Algebraic sets associated to isoparametric submanifolds.

SÁNCHEZ CRISTIÁN URBANO

New Developments in Lie Theory and Geometry

American Mathematical Society

Lugar: Providence Rhode Island; Año: 2009; p. 37 - 56

Abstract. In this paper we present some results about the algebraic setsof normal sections associated to compact spherical submanifolds of Euclideanspaces. In the first part we indicate some general facts about these algebraicsets and then concentrate on the case of isoparametric submanifolds. We introduce the polynomials defining the sets and describe a way to compute them forisoparametric submanifolds and in particular for isoparametric hypersurfacesin spheres. In the second part we apply these facts to study the particularcase of  Cartan isoparametric hypersurfaces obtaining a characterization ofthem among compact irreducible isoparametric submanifolds of any rank inEuclidean spaces. We also show that the non-planar normal sections are nat-urally divided, for these hypersurfaces, by the product of the square of theircurvature and torsion at the origin of the section, into smooth hypersurfacesof the tangent unit spheres.