Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Homogeneous hypersurfaces in hyperbolic spaces

Homogeneous hypersurfaces in hyperbolic spaces Math. Z. 229, 589–600 (1998) c Springer-Verlag 1998 Jur ¨ gen Berndt University of Hull, Department of Mathematics, Hull HU6 7RX, United Kingdom (e-mail: j.berndt@maths.hull.ac.uk) Received April 3, 1997; in final form November 19, 1997 1 Introduction Suppose a submanifold of some space has one or several specified geometric features. A natural problem, often studied in submanifold geometry and known as rigidity problem, is to decide whether this submanifold admits non-trivial deformations preserving these features. A well-known example is the deformation of the helicoid into the catenoid preserving minimality. In this paper we construct deformations of horospheres in the hyperbolic spaces over C, H and O, as well as in some other homogeneous spaces of non-positive curvature. These deformations preserve homogeneity - recall that a submanifold M of a Riemannian manifold M is said to be homoge- neous if there exists a closed subgroup of the isometry group of M whose action on M has M as an orbit. These deformations are of interest for two reasons. Firstly, they provide new examples of homogeneous hypersurfaces in the hyperbolic spaces over C, H and O. Their extrinsic geometry is quite interesting. In particular we will show that the way they http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Homogeneous hypersurfaces in hyperbolic spaces

Berndt, Jürgen
Mathematische Zeitschrift , Volume 229 (4) – Dec 1, 1998
Download PDF
12 pages

Loading next page...
 
/lp/springer-journals/homogeneous-hypersurfaces-in-hyperbolic-spaces-FPWCpq2jFy

References (7)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004673
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 229, 589–600 (1998) c Springer-Verlag 1998 Jur ¨ gen Berndt University of Hull, Department of Mathematics, Hull HU6 7RX, United Kingdom (e-mail: j.berndt@maths.hull.ac.uk) Received April 3, 1997; in final form November 19, 1997 1 Introduction Suppose a submanifold of some space has one or several specified geometric features. A natural problem, often studied in submanifold geometry and known as rigidity problem, is to decide whether this submanifold admits non-trivial deformations preserving these features. A well-known example is the deformation of the helicoid into the catenoid preserving minimality. In this paper we construct deformations of horospheres in the hyperbolic spaces over C, H and O, as well as in some other homogeneous spaces of non-positive curvature. These deformations preserve homogeneity - recall that a submanifold M of a Riemannian manifold M is said to be homoge- neous if there exists a closed subgroup of the isometry group of M whose action on M has M as an orbit. These deformations are of interest for two reasons. Firstly, they provide new examples of homogeneous hypersurfaces in the hyperbolic spaces over C, H and O. Their extrinsic geometry is quite interesting. In particular we will show that the way they

Journal

Mathematische Zeitschrift – Springer Journals

Published: Dec 1, 1998

There are no references for this article.