow theory for time-dependent functionals defined in abstract metric spaces. Global well-posedness and asymptotic behavior of solutions are provided. Conditions on functionals and metric spaces allow to consider the…