55 research outputs found
Polar actions on compact rank one symmetric spaces are taut
We prove that the orbits of a polar action of a compact Lie group on a
compact rank one symmetric space are tautly embedded with respect to
Z_2-coefficients.Comment: 7 pages; February 21st, 2006: Statement of main result corrected;
other minor change
Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space
We study isoparametric submanifolds of rank at least two in a separable
Hilbert space, which are known to be homogeneous by a result of Heintze and
Liu, and associate to such a submanifold M and a point x in M a canonical
homogeneous structure (a certain bilinear map on the tangent space). We prove
that the homogeneous structure together with the second fundamental form
encodes all the information about M, and deduce from this the rigidity result
that M is completely determined by the second fundamental form and its
covariant derivative, thereby making such submanifolds accessible to
classification. As an essential step, we show that the one-parameter groups of
isometries constructed by Heintze and Liu to prove their homogeneity induce
smooth and hence everywhere defined Killing fields, implying the continuity of
the homogeneous structure. Here an important tool is the introduction of affine
root systems of isoparametric submanifolds
Polar symplectic representations
We study polar representations in the sense of Dadok and Kac which are
symplectic. We show that such representations are coisotropic and use this fact
to give a classification. We also study their moment maps and prove that they
separate closed orbits. Our work can also be seen as a specialization of some
of the results of Knop on multiplicity free symplectic representations to the
polar case.Comment: 19 page
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