## 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.

### MSC:

 53C40 Global submanifolds 53C30 Differential geometry of homogeneous manifolds 58B25 Group structures and generalizations on infinite-dimensional manifolds

### Citations:

Zbl 0727.57028; Zbl 0791.53051; Zbl 0674.58004; Zbl 0845.53040
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