arXiv subjects

Victoria Bencheva

Publications and source records attributed to Victoria Bencheva.

5 recordsLinked to original sources

arXiv2026

Timelike meridian surfaces of hyperbolic type in the Minkowski 4-space

We consider a special class of timelike surfaces in the four-dimensional Minkowski space which are one-parameter systems of meridians of rotational hypersurfaces with spacelike axis and call them meridian surfaces of hyperbolic type. We show that all timelike meridian surfaces of hyperbolic type are surfaces with flat normal connection and give the complete classification of those surfaces with constant Gauss curvature. We also classify all minimal timelike meridian surfaces of hyperbolic type and all timelike meridian surfaces of hyperbolic type with non-zero constant mean curvature (CMC-surfaces). We show that there are no timelike meridian surfaces of hyperbolic type with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. Finally, we describe the class of timelike meridian surfaces of hyperbolic type with parallel normalized mean curvature vector field, but non-parallel mean curvature vector.

math.DG↗

arXiv2025

Timelike Meridian Surfaces of Elliptic type in the Minkowski 4-Space

We consider a special family of 2-dimensional timelike surfaces in the Minkowski 4-space $\mathbb{R}^4_1$ which lie on rotational hypersurfaces with timelike axis and call them meridian surfaces of elliptic type. We study the following basic classes of timelike meridian surfaces of elliptic type: with constant Gauss curvature, with constant mean curvature, with parallel mean curvature vector field, with parallel normalized mean curvature vector field. The results obtained for the last class are used to give explicit solutions to the background systems of natural PDEs describing the timelike surfaces with parallel normalized mean curvature vector field in $\mathbb{R}^4_1$.

math.DG↗

arXiv2024

Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative formulas with respect to this moving frame field and using the integrability conditions, we obtain a system of six functions satisfying some natural conditions. In the general case, we prove a Fundamental Bonnet-type theorem (existence and uniqueness theorem) stating that these six functions, satisfying the natural conditions, determine the surface up to a motion. In some particular cases, we reduce the number of functions and give the fundamental theorems.

math.DG↗

arXiv2024

Timelike Surfaces with Parallel Normalized Mean Curvature Vector Field in the Minkowski 4-Space

In the present paper, we study timelike surfaces with parallel normalized mean curvature vector field in the four-dimensional Minkowski space. We introduce special isotropic parameters on each such surface, which we call canonical parameters, and prove a fundamental existence and uniqueness theorem stating that each timelike surface with parallel normalized mean curvature vector field is determined up to a rigid motion in the Minkowski space by three geometric functions satisfying a system of three partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, thus solving the Lund-Regge problem for this class of surfaces.

math.DG↗

arXiv2022

Basic Classes of Timelike General Rotational Surfaces in the Four-dimensional Minkowski Space

In the present paper, we consider timelike general rotational surfaces in the Minkowski 4-space which are analogous to the general rotational surfaces in the Euclidean 4-space introduced by C. Moore. We study two types of such surfaces (with timelike and spacelike meridian curve, respectively) and describe analytically some of their basic geometric classes: flat timelike general rotational surfaces, timelike general rotational surfaces with flat normal connection, and timelike general rotational surfaces with non-zero constant mean curvature. We give explicitly all minimal timelike general rotational surfaces and all timelike general rotational surfaces with parallel normalized mean curvature vector field.

math.DG↗