arXiv research

Research explores Lorentzian distances on a specific geometric plane.

We give a sharp lower bound on the area of a domain that can be enclosed by a closed embedded $λ$-convex curve of a given length on the Lobachevsky plane.

Study finds optimal loops in hyperbolic space with Finsler structure.

This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.

We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional Minkowski half…

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

The study counts triangulations of a projective plane with specific vertex valencies.

We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space

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

We prove a theorem about an extremal property of Lobachevsky space among simply connected Riemannian manifolds of nonpositive curvature

After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here…

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum vo…

We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky $(n+1)$ s…

New formula calculates volumes of ideal hyperbolic drums.

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a m…

A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge s…

Study on planar graphs in Poincare model of hyperbolic geometry.

We give a unified description of tetrahedra with lightlike faces in 3d anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra are determined by a generalized cross-ratio with values in a commutative 2d real algebra t…

Historical notes on Thurston's 3-manifold geometry.

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic…

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

Stable planes are locally isomorphic to classical projective planes.

Only vertical planes are asymptotic to other planes in 3D space.

Study of elementary planes in Apollonian orbifold with unusual equidistribution.

Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…

Criteria for sharksfin and deltoid singularities from plane to plane, with applications.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

The study explores conformal planes with finite areas.

For a pair of points in a smooth locally convex surface in 3-space, its mid-plane is the plane containing its mid-point and the intersection line of the corresponding pair of tangent planes. In this paper we show that the limit of mid-planes when one point tends to the other along a direction is the Transon plane of th…

We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.

This thesis explores bifoliated planes and their group actions from Anosov flows.

We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…

Two proofs show that removing a loop from a plane circuit splits the plane.

The paper classifies singularities of plane congruences and affine distance functions.

Computed distortion coefficients for the α-Grushin plane.

New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.

New harmonic maps to hyperbolic plane via Bäcklund transformation.

Study compact plane waves, showing they are essentially standard.

The paper examines asymptotic lines of plane fields in 3D space.

A 6-regular triangulation for hyperbolic plane created.

Study local and global aspects of complex plane curve embeddings.

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and by considering the complexifications of these four projective planes.

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degene…

We study a moduli stratum of A-orbits of plane-to-plane germs of corank 2 with codimension 3. We describe explicitly the bifurcation diagram of its topologically A-versal unfolding. Two geometric applications to parabolic objects are presented.

We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.

Proves conjecture about geodesic foliations in Riemannian planes.