Zermelo deformation preserves geodesics and curvature in Finsler metrics.
problem Behavior of geodesics and curvature in Finsler metrics under Zermelo deformation.
method Zermelo deformation with Killing vector fields.
result Zermelo deformation preserves local symmetry in locally symmetric Finsler metrics.
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.
Study geodesics on a modified cotangent bundle over Kählerian manifolds.
problem Investigate geodesics on a modified cotangent bundle.
method Introduced Berger-type deformed Sasaki metric, investigated Levi-Civita connections, and studied geodesics.
result Geodesic properties on modified cotangent bundles.
Space of Zoll Finsler metrics on projective plane deformation retracts to round metric.
problem Understanding the structure of Zoll Finsler metrics on projective planes.
method Geodesic flow deformation and curvature flow.
result Space of Zoll Finsler metrics on projective plane is connected and deformation retracts to the round metric.
The paper explores non-metrizability of projective deformations of Finsler sprays.
problem Investigating non-metrizability of projective deformations of Finsler sprays.
method Analyzing projective deformation of Finsler sprays by holonomy invariant functions and proving non-metrizability for most cases.
result For most values of λ and holonomy invariant nontrivial functions P, the projective deformation is not Finsler metrizable.
Study projective deformations of hyperbolic 3-orbifolds with turnover ends.
problem Deformations of hyperbolic 3-orbifolds with turnover ends in projective geometry.
method Projective deformations of hyperbolic 3-orbifolds with turnover ends, focusing on totally geodesic generalized cusps.
result Turnover funnels remain totally geodesic and the deformed projective 3-orbifold remains properly convex.
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.
Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined…
We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomolog…
This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Generalizes Wolpert's formula for geodesic graphs on hyperbolic surfaces.
problem Computing symplectic forms on Teichmüller space for complex deformations.
method Defines a new infinitesimal deformation for balanced geodesic graphs and proves a generalized formula.
result Reproduces Wolpert's formula for simple closed curves and extends it to more complex graphs.
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…
Holomorphic deformations of weighted projective spaces yield Finsler spheres with closed geodesics.
problem Finding Finsler 2-spheres with constant curvature and closed geodesics.
method Establishing a correspondence between Finsler structures and Weyl connections on orbifolds.
result Holomorphic deformations of Veronese embeddings provide examples of Finsler 2-spheres with constant curvature and closed geodesics.
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 …
Proved contractibility of geodesic triangulation space on hyperbolic surfaces.
problem Open problem on contractibility of geodesic triangulations on hyperbolic surfaces.
method Generalized Tutte's embedding theorem for negative curvature surfaces.
result Contractibility of geodesic triangulation space proved.
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, ou…
Researchers prove spectral rigidity of Liouville tori under specific conditions.
problem Spectral rigidity of Liouville tori under generic conformal classes.
method Noncancellation of wave trace and analysis of second order variational formula for energy.
result Laplace isospectral deformations of Liouville metrics on torus are trivial.
Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
The paper surveys partial results on convex real projective orbifolds with specific ends.
problem Characterizing convex real projective orbifolds with radial or totally geodesic ends.
method Proving homeomorphisms between deformation spaces and strata of projective structures.
result A homeomorphism between deformation spaces of convex real projective structures on orbifolds with radial or totally geodesic ends.
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.
Study earthquake deformations on a once-punctured torus.
problem Understanding earthquake deformations on Teichmüller space.
method Two methods: linear recurrence relations and hyperbolic geometry.
result Algebraic and geometric interpretations of earthquake deformations.
We analyze the horizon and geodesic structure of a class of 4D off--diagonal metrics with deformed spherical symmetries, which are exact solutions of the vacuum Einstein equations with anholonomic variables. The maximal analytic extension of the ellipsoid type metrics are constructed and the Penrose diagrams are analyz…
Solves geodesics in homogeneous manifolds using one-parameter subgroups.
problem Finding geodesics in homogeneous manifolds with various symmetries.
method Explicitly solving the geodesic equation for a wide class of homogeneous manifolds, proving geodesic completeness, and showing geodesics as orbits of one-parameter subgroups.
result Geodesics in homogeneous manifolds are orbits of one-parameter subgroups, and metrics can be N-parameter deformations of a Riemannian metric with special symmetries.
Flat torus triangulations' space is homotopy equivalent to a torus.
problem Proving homotopy equivalence of triangulations of flat tori.
method Generalization of Tutte's embedding theorem for flat tori.
result Deformation space of geodesic triangulations is homotopy equivalent to a torus.
Study Einstein-Weyl spaces from Segre quartic surfaces, finding unique geodesics and deformations.
problem Characterize Einstein-Weyl spaces associated with Segre quartic surfaces.
method Explicit construction and analysis of minitwistor spaces, focusing on singularities and geodesics.
result Found unique closed geodesics on Einstein-Weyl spaces, showing deformations and non-compactifications.
A real projective orbifold is an n-dimensional orbifold modeled on RPn with the group PGL(n+1,R). We concentrate on an orbifold that contains a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed (n−1)-dimensional orbifolds times …
Sprays with vanishing X-curvature are studied in this paper.
problem Characterizing sprays with specific curvature properties.
method Analyzing expressions for X-curvature and using projective deformations.
result Sprays obtained by projective deformation using S-curvature have vanishing X-curvature.
Constructs deformations of hyperbolic hexagons for new geodesics in Teichmüller spaces.
problem Finding new geodesics in Teichmüller spaces.
method One-parameter family of right-angled hexagons with Lipschitz maps.
result New geodesics for arc and Thurston metrics on Teichmüller spaces.
Study deformed Hermitian-Yang-Mills equation via GIT and prove existence of geodesics.
problem Existence and regularity of solutions to deformed Hermitian-Yang-Mills equation.
method Variational approach via infinite dimensional GIT problem, scale estimates, Fourier-Mukai transform.
result Existence of smooth and weak geodesics with C1,α regularity. We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are equivalent.
The horizon and geodesic structure of static configurations generated by anisotropic conformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic extension of such off--diagonal vacuum metrics and conclude that for small deformations there are different classes of vacuum solutions of …
Let S be a torus with a hyperbolic metric admitting one puncture or cone singularity. We describe which infinitesimal deformations of S lengthen (or shrink) all closed geodesics. We also study how the answer degenerates when S becomes Euclidean, i.e. very small.
Develops a method to define and characterize geodesics on hyperbolic surfaces.
problem Characterizing closed geodesics on hyperbolic surfaces without self-intersection.
method Constructive definition of the Goldman bracket using closed geodesics.
result Algebraic characterization of geodesics on hyperbolic surfaces.
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…
New patterns deform Farey triangulation in symmetric space.
problem Deforming Farey triangulation in symmetric space.
method Realized representations of modular group as isometry groups of geodesic patterns in SL3(R)/SO(3). result 2-parameter family of deformations of Farey triangulation.
Derives curvature formulas for convex metric sums and conditions for positive average variation.
problem Understanding how the curvature of a convex sum of metrics changes and whether it can increase the average curvature.
method Explicit formulae for curvature of convex sums of Riemannian metrics, studying total geodesic flat torus.
result Necessary and sufficient conditions for positive average variation of curvature of \(g_t\).
Proves rigidity of geodesic balls in spheres under certain deformations.
problem Rigidity of geodesic balls in spheres under smooth deformations.
method Real Killing connection and solution of Dirac operator boundary value problem.
result Rigidity result for geodesic balls in spheres fails for hemispheres.
Given an Anosov representation $ρ\colon π_1(S) \to \PSL_{n}(\mathbb{R})$ and a maximal geodesic lamination λ in a surface S, we construct shear deformations along the leaves of the geodesic lamination λ endowed with a certain flag decoration, that is provided by the associated flag curve $\mathcal{F}_ρ\colon \Sin…
We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…
The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.
problem Constructing foliations of minimal surfaces in negatively curved 3-manifolds.
method Deformations of totally geodesic foliations, using Grassmann bundle and negatively curved metrics.
result The foliations of minimal surfaces are deformations of totally geodesic foliations.
New counterexample shows curved surfaces can deform geodesics without diffeomorphism.
problem Can curved surfaces deform geodesics without changing their lengths?
method Constructs a perturbed surface with longer geodesics but no contracting diffeomorphism.
result No diffeomorphism can contract all tangent vectors on a surface with longer geodesics.
Develops a Riemannian archetypal analysis for interpretable non-linear data.
problem Limited performance of classical archetypal analysis on non-linear data.
method Riemannian geometry for data-driven pullback, geodesic convex combinations, convex relaxation followed by non-convex refinement.
result Combines interpretability of classical archetypal analysis with expressive power of modern non-linear models.
Study on bending deformations in hyperbolic manifolds, generalizing Johnson and Millson's work.
problem Understanding infinitesimal deformations in branched bending complexes.
method Defining branched bending deformations, giving lower bounds, and constructing examples.
result Lower bounds on the dimension of deformation spaces and examples of specific deformations.
For an oriented isometric immersion f:M→Sn the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM⊥ into the unit sphere subbundle of TSn, and the geodesic Gauss map γ projects this into the manifold of oriented geodesics in Sn (the Grassmannian of oriented 2-planes in $\ma…
Abstract: Investigates octonion product deformations and related geometries.
problem Exploring geometries and deformations from the 7-sphere S7. method Analyzing the spontaneous compactification M4imesS7 and solutions of Lagrangian equations. result Obtains a family of geometries including those with torsion and G2-structures. We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…
A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…