Vasyl Gorkavyy

Vasyl Gorkavyy
B.Verkin Institute for Low Temperature Physics and Engineering · Differential equations and geometry

Dr. Sc.

About

40
Publications
669
Reads
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74
Citations
Additional affiliations
September 2002 - present
National Academy of Sciences of Ukraine
Position
  • Senior Researcher
February 2000 - present
V. N. Karazin Kharkiv National University
Position
  • Professor (Associate)
November 1995 - September 2002
B.Verkin Institute for Low Temperature Physics and Engineering
Position
  • Researcher
Education
September 1987 - June 1992
V. N. Karazin Kharkiv National University
Field of study
  • Mathematics and Mechanics

Publications

Publications (40)
Article
Full-text available
We study three-dimensional pseudo-spherical submanifolds in R5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^5$$\end{document}, whose Bianchi transformat...
Preprint
Three-dimensional pseudo-spherical submanifolds in $\mathbb R^5$, whose Bianchi transformations are degenerate of rank 2, are studied. A complete description of such submanifolds is obtained in the case where the Bianchi transformations are holonomically degenerate.
Article
Full-text available
Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.
Article
A new family of polyhedra called birosettes is presented. The geometric features of birosettes are analyzed. The model flexibility of birosettes is explained.
Preprint
The optimality of the integral inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds>2\pi$ for closed curves with non-vanishing curvatures in $\mathbb R^4$ is discussed. We prove that an arbitrary closed curve of constant positive curvatures in $\mathbb R^4$ satisfies the inequality $\int\limits_\gamma\sqrt{k_1^2+k_2^2+k_3^2}ds \geq 2\sqrt{5}\pi...
Article
A concept of degenerate B¨acklund transformation is introduced for two-dimensional surfaces in many-dimensional Euclidean spaces. It is shown that if a surface in Rn,n≥4, admits a degenerate B¨acklund transformation, then this surface is pseudospherical, i.e., its Gauss curvature is constant and negative. The complete classification of pseudospheri...
Article
The Jessen orthogonal icosahedron is known to be non-flexible however its physical models behave like physical models of flexible polyhedra. Adapting ideas of Milka, we propose two mathematical approaches for explaining the flexibility of physical models of the Jessen orthogonal icosahedron.
Article
In this paper, we discuss generalizations of the classical Bianchi–Bäcklund transformation of two-dimensional pseudo-spherical surfaces in three-dimensional spaces of constant curvature to the case of submanifolds of arbitrary codimension in spaces of constant curvature and in homogeneous Riemannian products.
Article
We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic plane.
Article
We present the classification of two-dimensional pseudospherical surfaces with degenerate Bianchi transformation in the multidimensional Euclidean space.
Article
Full-text available
We construct examples of two-dimensional pseudo-spherical surfaces, which admit Bianchi transformations, in the four-dimensional Euclidean space.
Article
We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in constant curvature spaces. Mathematics Subject Classification (2010)Primary 53A07
Article
Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative intrinsic curvature in the space S 3 × ℝ1. KeywordsBianchi transformation–2-surface negative intrinsic curvature–pseudospherical surface–pseudospherical congruence–Gaussian curvature–Bäcklund transformation
Article
We discuss the inflating of a closed thin shell made of inextensible flexible material like mylar. The problem is to determine the extremal form of the shell, when it is inflated to the maximal possible volume. We introduce a variational problem which describes the inflating of rotationally symmetric shells. The main result presents a criteria for...
Article
Full-text available
We construct particular linear bending of an arbitrary right pyramid, which increase the volume of pyramid. Consequently, for every convex right polyhedron we construct an iterating linear bending, which increase the volume of polyhedron. This improves previous numerical results obtained by D.D. Bleecker.
Article
Full-text available
Two-dimensional ruled surfaces in the spaces of constant curvature R n , S n , H n and in the Riemannian products S n × R 1 , H n × R 1 are considered. A ruled surface is proved to represent a pseudospherical congruence if and only if it is either an intrinsically flat surface in S n , or an intrinsically flat surface with constant extrinsic curvat...
Article
We distinguish two particular classes of lightlike surfaces in the Minkowski space Mn, which may be viewed as natural analogues of space- and timelike minimal surfaces.
Article
Pseudospherical Bianchi congruences in Euclidean 4-space are considered. The focal surfaces of such congruences are shown to have a constant negative Gaussian curvature. A geometric and an analytic description of special pseudo- spherical surfaces in admitting a Bianchi congruence are obtained.
Article
Full-text available
It is demonstrated that there exist surfaces of constant negative Gauss curvature in E 4 whose Grassmann image consists of either hyperbolic or parabolic or elliptic points. As a consequence, there exist surfaces of constant negative Gauss curvature in E 4 which do not admit Bäcklund transformations with help of pseudospherical congruences. A geome...
Article
A new type of electronic topological transition due to an abrupt change in the differential-geometric characteristics of the Fermi surface, whose topology remains unchanged, at some critical energy εd is predicted. This type of electronic topological transition differs from known electronic topological transitions due to a change in the topology of...
Article
Full-text available
Geometric Bäcklund transformations of pseudospherical surfaces in four-dimensional Euclidean space are studied. An analog of the classical theorems by Bäcklund, Tenenblat and Terng is proven.
Article
Geometric Bäcklund transformations of pseudospherical surfaces in four-dimensional Euclidean space are studied. An analog of the classical theorems by Bäcklund, Tenenblat and Terng is proved.
Article
We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifold G(m,m + 3) to be the Gauss image of a regular 3-submanifold in (m + 3)-dimensional Euclidean space for m > 4.
Article
We study the analogue of the Grassmannian image for submanifolds of a sphere, as described by Obata, and investigate the problem of reconstructing a submanifold from its Grassmannian image.
Article
We study the existence of a submanifold Fn of Euclidean space En+p with prescribed Grassmannian image that degenerates into a line. We prove that Γ is the Grassmannian image of a regular submanifold Fn of Euclidean space En+p if and only if the curve Γ in the Grassmann manifold G+(p, n + p) is asymptotically Cr-regular, r > 1. Here G+(n, n + p) is...