Geodesic concavity and hypersymplectic structures in G2-structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G2-structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G2 Laplacian flow decreases the length. Study connects contact structures to cone geodesics and contactomorphisms.
problem Understanding contact structures on cone geodesics.
method Review and generalize cone geodesics to contact manifolds, establish correspondence with contactomorphisms.
result Established correspondence between contactomorphisms and cone structures.
Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown, if the cone structure is regarded as a control system, then, the space of abnormal geodesics of the cone structure is naturally identified with the origi…
New approach constructs symplectic structure on pseudo-Riemannian geodesics.
problem Dimensional mismatch in classical symplectic structure construction for pseudo-Riemannian geodesics.
method Directly constructs geodesic space as quotient, introduces conformal co-symplectic structure.
result Conformal co-symplectic structure globally describes geometric distribution of geodesics.
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.
A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to Rn. Along the way, we establish various structural properties …
Study on geodesics in a specific sub-Riemannian structure with two types of behavior.
problem Symmetries and geodesics in a sub-Riemannian structure with growth vector (4,7).
method Symmetry reduction to analyze geodesics.
result Existence of two distinct types of geodesics: those that avoid the fixed point set and those that are contained within it.
This notes explores angle structures on ideally triangulated compact 3-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic 3-manifold with totally geodesic boundary has an ideal…
Geodesic vector fields on flat 3-manifolds are related to contact structures.
problem Understanding geodesic vector fields on flat 3-manifolds.
method Analyzing geodesic and Reeb vector fields on flat 3-manifolds.
result Geodesic vector fields on closed flat 3-manifolds are Reeb vector fields of contact forms.
Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.
Lightlike hypersurfaces in cone structures minimize time.
problem Finding time-minimizing paths in cone structures.
method Defining lightlike hypersurfaces and proving their foliation by cone geodesics.
result Lightlike hypersurfaces in globally hyperbolic spacetimes are time-minimizing.
Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined by how it interacts with the standard differential structure in this space -- the…
We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manif…
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine struc…
Researchers compute contact structures for null geodesics on specific spacetimes.
problem Understanding the canonical contact structure of null geodesics in spacetimes.
method Explicit calculations for specific spacetimes, including lens spaces and three-dimensional spacetimes.
result Contact structures on null geodesics are derived from the Lorentz prolongation of spacetimes.
The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…
Study geodesics on infinite-dimensional manifolds using Finsler structures.
problem Geodesics on Fréchet manifolds of Riemannian metrics.
method Establish Riemann-Finsler structures, prove existence and minimality of geodesics, derive Euler-Lagrange equations.
result Geodesics on Fréchet manifolds of Riemannian metrics are length minimizing and satisfy Euler-Lagrange equations.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
Geodesics and boundaries found for metric structures on hyperbolic groups.
problem Understanding the space of metric structures on hyperbolic groups.
method Outer automorphism invariant geodesic bicombing and boundary construction.
result Boundary contains well-known pseudo metrics and rigidity results.
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.
Study bounds changes in hyperbolic 3-manifold structures after drilling short geodesics.
problem Bounding changes in complex projective structures after drilling short geodesics.
method Analyzes L2-bounds on changes in conformally compact hyperbolic 3-manifolds. result Change is bounded by a universal constant times the square root of the length of the drilled geodesics.
We construct point invariants of ordinary differential equations that generalise the Cartan invariants of equations of order two and three. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consist of a paraconformal structure, an adapted GL(2,R)-conne…
The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
problem Finding a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.
method Defining a complex projective structure and using accessory parameters to characterize the curve.
result The accessory parameters are the residues of the quadratic differential comparing the projective structure to the trivial one.
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.
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally geodesic boundary. The purpose of this paper is to announce some partial result…
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.
Introduces new geodesic fields for Finsler manifolds.
problem Finding optimal paths in Finsler manifolds.
method Introduced Pontryagin type C0-Finsler structures and defined geodesic fields. result Extended geodesic field E provides more natural geodesics. Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure.…
Researchers find shortest paths on a special group structure.
problem Finding shortest paths on a specific group structure.
method Symmetry reduction to a simpler problem, then solving on a quotient space.
result Explicitly described length-minimizing geodesics.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C1. Classifies left invariant Kundt structures on 3D Lie groups.
problem Understanding Kundt spacetimes and their properties.
method Analyzes local structure and properties of left invariant Kundt structures.
result Classifies all left invariant Kundt structures on 3D simply connected unimodular Lie groups.
The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting) geometry as well as in (2+1) Lorentzian one (globally hyperbolic spacetimes of …
New maximal families of compatible Poisson structures derived from geodesically equivalent metrics.
problem Constructing maximal families of compatible Poisson structures.
method Connecting geodesically equivalent metrics and compatible Poisson structures of hydrodynamic type.
result Maximal families of compatible Poisson structures of dimension (n+1)(n+2)/2 are constructed. We extend Agrachev-Brockett-Jurjdevic's solution to normal sub-Riemannian geodesics.
problem Finding normal sub-Riemannian geodesics in Lie group structures.
method Constructing sub-Riemannian structures from Lie subalgebra filtrations and applying to homogeneous spaces.
result Explicit solutions for normal geodesics in general chains of Lie subgroups.
Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.
problem Properties of Lorentzian manifolds influenced by totally geodesic null hypersurfaces.
method Coupling rigging technique with null foliation existence to prove Riemann flow structure.
result Proves curvature conditions restrict causal structure of spacetime.
We introduce elastic geodesic grids for easy-to-fabricate, deployable structures.
problem Approximating freeform surfaces with deployable structures.
method Geodesic curves on target surfaces, kinematic mechanism, differential geometry.
result Elastic geodesic grids can approximate freeform surfaces easily and deployably.
Study and classify totally geodesic submanifolds in nearly Kaehler flag manifold.
problem Classifying totally geodesic submanifolds in nearly Kaehler flag manifold.
method Developed structural approach to nearly Kaehler flag manifold, expressed curvature tensor in terms of nearly Kaehler structure and canonical complex structures.
result Classified almost complex totally geodesic submanifolds of nearly Kaehler flag manifold and its semi-Riemannian counterpart.
Study of CR-submanifolds in various Lorentzian manifolds.
problem Exploring CR-submanifolds in different Lorentzian structures.
method Analyzing properties and results of CR-submanifolds in LCS, LP-cosymplectic, S, and GKM manifolds.
result Obtained results on totally umbilical and geodesic CR-submanifolds.
Since their introduction by Thurston, geodesic laminations on hyperbolic surfaces occur in many contexts. In this paper, we propose a generalization of geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose boundary at infinity of the universal cover is endowed with a invariant total cyclic ord…
We prove that a reversible Berwaldian Finsler structure (M,F) with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.
Integrable geodesics found on special orthogonal group.
problem Analyzing normal geodesics on the special orthogonal group.
method Adapted Lax pair and bi-Hamiltonian structure.
result Almost all normal geodesics are completely integrable.
Study geodesics and PSL(2,C) structures on hyperbolic 3-space.
problem Geometric properties of PSL(2,C) and geodesics in hyperbolic 3-space.
method Analyzes structures of Riemannian holomorphic manifolds and uses symmetry properties.
result G is a symmetric space for PSL(2,C) and has holomorphic Riemannian metrics.
We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web …
We consider real isotropic geodesics on manifolds endowed with a pseudoconformal structure and their applications to the theory of lightlike hypersurfaces on such manifolds, the geometry of four-dimensional conformal structures of Lorentzian type, and a classification of the Einstein spaces.
In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polari…