Searcharxiv⌕ Search

arXiv subjects

Georgi Ganchev

Publications and source records attributed to Georgi Ganchev.

At least 19 recordsLinked to original sources
arXiv2019

Canonical Coordinates and Natural Equations for Minimal Time-like Surfaces in $R^4_2$

We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature $K$ and the curvature of the normal connection $\varkappa$, satisfying the system of natural equations for these surfaces. We prove the following: If $(K, \varkappa), \, K^2- \varkappa^2 > 0 $ is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the first type and exactly one minimal time-like surface of the second type with invariants $(K, \varkappa)$; if $(K, \varkappa),\, K^2- \varkappa^2 < 0 $ is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants $(K, \varkappa)$.

math.DG↗
arXiv2019

Canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface we introduce a special complex function over the algebra of the double numbers and apply the analysis over the algebra of the double numbers as a convenient tool to study these surfaces. Then the canonical coordinates on minimal time-like surfaces are characterized by a natural condition for this complex function. We consider the hyperbola of the normal curvature of any minimal time-like surface and give a geometric interpretation of the canonical coordinates in terms of the elements of this hyperbola.

math.DG↗
arXiv2019

Canonical Weierstrass Representations for Maximal Space-like Surfaces in $\RR^4_2$

It is known that any maximal space-like surface without isotropic points in the four-dimensional pseudo-Euclidean space with neutral metric admits locally geometric parameters which are special case of isothermal parameters. With respect to such parameters the surface is determined uniquely up to a motion by the Gauss curvature and the curvature of the normal connection, which satisfy a system of two PDE's (the system of natural PDE's). For any maximal space-like surface parametrized by canonical parameters we obtain a special Weierstrass representation -- canonical Weierstrass representation. These Weierstrass formulas allow us to solve explicitly the system of natural PDE's by virtue of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holomorphic functions generating one and the same solution to the system of natural PDE's. We establish a geometric correspondence between the maximal space-like surfaces of general type in $\RR^4_2$, the solutions to the system of natural PDE's and the pairs of holomorphic functions in the Gauss plane. We prove that any maximal space-like surface in the four-dimensional pseudo-Euclidean space with neutral metric generates two maximal space-like surfaces in the three-dimensional Minkowski space and vice versa.

math.DG↗
arXiv2018

Surfaces with Parallel Normalized Mean Curvature Vector Field in Euclidean or Minkowski 4-Space

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

math.DG↗
arXiv2017

Canonical Weierstrass representations for minimal space-like surfaces in $\RR^4_1$

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations via two holomorphic functions. We find the expressions of the Gauss curvature and the normal curvature of the surface with respect to this pair of holomorphic functions. We find the relation between two pairs of holomorphic functions generating one and the same minimal space-like surface of general type. The canonical Weierstrass formulas allow us to establish geometric correspondence between minimal space-like surfaces of general type and classes of pairs of holomorphic functions in the Gauss plane.

math.DG↗
arXiv2016

Explicit Solving of the System of Natural PDE's of Minimal Space-like Surfaces in Minkowski Space-time

A minimal space-like surface in Minkowski space-time is said to be of general type if it is free of degenerate points. The fact that minimal space-like surfaces of general type in Minkowski space-time admit canonical parameters of the first (second) type implies that any minimal space-like surface is determined uniquely up to a motion in space-time by the Gauss curvature and the normal curvature, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal space-like surfaces. Using canonical Weierstrass representations of minimal space-like surfaces of general type in Minkowski space-time we solve explicitly the system of natural PDE's, expressing any solution by means of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holomorphic functions (i.e. the class of pairs of holomorphic functions) generating one and the same solution to the system of natural PDE's, i.e. generating one and the same minimal space-like surface in Minkowski space-time.

math.DG↗
arXiv2016

Canonical Weierstrass representations for minimal surfaces in Euclidean 4-space

Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such parameters the Gauss curvature and the normal curvature satisfy a system of two natural partial differential equations and determine the surface up to a motion. For any minimal surface parameterized by canonical parameters we obtain Weierstrass representations - canonical Weierstrass representations. These Weierstrass formulas allow us to solve explicitly the system of natural partial differential equations and to establish geometric correspondence between minimal surfaces of general type, the solutions to the system of natural equations and pairs of holomorphic functions in the Gauss plane. On the base of these correspondences we obtain that any minimal surface of general type in Euclidean 4-space determines locally a pair of two minimal surfaces in Euclidean 3-space and vice versa. Finally some applications of this phenomenon are given.

math.DG↗
arXiv2016

Explicit Solving of the System of Natural PDE's of Minimal Surfaces in the Four-Dimensional Euclidean Space

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal surfaces. Using the corresponding result for minimal surfaces in the three-dimensional Euclidean space, we solve explicitly the system of natural PDE's, expressing any solution by virtue of two holomorphic functions in the Gauss plane. We find the relation between two pairs of holomorphic functions (i.e. the class of pairs of holomorphic functions) generating one and the same solution of the system of natural PDE's.

math.DG↗
arXiv2016

Meridian Surfaces of Parabolic Type in the Four-dimensional Minkowski Space

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the invariants of these surfaces we give the complete classification of the meridian surfaces of parabolic type with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces of parabolic type and the meridian surfaces of parabolic type with parallel normal bundle.

math.DG↗
arXiv2015

On the Theory of Lorentz Surfaces with Parallel Normalized Mean Curvature Vector Field in Pseudo-Euclidean 4-Space

We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.

math.DG↗
arXiv2014

Meridian Surfaces of Elliptic or Hyperbolic Type in the Four-dimensional Minkowski Space

We consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. We call these surfaces meridian surfaces of elliptic or hyperbolic type, respectively. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants and give the complete classification of the meridian surfaces with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces and the meridian surfaces with parallel normal bundle.

math.DG↗
arXiv2014

Special Classes of Meridian Surfaces in the Four-dimensional Euclidean Space

Meridian surfaces in the Euclidean 4-space are two-dimensional surfaces which are one-parameter systems of meridians of a standard rotational hypersurface. On the base of our invariant theory of surfaces we study meridian surfaces with special invariants. In the present paper we give the complete classification of Chen meridian surfaces and meridian surfaces with parallel normal bundle.

math.DG↗
arXiv2013

General Rotational Surfaces in the Four-dimensional Minkowski Space

General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with special invariants. We describe analytically the flat general rotational surfaces and the general rotational surfaces with flat normal connection. We classify completely the minimal general rotational surfaces and the general rotational surfaces consisting of parabolic points.

math.DG↗
arXiv2013

Directed Riemannian manifolds of pointwise constant relative sectional curvature

We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hypersurfaces are directed and find the rotational hypersurfaces of pointwise constant relative sectional curvature. For the class of directed Riemannian manifolds of pointwise constant relative sectional curvature having a totally umbilical scalar distribution we prove a structural theorem and a theorem of Schur's type.

math.DG↗
arXiv2012

Holomorphic hypersurfaces of Kaehler manifolds with Norden metric

We study Kaehlerian manifolds with Norden metric $g$ and develop the theory of their holomorphic hypersurfaces with constant totally real sectional curvatures. We prove a classification theorem for the holomorphic hypersurfaces of $(\mathbb{R}^{2n+2}, g, J)$ with constant totally real sectional curvatures.

math.DG↗
arXiv2012

Points, whose pedal triangles are similar to the given triangle

We study the eleven points in the plane of a given triangle, whose pedal triangles are similar to the given one. We prove that the six points whose pedal triangles are positively oriented, lie on a single circle, while the five points, whose pedal triangles are negatively oriented, lie on a common straight line.

math.HO↗
arXiv2012

Quasi-minimal Rotational Surfaces in Pseudo-Euclidean Four-dimensional Space

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all quasi-minimal rotational surfaces of elliptic, hyperbolic and parabolic type, respectively.

math.DG↗