arXiv research

Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.

The study proves residual finiteness for certain lattice extensions and negatively curved projective varieties.

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric p…

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions…

The curve graphs are not locally finite. In this paper, we show that the curve graphs satisfy a property which is equivalent to graphs being uniformly locally finite via Masur--Minsky's subsurface projections. As a direct application of this study, we show that there exist computable bounds for Bowditch's slices on tig…

The Masur domain is a subset of the space of projective measured geodesic laminations on the boundary of a 3-manifold M. This domain plays an important role in the study of the hyperbolic structures on the interior of M. In this paper, we define an extension of the Masur domain and explain that it shares a lot of prope…

This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.

Geodesic flow mixing on convex projective manifolds proven.

New metric on geodesic currents connects different surface genera.

Bounding geodesic length variation for surface projective structures.

Entropy study of geodesic flow on convex projective surfaces.

Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.

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…

A projection maps geodesic currents to Teichmüller space.

Classifies geodesic flows on projective plane with potential field.

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.

We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only…

Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we tu…

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. 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 …

Geodesic flows on Kähler manifolds are quantum integrable when metrics are c-projectively equivalent.

We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geo…

Describes geodesic scattering on hyperboloids using quadrics results.

Study proves existence of closed geodesics on spheres and projective spaces.

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports s…

Formula for projecting geodesics in hyperbolic 3-manifolds, relating lengths to subsurface projections.

Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorph…

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…

Here we treat the problem: given a torsion-free connection do its geodesics, as unparametrised curves, coincide with the geodesics of an Einstein metric? We find projective invariants such that the vanishing of these is necessary for the existence of such a metric, and in generic settings the vanishing of these is also…

Proves lower bounds on Hausdorff dimension of projections of invariant sets.

Geodesic graphs for special Finsler metrics on spheres are studied.

Study projective deformations of hyperbolic 3-orbifolds with turnover ends.

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

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…

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain …

Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.

Study of hyperbolic directions in convex projective geometry.

Researchers classify geodesic orbit spaces for compact Lie groups of rank two.

Researchers compute c-projective symmetry algebras for Kähler surfaces.

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 …

New upper bound for geodesic complexity derived from cut locus decompositions.

Paper tackles online learning on curved spaces without projections.

We studied rules of transformations of Christoffel symbols under third type almost geodesic mappings in this paper. From this research, we obtained some new invariants of these mappings. These invariants are analogies of Thomas projective parameter and Weyl projective tensor.

Unique entropy measure found for convex projective manifolds.

Solves Lie's 3D metric problem for projective vector fields.

Develops a Riemannian archetypal analysis for interpretable non-linear data.

We consider the group of sense-preserving diffeomorphisms $\Diff S^1$ of the unit circle and its central extension, the Virasoro-Bott group, with their respective horizontal distributions chosen to be Ehresmann connections with respect to a projection to the smooth universal Teichmüller space and the universal Teichmül…

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.