Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

Abstract

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by the main result in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149–181], and with such a submanifold M and a point x in M we associate a canonical homogeneous structure Γ x (a certain bilinear map defined on a subspace of T x M × T x M). We prove that Γ x , together with the second fundamental form α x , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by α x and (Δα) x , thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149–181] to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of Γ (this result also seems to close a gap in [U. Christ, J. Differential Geom., 62 (2002), 1–15]). Here an important tool is the introduction of affine root systems of isoparametric submanifolds.