Spherical maximal operator on symmetric spaces of constant curvature

We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension . More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function ,

The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.

     

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.     

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.     

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 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...     

Width of convex bodies in spaces of constant curvature

We consider the measure of points, the measure of lines and the measure of planes intersecting a given convex body K in a space form. We obtain some integral formulas involving the width of K and the curvature of its boundary ∂K. Also we study the special case of constant width. Moreover we obtain a generalisation of the Heintze–Karcher inequality to space forms. Work partially supported by grant number ACI2003-44 Joint action Catalonia–Baden-Württemberg and by FEDER/MEC grant number MTM2006-04353. The third author was also supported by the program Ramón y Cajal, MEC.     

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.     

局部对称空间中具有常数量曲率的超曲面

利用自伴算子研究局部对称空间中具有常数量曲率的紧致超曲面,得到了这类超曲面中的某些刚性定理,推广了已有的结果.