Algebraic sets associated to isoparametric submanifolds

SÁNCHEZ CRISTIÁN URBANO

Contemporary Mathematics

American Mathematical Society

Lugar: Boston EEUU; Año: 2008; p. 21 - 40

Abstract. In this paper we present some results about the algebraic setsof normal sections associated to compact spherical submanifolds of Euclideanspaces. In the …rst part we indicate some general facts about these algebraicsets and then concentrate on the case of isoparametric submanifolds. We intro-duce the polynomials de…ning 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’s 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.