Geometrically interprets integrability of geodesic flow using web theory.
problem Integrability of geodesic flow by quadratic integrals.
method Geometric interpretation through web theory and integrable billiards construction.
result Constructs integrable billiards on surfaces with quadratic geodesic integrals.
Study geodesics on strong Kropina spaces, global and local aspects.
problem Geodesics behavior on strong Kropina spaces.
method Global and local analysis of geodesics.
result Illustrated geodesics with examples.
We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standar…
Study develops geodesic theory for foliations, proving Laplacian comparison theorems.
problem Comparing Laplacians on totally geodesic Riemannian foliations.
method Variational theory of geodesics, limit of Riemannian distance approximations.
result Sharp comparison theorems for sub-Riemannian distance in Sasakian foliations.
Optimizing quantum graphs yields geodesic nets on surfaces.
problem Finding optimal quantum graphs for geodesic nets.
method Optimizing functionals from spectral theory to find geodesic nets.
result Critical metrics for eigenvalues give rise to geodesic nets.
Proves min-max theory for constant geodesic curvature curves on closed surfaces.
problem Prescribing mean curvature on surfaces with constant geodesic curvature.
method Min-max theory applied to classify blowups and ensure almost embedded solutions.
result Produces a solution with constant geodesic curvature c on closed surfaces. Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
The study proves the existence of many geodesics on complex manifolds.
problem Existence of closed geodesics on manifolds with non-trivial first Betti number.
method Combining Mañé's theorem with a new theorem about minimal geodesics and transverse homoclinic points.
result Proves the existence of infinitely many closed geodesics of arbitrary large length on manifolds with non-trivial first Betti number.
Study geodesic string counts on Riemann-Finsler manifolds, linking to KAM theory.
problem Counting geodesic strings on Riemann-Finsler manifolds.
method Using Fuller index and KAM theory, derive product formula for counts.
result Arithmetic constraints on geodesic string counts and existence of negative curvature metrics.
Develops Morse theory for uniform energy using geodesics.
problem Minimizing properties of closed geodesics.
method One-sided directional derivative of distance function, gradient-like vectors, restarted negative gradient flow.
result Improved minimizing properties of closed geodesics.
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.
New method calculates winding of geodesics on surfaces.
problem Understanding the distribution of geodesics on surfaces.
method Introducing a new construction of winding numbers for geodesics on cusped hyperbolic orbifolds.
result Winding numbers can be expressed by Rademacher symbols for various arithmetic families of surfaces.
A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
The paper finds infinitely many magnetic geodesics on non-compact manifolds.
problem Existence and multiplicity of periodic orbits of magnetic flows.
method Morse theory applied to non-compact manifolds with energy levels above the Mañé critical value.
result Infinitely many noncontractible closed magnetic geodesics found.
Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.
Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.
problem Answering a problem posed by Haefliger and Li about geodesic flow foliations.
method Unitary representation theory of PSL(2, R) and Hodge decompositions of de Rham complexes.
result Computed de Rham cohomology of weak stable foliations for various coefficients.
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. Study proves existence of multiple geodesics in a specific metric space.
problem Existence of multiple geodesics in a manifold with a Randers-Kropina metric.
method Lusternik-Schnirelman theory applied to a homotopy type of solutions of an affine control system.
result Proves existence of infinitely many geodesics between two points in a non-contractible manifold.
This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the …
Long geodesics imply a special shape of convex bodies.
problem Understanding the geometry of convex surfaces.
method Intrinsic geometry of convex surfaces and proof by contradiction.
result Long geodesics on a convex surface imply the shape is an isosceles tetrahedron.
Paper constructs Morse-Floer homology for Dirac-geodesics.
problem Existence of Dirac-geodesics under super-quadratic perturbation.
method Morse-Floer theory applied to Dirac-geodesics with spectral sequence computation.
result Explicit computation of Dirac-geodesics' homology leading to existence results.
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and p…
Totally geodesic limit of quasi-Fuchsian groups with converging critical exponent.
problem Understanding convergence of quasi-Fuchsian groups.
method Critical exponent convergence analysis.
result Convergence of quasi-Fuchsian groups to totally geodesic representations.
Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.
problem Counting ambiguous geodesics in curved spaces.
method Asymptotic formula for common perpendiculars in negatively curved spaces, applying to modular orbifolds and number fields.
result Confirms and extends Motohashi's conjecture on binary additive divisor problem.
Totally geodesic sections found in polar actions.
problem Understanding sections of polar actions on Riemannian manifolds.
method Elementary proof of a folklore result.
result Sections of polar actions are totally geodesic.
Generalizes Newton's Second Law for field theory.
problem Applying Newton's Second Law to higher-dimensional parameterized submanifolds.
method Introducing geodesic k-vector field and deriving Hamilton's equations.
result Different forces can lead to the same Hamilton's equations.
For odd-dimensional spheres, there's always a second short geodesic.
problem Finding the second shortest closed geodesic on odd-dimensional spheres.
method Analyzing non-reversible Finsler metrics on spheres of odd dimension.
result There is a second closed geodesic with Morse index ≤ 4(m+2)(m-1)+2.
Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…
A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number theory (class number asymptotics) it is, however, necessary to consider this case, t…
In this paper we study geodesic mappings of n-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such n-dimensional ellipsoids admit non tri…
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
Quantizes geodesic lengths in Teichmüller spaces using algebraic methods.
problem Constructing quantized geodesic lengths for Teichmüller spaces.
method Developed quantum trace maps and investigated algebraic structures.
result Showed a recursion relation and commutation properties for quantized trace-of-monodromy.
GEORCE computes geodesics quickly and accurately.
problem Computing geodesics on Riemannian and Finsler manifolds is difficult and inefficient.
method GEORCE transforms geodesic computation into a discrete control problem.
result GEORCE achieves global convergence and quadratic local convergence.
The paper shows that the curvature of RP2 is constant iff all geodesics are closed. Therefore RP2 is the first known manifold with only one G-structure. It took quiete a long time to find such a manifold. The author shows only that if all geodesics are closed then there are infinitely many simple closed geodesics. This…
Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a…
The study proves semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
problem Semisimplicity of totally geodesic subvarieties in moduli spaces of Riemann surfaces.
method Intertwining results from dynamics, algebraic geometry, geometric group theory, and Teichmüller theory.
result Each component of the boundary is a product of simple factors, each behaving like a diagonal embedding.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
An example from Almgren and Federer shows geodesics that are not always the shortest.
problem Illustrating the subtleties of geodesic minimization in complex metrics.
method Exposition of a specific example in S1imesS2 to clarify definitions. result Found geodesics that are not minimizers in their homotopy classes.
The paper studies entropy and mass loss in geodesic flows on curved spaces.
problem Entropy and mass loss in geodesic flows on negatively curved manifolds.
method Ergodic theory, critical exponents of parabolic subgroups, pressure of potentials.
result Entropy is upper semicontinuous with no mass loss, but fails with mass loss due to critical exponents.
It is shown that geodesics in the space of Kähler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for Kähler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to Kähler man…
Finite geodesic submanifolds found in certain hyperbolic manifolds.
problem Maximal geodesic submanifolds in hyperbolic hybrids.
method Structure theory of arithmetic groups, dynamics, and geometry in negative curvature.
result Finiteness of maximal geodesic submanifolds in hyperbolic hybrids.
Study weak geodesics in deformed Hermitian-Yang-Mills equation space.
problem Geodesics in the space of potentials for deformed Hermitian-Yang-Mills equation.
method Formulated as degenerate elliptic equation, used nonlinear Dirichlet duality theory, constructed continuous solutions.
result Continuous solutions constructed for Dirichlet problem.
Formula connects linking number to spectral theory on 3-torus.
problem Computing linking number of multi-geodesics on 3-torus.
method Spectral theory of Laplace operator on differential forms.
result Formula for linking number of multi-geodesics on 3-torus.
This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…