Homogeneity of infinite dimensional isoparametric submanifolds. (English) Zbl 0931.53027

The main result of the paper is the following remarkable Theorem. Let \(M\) be a complete, connected, irreducible isoparametric submanifold in a Hilbert space \(V\). Assume that the set of all the curvature normals of \(M\) at some point is not contained in any affine line. Then, \(M\) is extrinscially homogeneous in the Hilbert space \(V\). The finite dimensional case of this theorem was first proved by the reviewer [Ann. Math., II. Ser. 133, 429-446 (1991; Zbl 0727.57028)] and later by C. Olmos [J. Differ. Geom. 38, 225-234 (1993; Zbl 0791.53051)]. Infinite dimensional isoparametric submanifolds were introduced by C.-L. Terng [J. Differ. Geom. 29, 9-47 (1989; Zbl 0674.58004)]. Besides being of obvious intrinsic interest, infinite dimensional isoparametric submanifolds of Hilbert spaces are closely related to equifocal submanifolds of symmetric spaces that were introduced by C.-L. Terng and G. Thorbergsson [J. Differ. Geom. 42, 665-718 (1995; Zbl 0845.53040)]. It is very likely that the main result of the paper under review can be used to prove a homogeneity theorem for equifocal submanifolds.


53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
58B25 Group structures and generalizations on infinite-dimensional manifolds
Full Text: DOI arXiv EuDML Link