Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.
problem Geodesic completeness of compact locally symmetric Lorentz manifolds.
method Proof in all remaining cases using completeness result.
result All compact, locally symmetric Lorentz manifolds are geodesically complete.
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.
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.
Type A surfaces are the locally homogeneous affine surfaces which can be locally described by constant Christoffel symbols. We address the issue of the geodesic completeness of these surfaces: we show that some models for Type A surfaces are geodesically complete, that some others admit an incomplete geodesic but model…
Characterizes geodesic completeness for landmark spaces.
problem Ensuring geodesics exist for all times in landmark spaces.
method Integrability criterion based on cometric kernel behavior.
result Full characterization of geodesic completeness for smooth Riemannian metrics.
The paper studies geodesic completeness for Lie groups and their metrics.
problem Geodesic completeness of pseudo and holomorphic Riemannian metrics on Lie groups.
method Euler-Arnold formalism, detailed study of geodesics, classification of metrics.
result Full classification of geodesic completeness for the Lie group SL(2, C).
Solves geodesic completeness on pseudo-homothetic Lie group.
problem Geodesic completeness on pseudo-homothetic Lie group.
method Exhibited a family of complete metrics with bounded geodesic velocity.
result Set of complete metrics is not closed.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g is geodesically rigid: if any other complete metric gˉ has the same (unparametrized) geodesics with g, then the Levi-Civita connections of g and gˉ coincide.
Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
problem Geodesic completeness and flow properties of compact Brinkmann spacetimes.
method Proof of geodesic completeness and flow properties of isotropic parallel vector fields in compact Brinkmann spaces.
result Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.
An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…
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 paper examines geodesic completeness in Lie groups with specific vector fields.
problem Investigating geodesic completeness in Lie groups with special vector fields.
method Analyzing left-invariant Lorentzian metrics on simple Lie groups with Killing vector fields.
result Conditions for geodesic completeness in Lie groups with specific vector fields.
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for q>3/2. The aim of this paper is to review and complete the study of geodesics on Gödel type spacetimes initiated in [8] and improved in [2] of the References. In particular, we prove some new results on geodesic connectedness and geodesic completeness for these spacetimes.
Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauc…
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
Given a hyperbolic surface §, a classic result of Birman and Series states that for each K, all complete geodesics with at most K self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of §. We define a self-intersection function for each complete geodesic, which bounds t…
Study geodesic curves on Heisenberg group, classify them, and compute first step of quadrature.
problem Classifying geodesic curves on the Heisenberg group.
method Completely integrable Hamiltonian system, classification of geodesic curves.
result Complete classification of geodesic curves on the Heisenberg group.
Study on extremal subsets in geodesically complete spaces with curvature constraints.
problem Characterizing extremal subsets in GCBA spaces.
method Introduced and analyzed extremal subsets in GCBA spaces, proving their properties.
result Set of topological singularities forms an extremal subset under additional assumptions.
Geodesic completeness proven for certain Lorentzian spaces.
problem Geodesic completeness of compact Lorentzian locally symmetric spaces.
method Using recent and earlier results on Lorentzian and de Rham-Wu decompositions.
result Geodesically complete if Lorentzian factor is of Cahen-Wallach type or maximal flat factor is one-dimensional and time-like.
The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.
problem Defining a metric for high energy spaces in Kähler manifolds.
method Endowing the high energy space with a metric that makes it a complete geodesic metric space.
result The geodesic metric space (Ep(X,θ),dp) is uniformly convex for p>1. CAT(0) spaces quasi-isometric to Euclidean spaces are homeomorphic to R^n.
problem Characterizing CAT(0) spaces quasi-isometric to Euclidean spaces.
method Analyzing properties of CAT(0) spaces and using quasi-isometry.
result CAT(0) spaces quasi-isometric to R^n are not necessarily homeomorphic to it.
We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.
Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.
problem Understanding the structure of Busemann spaces with measures.
method Analyzing geodesic completeness and non-collapse assumptions.
result Rigidity and structure theorems for Busemann spaces with MCP.
Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.
problem Classify complete totally geodesic subsets of complex hyperbolic plane.
method Two new proofs: one algebraic and one geometric.
result Only complex geodesics and real planes are non-trivial complete totally geodesic subsets.
Kähler manifold loop space inherits Kähler structure and is complete.
problem Characterizing the geometric properties of loop spaces of Kähler manifolds.
method Proving the Kähler structure and completeness of the loop space.
result The loop space of a Kähler manifold is complete and inherits a Kähler structure.
We review geometrical properties of a static spacetime (M,g), including geodesic completeness, causality, standard splittings, compact M, closed geodesics and geodesic connectedness. We pay special attention to the critical quadratic behavior at infinity of the coefficients β, β−1 (β=−g(K,K), being K a …
This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.
Totally geodesic minimal hypersurfaces in H5 with specific curvature properties.
problem Characterizing minimal hypersurfaces in hyperbolic space with certain curvature conditions.
method Analyzing properties of minimal hypersurfaces in H5 with constant scalar curvature and zero Gauss-Kronecker curvature. result Any complete minimal hypersurface in H5 with constant scalar curvature and zero Gauss-Kronecker curvature is totally geodesic. I show that if a geodesic space has curvature bounded below locally in the sense of Alexandrov then its completion has the same lower curvature bound globally.
We clarify the relationship between the null geodesic completeness of an Einstein Lorentz manifold and its conformal Kobayashi pseudodistance. We show that an Einstein manifold has at least one incomplete null geodesic if its pseudodistancfe is nontrivial. If its pseudodistance is nondegenerate, all of its null geodesi…
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to id…
Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.
problem Analyzing the properties of geodesically complete spaces with curvature constraints.
method Geometric perturbations of geodesics to curves with zero length on singular sets.
result Every Sobolev map in W1,∞ space has a Lipschitz representative with the same Lipschitz constant as its infinity energy. Almost Zoll affine surface found on cylinder.
problem Finding surfaces with special geodesic properties.
method Exhibited an affine structure on a cylinder.
result Affine structure on cylinder is almost Zoll.
Completeness of surface metrics established for Sobolev spaces.
problem Ensuring completeness of reparametrization-invariant Sobolev metrics on surface spaces.
method Recasting completeness criteria for infinite-dimensional Riemannian manifolds and applying geometric estimates based on the Michael--Simon--Sobolev inequality.
result Established metric and geodesic completeness for specific Sobolev metrics on immersed surfaces, validating Mumford's conjecture.
Optimizes tensor completion using geodesics on Segre manifolds.
problem Incomplete tensor data in recommender systems and spectroscopy.
method Riemannian conjugate gradient optimization with explicit geodesic expressions.
result Recovery of tensor decomposition from as little as 10% of data.
Suppose S is a semispray on a manifold M. We know that the complete lift Sc of S is a semispray on TM with the property that geodesics of Sc correspond to Jacobi fields of S. In this note we generalize this result and show how geodesic variations of k-variables are related to geodesics of the kth it…
The study extends stochastic completeness to landmark spaces with any number of landmarks.
problem Stochastic completeness for landmark spaces with arbitrary numbers of landmarks.
method Volume growth criterion and eigenvalue bounds for geodesic balls.
result Stochastic completeness for landmark spaces with any number of landmarks is proven.
Minimal surfaces can't have certain epitrochoid geodesics.
problem Minimal surfaces and geodesics.
method Analyzing minimal immersions and epitrochoid properties.
result Minimal surfaces cannot contain certain epitrochoid geodesics.
Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.
problem Identifying Clairaut constants from Fermat constants for specific geodesics.
method Analytical proof for a specific class of geodesics on a surface of revolution.
result Fermat constants do not fully determine Clairaut constants for some geodesics, except for a standard sphere.
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing …
New model for visual cortex border completion using bicycle wheel motions.
problem Understanding border completion in the visual cortex V1.
method Sub-Riemannian Hamiltonian formalism and bicycle wheel analogy.
result Analogies between visual cortex border completion and bicycle wheel motions.
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order n≥2. result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
We create a smooth manifold of triangular meshes with a geodesically complete metric.
problem Representing and manipulating 2D shapes as triangular meshes.
method Developed a geodesically complete Riemannian metric for triangular meshes.
result The metric preserves mesh connectivity and avoids mesh degradation.