Homogeneity of infinite dimensional isoparametric submanifolds

Ernst Heintze and Xiaobo Liu

Review from Zentralblatt MATH:

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 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.

Reviewed by G.Thorbergsson

Keywords: isoparametric submanifolds; proper Fredholm submanifolds in Hilbert spaces; homogeneity theorem; equifocal submanifolds

Classification (MSC2000): 53C40 53C30

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