On ricci-pseudosymmetric hypersurfaces in spaces of constant curvature

Results in Mathematics - This article studies the conditions of pseudosymmetry and Ricci-pseudosymmetry, realizedon hypersurfaces of semi-Riemannian spaces of constant curvature. In particular, we...     

Simplexes in spaces of constant curvature

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form _n (l)l _ijl which guarantees the existence of an n-simplex p ₁ p ₂ ... p _n+1 with edge-lengths p _ip_j=l _ij. (In spherical geometry of curvature 1 the interval is empty unless l2 arcsin ) The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.This work was supported by Canadian NSERC grants.     

Geodesics in Randers spaces of constant curvature

Geodesics in Randers spaces of constant curvature are classified.

     

Incidence theorems in spaces of constant curvature

Certain analogs of the classic theorems of Menelaus and Ceva are considered for a hyperbolic surface, a sphere, and for three-dimensional hyperbolic and spherical spaces.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 67–74, 1992.     

Spherical means in spaces of constant curvature

Summary Extension of theorems ofF. Riesz on subharmonic functions to spaces of constant curvature by the use of hyperbolic partial differential equations. To Enrico Bompiani on his scientific Jubiles This research was supported in part by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command under ontract No. AF 49 (638)–228.     

Integral geometry in constant curvature Lorentz spaces

We extend classical Cauchy formulas and Crofton formulas to the Lorentz-Minkowski space, and to constant curvature Lorentz spaces.     

On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in Received: 30 June 2005     

Radon transform on spaces of constant curvature

A correspondence among the totally geodesic Radon transforms-as well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.

     

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.     

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q _c ⁴.     

Orthogonal curvilinear coordinate systems corresponding to singular spectral curves

We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that when the curve is reducible and all its irreducible components are rational curves, the construction procedure reduces to solving systems of linear equations and to simple computations with elementary functions. We also demonstrate how well-known coordinate systems, such as polar coordinates, cylindrical coordinates, and spherical coordinates in Euclidean spaces, fit in this scheme. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 180–196.