arXiv research

The Funk-Finsler structure is constructed in hyperbolic models, including the Klein unit disc.

Study finds optimal loops in hyperbolic space with Finsler structure.

The paper solves optimal control problems for various convex sets using convex trigonometry.

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 express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmann manifold. Some applications are given.

Study on isoperimetric problem in Randers planes achieving maximum area.

The study proves surfaces in a specific Heisenberg group must be simple planes.

Extends Penrose limit to Finsler spacetimes.

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectr…

The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.

We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law "angle of incidence equals angle of reflection" are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. Thi…

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

Sharp estimates for Finsler metrics in convex domains.

Three models are shown to be isometrically equivalent, with a gapless first eigenvalue.

The paper finds the Finsler structure of Apollonian weak metric on unit disc.

We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes through 0 such that all sections of $B$ by these planes are linearly equivalent. Then…

The paper defines a metric on Euclidean triangles and polygons, proving properties and completeness.

The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such me…

New harmonic maps to hyperbolic plane via Bäcklund transformation.

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily…

A 6-regular triangulation for hyperbolic plane created.

New conditions found for hyperbolic bicycle tracks.

Study on travel time formulas in a lake with wind flow.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

The paper shows caustics by reflection in projective Finsler metrics have at least four cusps.

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.

Harmonic maps from hyperbolic planes to hyperbolic space exist with given boundary data.

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (i…

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

The paper explores how the Gauss curvature of Riemannian surfaces can be represented as the divergence of a vector field.

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^n,F), n\ge 3$ with reversibility $λ$ and flag curvature $K$ satisfying $\left(\fracλ{1+λ}\right)^2<K\le 1$, there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is fini…

New splitting theorem for weighted Finsler spacetimes without Berwald condition.

We prove that for every $\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.

Study inverse curve shortening flow on hyperbolic plane, classifying solitons.

The paper defines catenary curves in spheres and hyperbolic planes.

Study on closed $p$-elastic curves in hyperbolic and de Sitter planes.

In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is …

Outer billiards studied in complex hyperbolic plane, proving smooth and symplectic properties.

The paper finds geodesics on specific Finsler spheres with unique properties.

Study compares hyperbolic and quasihyperbolic metrics in plane domains.

Characterizes hyperbolic links with stable maps to the plane.

Classifies tilings of hyperbolic plane by regular polygons.

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F + df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is poss…

Crochet models of a hyperbolic plane is a popular educational tool as they help to visualize complicated objets in hyperbolic geometry. We present another way how to make crochet models when we view them as a part of a triangulated hyperbolic plane. We also provide a model of a cylinder in a hyperbolic space. This appr…

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.