关于SOL流形具有常平均曲率旋转曲面的注记

研究非欧流形SOL 空间上共形平均曲率方程的可解性,通过研究轮廓曲线对具有平均曲率的旋转曲面进行分类。当这些旋转曲面的平均曲率为给定函数时,计算出相应轮廓曲线的微分方程。通过求解这些微分方程,给出旋转函数是其上共形平均曲率的充分条件。     

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.     

On the total mean curvature of piecewise smooth surfaces under infinitesimal bending

Variation of the total mean curvature of piecewise smooth surfaces in Euclidean 3-spaces under infinitesimal bending is discussed and reduced to a sum of line integrals of a rotation vector field.     

Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

We consider the Kepler problem on surfaces of revolution that are homeomorphic to S² and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lenz vector. Then, using such first integrals, we determine the class of surfaces that lead to block-regularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.     

Integrable billiards on surfaces of constant curvature

Translated from Matematicheskie Zametki, Vol. 51, No. 2, pp. 20–28, February, 1992.     

Closed geodesics on orbifolds

In this paper, we try to generalize to the case of compact Riemannian orbifolds Q some classical results about the existence of closed geodesics of positive length on compact Riemannian manifolds M. We shall also consider the problem of the existence of infinitely many geometrically distinct closed geodesics.In the classical case the solution of those problems involve the consideration of the homotopy groups of M and the homology properties of the free loop space on M (Morse theory). Those notions have their analogue in the case of orbifolds. The main part of this paper will be to recall those notions and to show how the classical techniques can be adapted to the case of orbifolds.     

Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.     

On Peterson surfaces of constant Gaussian curvature

The surfaces of constant Gaussian curvature bearing conjugate networks of conic lines are found.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 3–5.     

A note on constant geodesic curvature curves on surfaces

In this paper we are concerned with the structure of curves on surfaces whose geodesic curvature is a large constant. We first discuss the relation between closed curves with large constant geodesic curvature and the critical points of Gauss curvature. Then, we consider the case where a curve with large constant geodesic curvature is immersed in a domain which does not contain any critical point of the Gauss curvature.