Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
38(2), 437-458 (1997)

A New Intrinsic Curvature Invariant for Centroaffine Hypersurfaces

C. Scharlach, U. Simon, L. Verstraelen, L. Vrancken

Fachbereich Mathematik, MA 8-3, Technische Universität Berlin
Strasse des 17. Juni 136, D - 10623 Berlin, Germany

Departement Wiskunde, Katholieke Universiteit Leuven
Celestijnenlaan 200 B, B - 3001 Leuven, Belgium

Abstract: We consider centroaffine hypersurfaces $M^n$ and introduce a new intrinsic curvature invariant of the centroaffine metric. For $n>2$ we derive an inequality between this invariant and the square norm of the Tchebychev field. We study the class of hypersurfaces which realize the equality and obtain a wide class of examples and a partial classification. Furthermore we discuss our results under polarization of the hypersurface and finally relate them to recent asymptotic spectral results on second order Laplace type operators in affine differential geometry.

Keywords: centroaffine hypersurfaces, curvature invariant, proper affine hyperspheres, polar hypersurfaces, asymptotic spectral geometry of Laplace type operators

Classification (MSC91): 53A15, 58G25

Full text of the article:

[Previous Article] [Table of Contents]