Homogeneity of infinite dimensional isoparametric submanifolds

A subset S of a Riemannian manifoldN is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N . In [Th], Thorbergsson proved the remarkable result that every complete, connected, full, irreducible isoparametric submanifold of a finite dimensional Euclidean space of rank at least 3 is extrinsically homogeneous. This result, combined with results of [PT1] and [Da], finally classified irreducible isoparametric submanifolds of a finite dimensional Euclidean space of rank at least 3. While Thorbergsson’s proof used Tits buildings, a simpler proof without using Tits buildings was given by Olmos (cf. [O2]). The main purpose of this paper is to extend Thorbergsson’s result to the infinite dimensional case. The study of infinite dimensional isoparametric submanifolds of a Hilbert space (always assumed to be separable) was initiated by Terng [T2]. Besides its intrinsic interest, the theory of infinite dimensional isoparametric submanifolds is a very useful tool in studying the submanifold geometry of compact symmetric spaces (cf. [TT] as well as [HL] and [Ew]). Although much progress has been made (especially, many basic properties of finite dimensional isoparametric submanifolds having been successfully extended to the infinite dimensional case (cf. [T2], [T5], and [HL])), the classification of infinite dimensional isoparametric submanifolds is far from being solved. To this end, the understanding of the homogeneity of infinite dimensional isoparametric submanifolds will certainly play a very important role, as it does in the finite dimensional case. Examples of non-homogeneous infinite dimensional isoparametric hypersurfaces have been found by Terng and Thorbergsson (cf. [TT]). In this paper, we will prove the following theorem which solves a long standing open problem (cf. [T3], [T4], [TT]).