Searcharxiv⌕ Search

arXiv subjects

Ognian Kassabov

Publications and source records attributed to Ognian Kassabov.

At least 19 recordsLinked to original sources
arXiv2026

Basic invariants for time-like surfaces in $\mathbb R^3_1$ with real asymptotic lines

The geometrically defined wide class of time-like surfaces in $\mathbb R^3$, admitting real asymptotic lines is considered. A fundamental theorem of Bonnet-type is obtained for these surfaces. It states that a surface in this class is determined (up to a motion) by four invariant functions, satisfying some natural PDEs. Then canonical parameters are defined for these surfaces and it is proved that such a surface is determined (up to a motion) in canonical parameters with only two invariant functions (which in particular can be the Gauss and the mean curvature), satisfying a partial differential equation, equivalent to the Gauss equation.

math.DG↗
arXiv2025

Canonical parameters on a surface in $\mathbb R^4$

In the present paper, we study surfaces in the four-dimensional Euclidean space $\mathbb{R}^4$. We define special principal parameters, which we call canonical, on each surface without minimal points, and prove that the surface admits (at least locally) canonical principal parameters. They can be considered as a generalization of the canonical parameters for minimal surfaces and the canonical parameters for surfaces with parallel normalized mean curvature vector field introduced before. We prove a fundamental existence and uniqueness theorem formulated in terms of canonical principal parameters, which states that the surfaces in $\mathbb{R}^4$ are determined up to a motion by four geometrically determined functions satisfying a system of partial differential equations.

math.DG↗
arXiv2024

Relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and $\mathbb R^3_1$

In this paper we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space $\mathbb R^3_1$ using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in $\mathbb R^4_2$ and the pairs of null curves in $\mathbb R^3_1$. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in $\mathbb R^4_2$ and the pairs of minimal Lorentz surfaces in $\mathbb R^3_1$.

math.DG↗
arXiv2023

Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters

We study minimal timelike surfaces in $\mathbb R^3_1$ using a special Weierstrass-type formula in terms of holomorphic functions defined in the algebra of the double (split-complex) numbers. We present a method of obtaining an equation of a minimal timelike surface in terms of canonical parameters, which play a role similar to the role of the natural parameters of curves in $\mathbb R^3$. Having one holomorphic function that generates a minimal timelike surface, we find all holomorphic functions that generate the same surface. In this way we give a correspondence between a minimal timelike surface and a class of holomorphic functions. As an application, we prove that the Enneper surfaces are the only minimal timelike surfaces in $\mathbb R^3_1$ with polynomial parametrization of degree 3 in isothermal parameters.

math.DG↗
arXiv2021

Canonical Coordinates and Natural Equation for Lorentz Surfaces in $\mathbb R^3_1$

We consider Lorentz surfaces in $\mathbb R^3_1$ satisfying the condition $H^2-K\neq 0$, where $K$ and $H$ are the Gauss curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces we introduce special isotropic coordinates, which we call canonical, and show that the coefficient $F$ of the first fundamental form and the mean curvature $H$, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of general type. Using this natural equation we prove a fundamental theorem of Bonnet type for Lorentz surfaces of general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

math.DG↗
arXiv2021

Explicit Solving of the System of Natural PDEs of Minimal Lorentz Surfaces in $\mathbb R^4_2$

A minimal Lorentz surface in $\mathbb R^4_2$ is said to be of general type if its corresponding null curves are non-degenerate. These surfaces admit canonical isothermal and canonical isotropic coordinates. It is known that the Gauss curvature $K$ and the normal curvature $\varkappa$ of such a surface considered as functions of the canonical coordinates satisfy a system of two natural PDEs. Using the Weierstrass type representations of the corresponding null curves, we solve explicitly the system of natural PDEs, expressing any solution by means of four real functions of one variable. We obtain the transformation formulas for the functions in the Weierstrass representation of a null curve under a proper motion in $\mathbb R^4_2$. Using this, we find the relation between two quadruples of real functions generating one and the same solution to the system of natural PDEs.

math.DG↗
arXiv2021

Weierstrass Representations of Lorentzian Minimal Surfaces in $\mathbb R^4_2$

The minimal Lorentzian surfaces in $\mathbb{R}^4_2$ whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy $K^2-\varkappa^2 >0$ are called minimal Lorentzian surfaces of general type. These surfaces admit canonical parameters and with respect to such parameters are determined uniquely up to a motion in $\mathbb{R}^4_2$ by the curvatures $K$ and $\varkappa$ satisfying a system of two natural PDEs. In the present paper we study minimal Lorentzian surfaces in $\mathbb{R}^4_2$ and find a Weierstrass representation with respect to isothermal parameters of any minimal surface with two-dimensional first normal space. We also obtain a Weierstrass representation with respect to canonical parameters of any minimal Lorentzian surface of general type and solve explicitly the system of natural PDEs expressing any solution to this system by means of four real functions of one variable.

math.DG↗
arXiv2019

Characterizing a surface by invariants

Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial differential equation equivalent to the Gauss equation. As such a pair of invariants we may use the principal curvatures or the Gauss and the mean curvature.

math.DG↗
arXiv2016

Polynomial minimal surfaces of degree five

The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees the problem is quite more complicated. Here we find a general form for the functions that generate a polynomial minimal surface of arbitrary degree via the Weierstrass formula and prove that any polynomial minimal surface of degree 5 in isothermal parameters may be considered as belonging to one of three special families.

math.DG↗
arXiv2015

Bi-quartic parametric polynomial minimal surfaces

Minimal surfaces with isothermal parameters admitting Bézier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of their generating functions in the Weierstrass representation formula. We apply an approach proposed by Ganchev to compute the normal curvature and show that, in contrast to the bi-cubic case, there is a variety of bi-quartic isothermal minimal surfaces. Based on the Bezier representation we establish some geometric properties of the bi-quartic harmonic surfaces. Numerical experiments are visualized and presented to illustrate and support our results.

math.DG↗
arXiv2015

On Bochner flat almost Kahler manifolds

The main purpose of this article is to prove that there exist no proper $AK_3$-manifold of dimension $2n\ge 6$ with vanishing Tricerri-Vanhecke Bochner curvature tensor and constant scalar curvature.

math.DG↗
arXiv2014

Congruence of minimal surfaces

An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for minimal surfaces. Here we explain not only how this method works but also how we can find the correspondence between the minimal surfaces, if they are congruent. We show that two families of minimal surfaces which are proved to be conjugate actually coincide and coincide with their associated surfaces. We also consider another family of minimal polynomial surfaces of degree 6 and we apply the method to show that some of them are congruent.

math.DG↗
arXiv2011

Hermitian Manifolds with Flat Associated Connection

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kaehler manifold of constant holomorphic sectional curvatures.

math.DG↗
arXiv2011

Schur's Theorem for Almost Hermitian Manifolds

The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.

math.DG↗
arXiv2010

A characterization of the extrinsic spheres in a Riemannian manifold

The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N or an extrinsic sphere of N.

math.DG↗
arXiv2010

On totally real submanifolds

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the normal bundle and of constant length of the second fundamental form (or equivalently of constant scalar curvature) of a complex space form N is totally geodesic in N or of positive scalar curvature. Moreover, if the scalar curvature of M vanishes, then M is flat. Theorem 3. A complete, compact totally real submanifold with parallel mean curvature vector, parallel f-structure in the normal bundle and commutative second fundamental forms of a simply connected complete complex space form is totally geodesic or a pythagorean product of circles. Note that if M is a totally real submanifold of a Kähler manifold N and the dimension of N is twice the dimension of M, then the f-structure in the normal bundle vanishes.

math.DG↗