Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

TL;DR

This paper demonstrates that isoparametric submanifolds of rank at least two in Hilbert spaces are uniquely determined by their second fundamental form and homogeneous structure, leading to advances in their classification.

Contribution

It establishes that the homogeneous structure and second fundamental form fully determine such submanifolds, providing a new approach to their classification.

Findings

Homogeneous structure and second fundamental form encode all information about the submanifold.

One-parameter isometry groups induce smooth Killing fields, ensuring homogeneous structure continuity.

Introduction of affine root systems as a key tool for analysis.

Abstract

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by a result of Heintze and Liu, and associate to such a submanifold M and a point x in M a canonical homogeneous structure (a certain bilinear map on the tangent space). We prove that the homogeneous structure together with the second fundamental form encodes all the information about M, and deduce from this the rigidity result that M is completely determined by the second fundamental form and its covariant derivative, thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed by Heintze and Liu to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of the homogeneous structure. Here an important tool is the introduction of affine root systems of isoparametric submanifolds.