Artículos de revistas
Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space
Fecha
2013-08-02Registro en:
JOURNAL OF FIXED POINT THEORY AND APPLICATIONS, BASEL, v. 11, n. 1, supl. 1, Part 2, pp. 93-136, MAR, 2012
1661-7738
10.1007/s11784-012-0079-y
Autor
Gorodski, Claudio
Heintze, Ernst
Institución
Resumen
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.