Concircular Hypersurfaces and Concircular Helices in Space Forms

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties known from their smooth counterparts. Special emphasis is paid to discrete flows built from Bäcklund transformations in the respective space forms. The invariants of the flows form a hierarchy of curves and we show that discrete elastic and constrained elastic curves can be characterized as elements of this hierarchy. This work also includes an introductory chapter on discrete curve theory in space forms, where we find discrete Frenet-type formulas and describe an associated family related to a fundamental theorem.

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.

The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures \(f(k_1,\ldots ,k_{n-1})\) satisfying suitable conditions. In this paper, we give sharp quantitative estimates of proximity to a single sphere for Alexandrov Soap Bubble Theorem in space forms when the curvature operator f is close to a constant. Under an assumption that prevents bubbling, the proximity to a single sphere is optimally quantified in terms of the oscillation of the curvature function f. Our approach provides a unified picture of quantitative studies of the method of moving planes in space forms.

In the three dimensional Riemannian space forms, we introduce a natural moving frame to define associate curve of a curve. Using the notion of associate curve we give a new necessary and sufficient condition of which a Frenet curve is a Mannheim curve or Mannheim partner curve in the three dimensional Euclidean space. Then we generalize these conclusions to the curves which lie on the three dimensional Riemannian sphere and the curves which lie in the three dimensional hyperbolic space. We also give the geometric characterizations of these curves. Our methods can be easily used to reveal the properties of the curves on space forms.

We classify minimal surfaces with constant Gaussian curvature in a 4 4 -dimensional space form without any global assumption. As a corollary of the main theorem, we show there is no isometric minimal immersion of a surface with constant negative Gaussian curvature into the unit 4 4 -sphere even locally. This gives a partial answer to a problem proposed by S. T. Yau.

Bertrand curves in 3-dimensional space forms

Let \( \wp_{1}, \wp_0 \) be two regular polygons of n sides in a space form M 2(κ) of constant curvature κ = 0,1 or − 1 such that \(\wp_0 \subset \wp_1\) and having the same center of mass. Suppose \( \wp_{0} \) is circumscribed by a circle C contained in \( \wp_{1} .\) We fix \(\wp_1\) and vary \(\wp_0\) by rotating it in C about its center of mass. Put \(\Omega=(\wp_{1} \setminus \wp_{0})^{0},\) the interior of \( \wp_{1} \setminus \wp_{0} \) in M 2(κ). It is shown that the first Dirichlet’s eigenvalue λ 1( Ω) attains extremum when the axes of symmetry of \( \wp_{0} \) coincide with those of \( \wp_{1}.\)

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gauss lifts of a compact surface to the twistor bundle are not vertically harmonic, then there exist at most three non-trivial congruence classes. We show that surfaces with a vertically harmonic Gauss lift possess a holomorphic quadratic differential, yielding thus a Hopf-type theorem. We prove that such surfaces allow locally a one-parameter family of isometric deformations with the same mean curvature. This family is trivial only if the surface is superconformal. For such compact surfaces with non-parallel mean curvature, we prove that the moduli space is the disjoint union of two sets, each one being either finite, or a circle. In particular, for surfaces in $$\mathbb {R}^4$$ we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.

We prove that a sphere-foliated minimal or constant mean curvature hypersurface in hyperbolic space of dimension > 5 is one of the following: hypersurface of rotation around a geodesic, geodesic hyperplane, horosphere, equidistant hypersurface, or a geodesic sphere in the upper half-space model. And we show that a sphere-foliated minimal or constant mean curvature hypersurface in sphere of dimension > 5 is either a hypersurface of rotation or a hypersphere. We also show that a hypersurface of nonzero constant mean curvature in Lorentz-Minkowski space foliated by spheres in space-like hyperplanes is either a hypersurface of rotation or a pseudo-hyperbolic space and that maximal space-like hypersurfaces foliated by spheres in hyperplanes are rotational if the ambient space has dimension > 4.

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with Lopez [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms.

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.

In his baic paper /1/ H.Einstein considered the probabilistic model of the motion of the bed-load material.He assumed, that the motion of the grain has diphase character:the moving period and the rest period. Einstei assumed, that the time of the moving period is very short compared to the duration of the rest period.This assumption leads us to the description of hydrotransportation process as a simple Poisson process. But in orginal Einstein paper, the transition from the motion of one particle to the motion of the group of particles is not clear.Einstein assumed, that the motion of the group of particles form the process of the same character.An interesting generalization of the Einstein model was presented by Shen and Cheong /7/. They suggested for the description of the hydro-transportation process a nonhomogenous Poisson process. Yang and Sayre/8/ obtained from experimental data gamma-distribution for the length of the rest period. In our paper /2/ we considered the motion of river bed-load material as a Markov chain. We found the estimators of transition probability in one step. In that model we assumed, that the motion of one particle in 3-dimensional space form 3 independent Markov chains : forward/backward, left/right and top/bottom.

We give explicit parametrizations for all parabolic geodesics in 3-dimensional Sasakian space forms.

Killing submersions and magnetic curves

Complete classification of parallel spatial surfaces in pseudo-Riemannian space forms with arbitrary index and dimension