The index of cubic focal manifolds

TL;DR

This paper computes the index and nullity of focal manifolds of isoparametric hypersurfaces in spheres, revealing their stability properties and geometric characteristics.

Contribution

It provides explicit calculations of index and nullity for focal manifolds with three principal curvatures, linking nullity to Killing vector fields.

Findings

Index equals the ambient space dimension.

Nullity is determined by normal Killing vector fields.

Veronese embeddings are highly stable.

Abstract

We calculate the index and nullity of the three orientable focal manifolds of isoparametric hypersurfaces in spheres with three distinct principal curvatures. It turns out that the index is equal to the dimension of the ambient Euclidean space and the nullity is completely determined by the normal part of Killing vector fields of the ambient sphere. In that sense, the Veronese embeddings of the projective planes are as stable as possible for non totally geodesic submanifolds of the sphere.