Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

Author(s)

Gorodski, Claudio ^[1] ; Heintze, Ernst ^[2]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, BR-05508 Sao Paulo - Brazil

^[2] Univ Augsburg, Math Inst, Augsburg - Germany

• Document type: Journal article
• Source: Journal of Fixed Point Theory and Applications; v. 11, n. 1, p. 93-136, MAR 2012.
• Web of Science Citations: 4

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 I{''} (x) (a certain bilinear map defined on a subspace of T (x) M x T (x) M). We prove that I{''} (x) , together with the second fundamental form alpha (x) , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by alpha (x) and (Delta alpha) (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 I{''} (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. (AU)

• FAPESP's process: 07/03192-7 - Submanifold geometry and Morse theory in finite and infinite dimensions
• Grantee: Claudio Gorodski
• Support Opportunities: Research Projects - Thematic Grants