Classifies geodesic-preserving bijections in Thurston geometries.
problem Identifying bijections that preserve geodesics in different geometries.
method Comprehensive classification and proof for various geometries.
result Complete classification of geodesic-preserving bijections in Thurston geometries.
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
FPP preserves sublinear Morse boundaries in geodesic graphs.
problem Preserving sublinear Morse boundaries in FPP.
method First passage percolation on geodesic graphs with i.i.d. passage times.
result Sublinear Morse boundaries are invariant under FPP.
Embeds directed graphs into statistical manifolds for better geodesic preservation.
problem Preserving global geodesic information in directed graphs.
method Global minimization of pairwise relative entropy and graph geodesics.
result Our embedding outperforms existing models in various evaluation metrics.
In this paper we study fundamental properties of geodesic mappings with respect to the smoothness class of metrics. We show that geodesic mappings preserve the smoothness class of metrics. We study geodesic mappings of Einstein spaces.
Flow preserves quermassintegrals, converging to a geodesic sphere.
problem Volume preservation issue in sphere mean curvature flow.
method Introduced a mean curvature flow with a global term to keep quermassintegrals fixed.
result Flow exists for all times and converges to a geodesic sphere.
Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.
problem Preserving distances on hyperbolic surfaces with boundaries.
method Proving bijections between geodesics are isometries.
result A bijection between geodesics is an isometry.
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
The two main topics of this text are as follows: Firstly, three modifications of the theorem of Beltrami will be presented for diffeomorphisms between Riemannian manifolds and a space form which preserve the geodesic circles, the geodesic hyperspheres, or the minimal surfaces, respectively. Secondly, it is defined what…
We prove: "If M is a compact hypersurface of the hyperbolic space, convex by horospheres and evolving by the volume preserving mean curvature flow, then it flows for all time, convexity by horospheres is preserved and the flow converges, exponentially, to a geodesic sphere". In addition, we show that the same conclus…
We determine the Riemannian manifolds for which the group of exact volume preserving diffeomorphisms is a totally geodesic subgroup of the group of volume preserving diffeomorphisms, considering right invariant L2-metrics. The same is done for the subgroup of Hamiltonian diffeomorphisms as a subgroup of the group of…
Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.
problem Maintain geodesic fibers and positive sectional curvatures in Riemannian submersions.
method Vary Riemannian metrics while keeping fibers totally geodesic and horizontal distribution fixed.
result Conditions for making sectional curvatures positive and existence of fat submersions.
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.
problem Characterizing strong Hamel functions and their symmetries in Finsler spaces.
method Analyzing geodesic spray, strong dual symmetries, and strong dynamical symmetries.
result Strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries.
Lattices in PSL(2,C) are omnipotent, acting on geodesics and homology.
problem Understanding the full isometry groups of hyperbolic 3-manifolds and their lattices.
method Analyzing geodesics and applying the Virtual Special Theorems.
result Every non-arithmetic lattice in PSL(2,C) is omnipotent, acting on homology.
In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox{(Cr,r≥1)}. Also, if the Einstein space Vn admits a non trivial geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold Vˉn∈C1, then Vˉn is an Einstein space. If …
A new method tracks retinal vessels more accurately than existing methods.
problem Tracking retinal vessels accurately in spherical images.
method Computing cusp-free, crossing-preserving geodesics on spherical positions and orientations.
result Crossing-preserving tracking shows clear advantages over non-crossing-preserving tracking.
Geometric framework for SPD matrices preserving subspace structures.
problem Processing SPD-valued data with preserved subspace structures.
method Thompson geometry of the semidefinite cone, extreme generalized eigenvalues, geodesic space structure.
result Novel inductive mean of SPD matrices based on Thompson geometry.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…
Over a compact oriented manifold, the space of Riemannian metrics and normalised positive volume forms admits a natural pseudo-Riemannian metric G, which is useful for the study of Perelman's W functional. We show that if the initial speed of a G-geodesic is G-orthogonal to the tangent space to the or…
We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.
We propose a special deformation of the Sasaki metric on tangent and unit tangent bundle of a Hermitian locally symmetric manifold. Geodesics of this deformed metric have different projections on a base manifold for tangent or unit tangent bundle cases in contrast to usual Sasaki metric. Nevertheless, the projections o…
Revises Gauss's Lemma using metrical distortion and differential slip.
problem Revising Gauss's Lemma in Riemannian geometry.
method Defining metrical distortion and differential slip, showing their geometric implications.
result Geodesically radial volume and length preservation properties.
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For this class of speeds we prove the exponential convergence to a geodesic sphere. …
The paper studies geodesic mappings in special Riemannian manifolds.
problem Investigating geodesic mappings in specific Riemannian manifolds.
method Proof of geodesic mappings properties and new results on geodesic mappings of various Riemannian manifolds.
result New results on geodesic mappings of quasi Einstein, Ricci recurrent, and Ricci symmetric manifolds.
We prove rigidity facts for groups acting on pseudo-Riemannian manifolds by preserving unparameterized geodesics.
Study of centralizer elements preserving geodesic flow foliations on covers.
problem Rigidity properties of centralizer elements in geodesic flows.
method Definition and study of foliated centralizer, proving rigidity properties.
result Foliated centralizer is a finite-dimensional Lie group, discrete unless metric is homothetic to real hyperbolic.
Let S be a closed Riemann surface of genus p>1 with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on S that preserve at least one bi-infinite geodesic in the curve complex.
Decomposes Busemann spaces into simpler structures.
problem Understanding the structure of Busemann spaces.
method Topological decomposition and isometry analysis.
result Busemann spaces can be decomposed into products of simpler spaces.
We show that a conformal connection on a closed oriented surface Σ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on Σ determine th…
Paper proves geodesics and focal points unchanged by conformal changes in pseudo-Finsler manifolds.
problem Understanding the behavior of lightlike geodesics in pseudo-Finsler manifolds.
method Used Chern connection, anisotropic calculus, and critical points of energy functional to prove invariance.
result Lightlike geodesics and focal points are preserved by anisotropic conformal changes.
First-passage percolation affects graph properties like curvature and geodesics.
problem Effect of first-passage percolation on graph curvature and geodesics.
method Randomly perturbs the metric of a graph by assigning random edge lengths.
result Non-positive curvature and geodesic properties are not preserved by first-passage percolation.
Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.
problem Behavior of mean curvature flows on graphs in warped product manifolds, especially at infinity.
method Analysis of curve shortening flow and mean curvature flow on geodesic graphs for various warping functions.
result Long-time existence of mean curvature flows and vanishing of curvature and derivatives at infinity.
Causal spacetimes with Ricci tensor have unique transformations.
problem Understanding transformations in viable causal spacetimes.
method Analyzing Ricci tensor and causal diffeomorphisms.
result Causal diffeomorphisms preserving Ricci tensor are homotheties.
We study the family of α-connections of Amari-Chentsov on the homogeneous space D(M)/Dμ(M) of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold M. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation …
We show that if two closed hyperbolic surfaces (not necessarily orientable or even connected) have the same Laplace spectrum, then for every length they have the same number of orientation-preserving geodesics and the same number of orientation-reversing geodesics. Restricted to orientable surfaces, this result reduces…
LIMP learns latent shapes with metric preservation, improving generative models.
problem Insufficient training data for high-fidelity latent representations.
method Metric preservation as a prior, geometric distortion criterion, geodesic loss.
result Synthetic samples of higher quality achieved through metric preservation.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. The paper proves that under certain conditions, solutions to a specific differential inequality are nonnegative.
problem Preserving positivity of solutions to a differential inequality on Riemannian manifolds.
method Analytic approach using Llocp norms and growth conditions over geodesic balls. result Nonnegative solutions to the inequality −Δu+λu≥0 are preserved under suitable growth conditions. Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.
problem Understanding geodesic flows on compact surfaces without conjugate points.
method Time-preserving semi-conjugation to a continuous expansive flow.
result Geodesic flows on compact surfaces without conjugate points of genus > 1 have a unique measure of maximal entropy.
The paper studies curvature conditions on manifolds with boundary.
problem Curvature preservation on manifolds with smooth boundaries.
method Constructing a family of metrics that agree with given metrics on the boundary and interior.
result Deforming metrics to ones with totally geodesic boundary while preserving curvature conditions.
Extends DCP framework to Hadamard manifolds for geodesically convex functions.
problem Verifying convexity in nonlinear programs on Hadamard manifolds.
method Introduces Disciplined Geodesically Convex Programming (DGCP) framework, defining compositions and transformations for geodesically convex functions.
result Allows verification of geodesic convexity for a broader range of functions, including statistical estimators and matrix-valued optimization.
Extends Paulin's result to relatively hyperbolic groups.
problem Proving quasi-isometric equivalence between relatively hyperbolic groups.
method Introducing relative quasi-Mobius maps and using coarsely cusp-preserving quasi-isometries.
result Establishes a homeomorphism between Bowditch boundaries inducing quasi-Mobius maps.
Billiard trajectories in curved spaces have predictable travel times.
problem Understanding travel times in billiard trajectories on curved surfaces.
method Analyzing geodesic flows and sectional curvature to prove time-preserving conjugacy.
result Billiard trajectories with almost identical obstacles have identical shapes.