Normal holonomy of orbits and Veronese submanifolds

TL;DR

This paper proves a conjecture relating the normal holonomy of homogeneous submanifolds in spheres to their geometric structure, specifically classifying certain cases as Veronese submanifolds or isoparametric, using geometric and topological methods.

Contribution

It confirms the conjecture for 3-dimensional submanifolds and maximal codimension cases, characterizes Veronese submanifolds via holonomy, and improves bounds on irreducible factors of the holonomy group.

Findings

Proved the conjecture for n=3, classifying the submanifolds as isoparametric or Veronese.

Established the conjecture for n≥3 with maximal codimension, characterizing Veronese submanifolds.

Derived a sharp bound on the number of irreducible factors of the local normal holonomy group.

Abstract

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then M^n must be an orbit of an irreducible s-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If n=2, then the normal holonomy is always transitive, unless M is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when n=3. In this case M^3 must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that uses information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for n>= 3 when the normal holonomy acts irreducibly and the codimension is the maximal possible n(n+1)/2. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case n=3, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold M^n, is less or equal than [n/2] (which is the rank of the orthogonal group SO(n)). This bound is sharp and improves the known bound n(n-1)/2.