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Papers by vasyl gorkavyy

Research paper thumbnail of Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

Journal of Mathematical Sciences

In this paper, we discuss generalizations of the classical Bianchi–Bäcklund transformation of two... more 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.

Research paper thumbnail of One integral inequality for closed curve in Euclidean space

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Ruled Surfaces as Pseudospherical Congruences

Journal of Mathematical Physics, Analysis, Geometry

Two-dimensional ruled surfaces in the spaces of constant curvature R n , S n , H n and in the Rie... more 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 curvature in S n × R 1 .

Research paper thumbnail of An Example of Bianchi Transformation in E4

Journal of Mathematical Physics, Analysis, Geometry

We construct examples of two-dimensional pseudo-spherical surfaces, which admit Bianchi transform... more We construct examples of two-dimensional pseudo-spherical surfaces, which admit Bianchi transformations, in the four-dimensional Euclidean space.

Research paper thumbnail of On pseudospherical congruences in E 4

On pseudospherical congruences in E 4

Geometric Bäcklund transformations of pseudospherical surfaces in four-dimensional Euclidean spac... more 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.

Research paper thumbnail of On pseudospherical survaces in E 4 with Grassmann image of prescribed type

Journal of Mathematical Physics, Analysis, Geometry

It is demonstrated that there exist surfaces of constant negative Gauss curvature in E 4 whose Gr... more 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 geometric representation for pseudospherical surfaces in E 4 with parabolic Grassmann image is proposed.

Research paper thumbnail of Deformability of surfaces F 2 in 𝔼 4 preservation of the Grassmannian image

Deformability of surfaces F 2 in 𝔼 4 preservation of the Grassmannian image

Siberian Advances in Mathematics

Research paper thumbnail of Barker–Larman Problem for Convex Polygons in the Hyperbolic Plane

Barker–Larman Problem for Convex Polygons in the Hyperbolic Plane

Results in Mathematics, 2015

ABSTRACT We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic ... more ABSTRACT We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic plane.

Research paper thumbnail of Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation

Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation

Ukrainian Mathematical Journal, 2012

ABSTRACT We present the classification of two-dimensional pseudospherical surfaces with degenerat... more ABSTRACT We present the classification of two-dimensional pseudospherical surfaces with degenerate Bianchi transformation in the multidimensional Euclidean space.

Research paper thumbnail of The analogue of the Grassmannian image for submanifolds of a sphere

The analogue of the Grassmannian image for submanifolds of a sphere

Sbornik: Mathematics, 1996

We study the analogue of the Grassmannian image for submanifolds of a sphere, as described by Oba... more 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.

Research paper thumbnail of Bianchi congruences of two-dimensional surfaces in $ E^4$

Bianchi congruences of two-dimensional surfaces in $ E^4$

Sbornik: Mathematics, 2005

Pseudospherical Bianchi congruences in Euclidean 4-space are considered. The focal surfaces of su... more 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.

Research paper thumbnail of Восстановление трехмерных подмногообразий евклидова пространства с большой коразмерностью по грассманову образу

Восстановление трехмерных подмногообразий евклидова пространства с большой коразмерностью по грассманову образу

Математические заметки, 1997

Research paper thumbnail of Восстановление подмногообразия евклидова пространства по вырожденному в линию грассманову образу

Восстановление подмногообразия евклидова пространства по вырожденному в линию грассманову образу

Математические заметки, 1996

Research paper thumbnail of An analog of bianchi transformations for two-dimensional surfaces in the space S 3 × ℝ1

Mathematical Notes, 2011

Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative ... more Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative intrinsic curvature in the space S 3 × R 1 .

Research paper thumbnail of Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

Results in Mathematics, 2011

ABSTRACT We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in cons... more ABSTRACT We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in constant curvature spaces. Mathematics Subject Classification (2010)Primary 53A07

Research paper thumbnail of On minimal lightlike surfaces in Minkowski space–time

Differential Geometry and its Applications, 2008

We distinguish two particular classes of lightlike surfaces in the Minkowski space M n , which ma... more We distinguish two particular classes of lightlike surfaces in the Minkowski space M n , which may be viewed as natural analogues of space-and timelike minimal surfaces.

Research paper thumbnail of On inflating closed mylar shells

Comptes Rendus Mécanique, 2010

Research paper thumbnail of Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

Journal of Mathematical Sciences

In this paper, we discuss generalizations of the classical Bianchi–Bäcklund transformation of two... more 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.

Research paper thumbnail of One integral inequality for closed curve in Euclidean space

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

Research paper thumbnail of Ruled Surfaces as Pseudospherical Congruences

Journal of Mathematical Physics, Analysis, Geometry

Two-dimensional ruled surfaces in the spaces of constant curvature R n , S n , H n and in the Rie... more 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 curvature in S n × R 1 .

Research paper thumbnail of An Example of Bianchi Transformation in E4

Journal of Mathematical Physics, Analysis, Geometry

We construct examples of two-dimensional pseudo-spherical surfaces, which admit Bianchi transform... more We construct examples of two-dimensional pseudo-spherical surfaces, which admit Bianchi transformations, in the four-dimensional Euclidean space.

Research paper thumbnail of On pseudospherical congruences in E 4

On pseudospherical congruences in E 4

Geometric Bäcklund transformations of pseudospherical surfaces in four-dimensional Euclidean spac... more 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.

Research paper thumbnail of On pseudospherical survaces in E 4 with Grassmann image of prescribed type

Journal of Mathematical Physics, Analysis, Geometry

It is demonstrated that there exist surfaces of constant negative Gauss curvature in E 4 whose Gr... more 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 geometric representation for pseudospherical surfaces in E 4 with parabolic Grassmann image is proposed.

Research paper thumbnail of Deformability of surfaces F 2 in 𝔼 4 preservation of the Grassmannian image

Deformability of surfaces F 2 in 𝔼 4 preservation of the Grassmannian image

Siberian Advances in Mathematics

Research paper thumbnail of Barker–Larman Problem for Convex Polygons in the Hyperbolic Plane

Barker–Larman Problem for Convex Polygons in the Hyperbolic Plane

Results in Mathematics, 2015

ABSTRACT We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic ... more ABSTRACT We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic plane.

Research paper thumbnail of Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation

Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation

Ukrainian Mathematical Journal, 2012

ABSTRACT We present the classification of two-dimensional pseudospherical surfaces with degenerat... more ABSTRACT We present the classification of two-dimensional pseudospherical surfaces with degenerate Bianchi transformation in the multidimensional Euclidean space.

Research paper thumbnail of The analogue of the Grassmannian image for submanifolds of a sphere

The analogue of the Grassmannian image for submanifolds of a sphere

Sbornik: Mathematics, 1996

We study the analogue of the Grassmannian image for submanifolds of a sphere, as described by Oba... more 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.

Research paper thumbnail of Bianchi congruences of two-dimensional surfaces in $ E^4$

Bianchi congruences of two-dimensional surfaces in $ E^4$

Sbornik: Mathematics, 2005

Pseudospherical Bianchi congruences in Euclidean 4-space are considered. The focal surfaces of su... more 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.

Research paper thumbnail of Восстановление трехмерных подмногообразий евклидова пространства с большой коразмерностью по грассманову образу

Восстановление трехмерных подмногообразий евклидова пространства с большой коразмерностью по грассманову образу

Математические заметки, 1997

Research paper thumbnail of Восстановление подмногообразия евклидова пространства по вырожденному в линию грассманову образу

Восстановление подмногообразия евклидова пространства по вырожденному в линию грассманову образу

Математические заметки, 1996

Research paper thumbnail of An analog of bianchi transformations for two-dimensional surfaces in the space S 3 × ℝ1

Mathematical Notes, 2011

Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative ... more Bianchi-type transformations are constructed for two-dimensional surfaces with constant negative intrinsic curvature in the space S 3 × R 1 .

Research paper thumbnail of Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

Results in Mathematics, 2011

ABSTRACT We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in cons... more ABSTRACT We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in constant curvature spaces. Mathematics Subject Classification (2010)Primary 53A07

Research paper thumbnail of On minimal lightlike surfaces in Minkowski space–time

Differential Geometry and its Applications, 2008

We distinguish two particular classes of lightlike surfaces in the Minkowski space M n , which ma... more We distinguish two particular classes of lightlike surfaces in the Minkowski space M n , which may be viewed as natural analogues of space-and timelike minimal surfaces.

Research paper thumbnail of On inflating closed mylar shells

Comptes Rendus Mécanique, 2010