arXiv subjects

Velichka Milousheva

Publications and source records attributed to Velichka Milousheva.

40 records · Page 3Linked to original sources

arXiv2010

Invariants of Lines on Surfaces in $\R^4$

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of this tangent. Asymptotic tangents (curves) are characterized by zero normal curvature. Considering the invariant of the pair of a given tangent and its orthogonal one, we introduce the geodesic torsion of this tangent. We obtain that principal tangents (curves) are characterized by zero geodesic torsion. The invariants $\varkappa$ and $k$ are introduced as the symmetric functions of the two principal normal curvatures. The geometric meaning of the semi-sum $\varkappa$ of the principal normal curvatures is equal (up to a sign) to the curvature of the normal connection of the surface. The number of asymptotic tangents at a point of the surface is determined by the sign of the invariant $k$. In the case $k=0$ there exists a one-parameter family of asymptotic lines, which are principal. We find examples of such surfaces ($k=0$) in the class of the general rotational surfaces (in the sense of Moore). The principal asymptotic lines on these surfaces are helices in the four-dimensional Euclidean space.

math.DG↗

arXiv2009

Geometric interpretation of the invariants of a surface in R^4 via the tangent indicatrix and the normal curvature ellipse

At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures. We give some non-trivial examples of surfaces from the classes in consideration.

math.DG↗

arXiv2008

Minimal Surfaces in the Four-Dimensional Euclidean Space

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for strongly regular minimal surfaces of general type in terms of their invariants. We introduce canonical parameters on strongly regular minimal surfaces of general type and prove that any such a surface is determined up to a motion by two invariant functions satisfying a system of two natural partial differential equations. On any minimal surface of the basic class of non strongly regular minimal surfaces we define canonical parameters and prove that any such a surface is determined up to a motion by two invariant functions of one variable satisfying a system of two natural ordinary differential equations. We find a geometric description of this class of non strongly regular minimal surfaces.

math.DG↗

arXiv2007

On the Theory of Surfaces in the Four-dimensional Euclidean Space

For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality kappa^2=k. The class of the surfaces with flat normal connection is characterized by the condition kappa = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas. We apply our theory to the class of the rotational surfaces, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.

math.DG↗