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Georgi Ganchev

Publications and source records attributed to Georgi Ganchev.

46 records · Page 3Linked to original sources
arXiv2008

Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space

We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Gauss plane). We also find a canonical representation of maximal strongly regular space-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic functions (in the Lorentz plane). This allows us to describe locally the solutions of the corresponding natural partial differential equations.

math.DG↗
arXiv2008

Canonical Weierstrass Representation of Minimal Surfaces in Euclidean Space

Using the fact that any minimal strongly regular surface carries locally canonical principal parameters, we obtain a canonical representation of these surfaces, which makes more precise the Weierstrass representation in canonical principal parameters. This allows us to describe locally the solutions of the natural partial differential equation of minimal surfaces.

math.DG↗
arXiv2008

On the Invariant Theory of Weingarten Surfaces in Euclidean Space

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature.

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↗
arXiv2007

Isogonal Conjugacy and Fermat Problems

We consider three types of isogonal conjugacy of two points with respect to a given triangle and characterize any of these types by a geometric equality. As an application to the Fermat problem with positive weights, we prove that in the general case the given weights determine uniquely a point X and the solution to the Fermat problem is the point Y, which is isogonally conjugate of type I to the point X. We obtain a similar characterization of the solution to the Fermat problem in the case of mixed weights as well.

math.GM↗
arXiv2006

Kaehler Metrics Generated by Functions of the Time-Like Distance in the Flat Kaehler-Lorentz Space

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kaehler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kaehler structures. We prove that these rotational hypersurfaces carry Kaehler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kaehler metrics are Bochner-Kaehler (especially of constant holomorphic sectional curvatures) are also described.

math.DG↗
arXiv2006

Kaehler manifolds admitting a flat complex conformal connection

We prove that any Kaehler manifold admitting a flat complex conformal connection is a Bochner-Kaehler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kaehler manifolds under consideration.

math.DG↗
arXiv2006

Warped product Kaehler manifolds and Bochner-Kaehler metrics

Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prove that any Kaehler manifold of quasi-constant holomorphic sectional curvatures with special distribution locally has the structure of a warped product Kaehler manifold whose base is an alpha-Sasakian space form. Considering the scalar distribution generated by the scalar curvature of a Kaehler manifold, we give a new approach to the local theory of Bochner-Kaehler manifolds. We study the class of Bochner-Kaehler manifolds whose scalar distribution is of special type. Taking into account that any manifold of this class locally is a warped product Kaehler manifold, we describe all warped product Bochner-Kaehler metrics. We find four families of complete metrics of this type.

math.DG↗
arXiv2005

Kaehler Manifolds of Quasi-Constant Holomorphic Sectional Curvatures

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kaehler metrics into Kaehler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kaehler metrics is shown to be exactly the class of Kaehler metrics whose potential function is only a function of the distance from the origin in complex Euclidean space. Finally we show that any rotational even dimensional hypersurface carries locally a natural Kaehler structure, which is of quasi-constant holomorphic sectional curvatures.

math.DG↗