Related papers: Escape Metrics and its Applications
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…
We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules…
The geodesic deviation equation (GDE) describes the tendency of objects to accelerate towards or away from each other due to spacetime curvature. The GDE assumes that nearby geodesics have a small rate of separation, which is formally…
Transport-based metrics and related embeddings (transforms) have recently been used to model signal classes where nonlinear structures or variations are present. In this paper, we study the geodesic properties of time series data with a…
We consider the metrics of the General Relativity, whose energy-momentum tensor has a bounded support where it is continuous except for a finite step across the corresponding boundary surface. As a consequence, the first derivative of the…
For metrology, geodesy and gravimetry in space, satellite based instruments and measurement techniques are used and the orbits of the satellites as well as possible deviations between nearby ones are of central interest. The measurement of…
In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the…
In the context of general relativity, the geodesic deviation equation (GDE) relates the Riemann curvature tensor to the relative acceleration of two neighboring geodesics. In this paper, we consider the GDE for the generalized hybrid…
We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the…
We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used in order to strengthen boundary…
We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the…
We consider the Weil--Petersson (WP) metric on the modular surface. We lift WP geodesics to the universal cover of the modular surface and analyse geometric properties of the lifts as paths in the hyperbolic metric on the universal cover.…
We generalize the notion of cusp excursion of geodesic rays by introducing for any $k \geq 1$ the $k^{th}$ excursion in the cusps of a hyperbolic $N$-manifold of finite volume. We show that on one hand, this excursion is at most linear for…
In this paper a new formalism based on exterior differential systems is derived for perfect-fluid spacetimes endowed with an abelian orthogonally transitive G2 group of motions acting on spacelike surfaces. This formulation allows…
Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a…
In this paper, we study Riemannian zeroth-order optimization in settings where the underlying Riemannian metric $g$ is geodesically incomplete, and the goal is to approximate stationary points with respect to this incomplete metric. To…
For the Newtonian N-body problem, we study the Jacobi-Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More…
Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $\epsilon>0$, as $g\to \infty$, for a generic surface in $\mathcal{M}_g$, the error term…
Gravity is a phenomenon which arises due to the space-time geometry. The main equations that describe gravity are the Einstein equations. To understand the consequences of these field equations we need to calculate the free particle…
The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…