Generates samples conditioned on labels using optimal transport.
problem Estimating conditional distributions for specific labels.
method Wasserstein geodesic generator based on optimal transport theory.
result Learned conditional distributions and optimal transport maps.
Totally geodesic null hypersurfaces found in Lorentzian manifolds.
problem Characterizing null hypersurfaces in Lorentzian manifolds.
method Analyzing light-like geodesically complete Lorentzian manifolds with null energy condition.
result Null hypersurfaces are totally geodesic.
New conditions ensure geodesic connectedness of affine manifolds.
problem Ensuring geodesic connectedness in affine manifolds.
method New sufficient conditions for geodesic connectedness, weaker than previous work.
result Elementary proof of geodesic connectedness.
Paper finds conditions for two geodesics on complex manifolds.
problem Existence of two distinct closed geodesics on manifolds with infinite fundamental group.
method Topological and metric conditions for existence of geodesics in Riemannian and Finsler metrics.
result Generic Finsler metrics have two distinct closed geodesics.
Conditions for geodesics to project to curves of constant curvature found.
problem Understanding geodesics in sub-Riemannian manifolds.
method Analyzing Riemannian submersions and geodesic curvature properties.
result Necessary and sufficient conditions for geodesics to project to curves of constant curvature.
Defines new geodesic semilocal E-preinvex functions and studies their properties.
problem Defines new functions to generalize existing convex and preinvex concepts.
method Introduces geodesic semilocal E-preinvex functions and proves their properties.
result Establishes sufficient optimality conditions for nonlinear fractional multiobjective programming.
The paper proves the existence of capillary geodesics on Riemannian 2-disks.
problem Existence of capillary geodesics on Riemannian 2-disks with specific conditions.
method Analytical proof and examples.
result Existence of capillary geodesics with contact angle θ ∈ (0, π/2).
The paper proves conditions for Anosov geodesic flows on non-compact manifolds.
problem Conditions for Anosov geodesic flows on non-compact manifolds.
method Analyzing sectional curvature and focal points to determine Anosov geodesic flows.
result A sufficient condition for Anosov geodesic flows on non-compact manifolds.
Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endp…
We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient condition…
Study geodesics on compact Lorentzian solvmanifolds, finding conditions for closedness.
problem Conditions for closed geodesics on compact Lorentzian solvmanifolds.
method Analyzing geodesics on Lorentzian homogeneous spaces of solvable Lie groups.
result Conditions for every lightlike geodesic to be closed on quotient spaces.
The paper proves the existence of multiple closed geodesics on certain Finsler manifolds.
problem Proving the existence of multiple closed geodesics on positively curved Finsler manifolds.
method Analyzing Finsler manifolds with specific curvature conditions to prove the existence of closed geodesics.
result On Finsler manifolds with certain curvature conditions, there exist multiple closed geodesics.
The study defines conditions for mean curvature of umbilical leaves in hyperbolic space.
problem Understanding mean curvature of umbilical leaves in hyperbolic space.
method Formulated necessary and sufficient conditions for a function along geodesics and hypercycles in hyperbolic space.
result Conditions for mean curvature of umbilical leaves are established for geodesics and hypercycles.
Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\om)$ with the Hofer L∞-norm. A path in $\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\Ll$. In this paper, we giv…
Geodesics found in spacetime satisfy curvature conditions.
problem Finding geodesics in spacetime satisfying specific curvature conditions.
method Proving existence of geodesics with entropic semiconvexity and uniform L∞ densities. result Existence of geodesics satisfying the timelike curvature-dimension condition.
New optimality conditions for sub-Riemannian geodesics derived.
problem Optimality conditions for sub-Riemannian geodesics.
method Geometric translation and ODE derivation.
result New second-order necessary optimality conditions.
We study the necessary and sufficient conditions for a Finsler surface with (α,β)-metrics to be with reversible geodesics.
We provide a combinatorial condition characterizing curves that are short along a Teichmueller geodesic. This condition is closely related to the condition provided by Minsky for curves in a hyperbolic 3-manifold to be short. We show that short curves in a hyperbolic manifold homeomorphic to S x R are also short in the…
Study on Riemannian submersions from nearly Kaehler manifolds.
problem Conditions for integrability and geodesic properties in Riemannian submersions.
method Investigation of various distributions and conditions for integrability and geodesic properties.
result Conditions for a generic Riemannian submersion to be a harmonic map.
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
The paper examines geodesics in special Finsler spaces.
problem Understanding geodesics in homogeneous Finsler spaces with specific metrics.
method Analyzes geodesics and geodesic vectors in homogeneous exponential and infinite series Finsler spaces.
result Identifies conditions for a vector to be a geodesic in these spaces.
The study examines geodesic orbit Finsler spaces with non-negative flag curvature and (FP) condition, proving they are compact.
problem Characterizing geodesic orbit Finsler spaces with specific curvature conditions.
method Analyzes the interaction between geodesic orbit property and flag curvature conditions.
result Compactness of geodesic orbit Finsler spaces with non-negative flag curvature and (FP) condition.
We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfa…
A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex …
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
The study finds homogeneous geodesics in homogeneous Kropina spaces.
problem Existence of homogeneous geodesics in homogeneous Kropina spaces.
method Analyzes homogeneous Kropina spaces and (α,β)-homogeneous spaces to find geodesics. result Homogeneous Kropina spaces admit at least one homogeneous geodesic through any point.
We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as possible with respect to this condition.
This survey explores compact geodesic orbit manifolds and their properties.
problem Classifying compact geodesic orbit manifolds.
method Review and analysis of existing results.
result Study of geodesic orbit condition for $\SU(5)/\s(\U(2) imes \U(2))$.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the …
Conditions for metallic structures to induce integrable distributions and geodesic invariance.
problem Conditions for metallic structures to induce integrable and geodesically invariant distributions.
method Analyzing tensor fields and adapted connections associated with metallic structures.
result Chen-type inequality for metallic distributions and conditions for metallic maps to preserve these distributions.
Paper constructs new non-Anosov Partially Hyperbolic Geodesic flows using conformal deformations.
problem Creating new non-Anosov Partially Hyperbolic Geodesic flows.
method Using conformal deformations to produce examples of partially hyperbolic geodesic flows.
result Proves ergodicity for the Liouville measure and uniqueness of the measure of maximal entropy.
Anosov geodesic flow proven in non-compact manifolds with negative curvature.
problem Proving Anosov geodesic flow in non-compact manifolds with negative curvature.
method Proving the geodesic flow is Anosov by showing average sectional curvature is negative and uniformly away from zero.
result Constructed a non-compact manifold with Anosov geodesic flow.
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.
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. Given two points of a Generalized Robertson-Walker spacetime, the existence, multiplicity and causal character of geodesic connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional …
Let (T^2, g) be a two-dimensional Riemannian torus. In this paper we prove that the topological entropy of the geodesic flow restricted to the set of initial conditions of minimal geodesics vanishes, independent of the choice of the Riemannian metric.
Geodesic rays prove key aspects of cscK metrics existence and stability.
problem Existence and stability of constant scalar curvature Kähler metrics.
method Reduction to regularization conjecture and analysis of geodesic rays.
result Uniform K-stability and JKX-stability are sufficient for cscK metrics existence. We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…
Totally geodesic submanifolds in product spaces imply special curvature properties.
problem Characterizing manifolds based on the existence of totally geodesic submanifolds.
method Analyzing totally geodesic submanifolds in products of non-positively curved manifolds with transversality conditions.
result If infinitely or just a single dense such submanifolds exist, the manifold must be locally symmetric.
The study reveals conditions for infinite closed geodesics on specific surfaces.
problem Conditions for infinite closed geodesics on complete surfaces.
method Analyzes geometric and homological properties of closed geodesics on cylinders and planes.
result Proves that complete cylinders with isolated geodesics have zero, one, or infinitely many homologically visible geodesics.
For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with a metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to c…
New invariants help find closed geodesics on curved spaces.
problem Finding closed geodesics on curved spaces.
method Constructing numerical homotopy invariants and applying them to functions on loop and sphere spaces.
result Proved new existence results for closed geodesics on Finsler manifolds.
We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for diver…
This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.
problem Classifying geodesics of projectively flat sprays and determining sprays based on geodesics.
method Introduction of a geodesic method to determine an n-dimensional spray based on a family of curves with 2(n-1) free parameters as geodesics.
result Classification of geodesics of projectively flat sprays and determination of sprays based on geodesics.
Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics …
Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.
problem Existence of periodic geodesics on Hilbert half-Lie groups
method Using completeness results and Lyusternik-Fet type theorem
result Periodic geodesics exist whenever the fundamental group is nontrivial