Research interests of
Claudio Gorodski

Isoparametric submanifolds

An important topic in my current research is, jointly with Ernst Heintze, a contribution to the conjecture that isoparametric submanifolds of codimension at least two in Hilbert space are essentially principal orbits of isotropy representations of symmetric spaces of affine Kac-Moody type.

A submanifold of an Euclidean space is called isoparametric if the eigenvalues of a locally defined parallel normal vector field are constant and its normal bundle is flat. In codimension one, the second condition is void and thus isoparametric hypersurfaces are just those having constant principal curvatures, namely those with the simplest local invariants (in particular, homogeneous hypersurfaces are always isoparametric). The subject had its origins in geometrical optics around the turn of the 20th century, but the main early contributions were made by the great É. Cartan during 1930-40. The subject was then forgotten for about 30 years when it was revived by the successive works of Nomizu, Terng, Harle, Carter and West and many others.

Homogeneous examples are supplied by the principal orbits of isotropy representations of Riemannian symmetric spaces. On the other hand, a theorem by Thorbergsson asserts that homogeneity is necessary if the codimension is different from two (inhomogeneous examples in codimension two have been constructed by Ozeki and Takeuchi, and Ferus, Karcher and Münzner). This result, combined with a remark by Palais and Terng and a classification theorem of Dadok, implies that in codimension different from two there are no further examples besides those produced by isotropy representations of symmetric spaces. Hence we see a close relation between isoparametric submanifolds and symmetric spaces. Further, in a similar way as symmetric spaces are test/model spaces in Riemannian geometry, also isoparametric submanifolds are test/model spaces in submanifold geometry.

Isoparametric submanifolds can also be considered in Hilbert space. By assumption, they are proper Fredholm, which essentially means that they have finite codimension and compact (self-adjoint) shape operators. Homogeneous examples are now provided by the isotropy representations of symmetric spaces of affine Kac-Moody type. Further, Thorbergsson's result has been generalized by Heintze and Liu to yield that homogeneity is necessary if the codimension is different from one and, similarly to the above, it is conjectured that in codimension different from one all examples come from symmetric spaces of affine Kac-Moody type. However, since there are neither suitable classifications of Hilbert-Lie groups nor of their representations, one cannot repeat the strategy that works finite dimensions. Our work in progress relies on the geometry of isoparametric submanifolds to attain results about their classification.

Geometry of orbit spaces

Together with Alexander Lytchak, we have investigated representations of compact Lie groups that have isometric orbit spaces and found invariants and a geometrical description of classes containing representations with very different algebraic porperties.

For an orthogonal representation ρ:G→O(V) of a compact (possibly disconnected) Lie group G, the quotient metric space V/G is the most important invariant of the action, at least from the metric point of view. Moreover, from the geometric-topologic point of view, some properties of the action can be recovered from the transverse geometry of the orbital foliation. We call two such representations quotient-equivalent if they have the same orbit space; and one of them a reduction of the other if the former has strictly smaller dimension. It is an interesting problem to understand when two actions can have the same orbit space, or, to find elements of minimal dimension in a given quotient-equivalence class (minimal reductions). On one hand, the existence of a reduction implies some, probably severe, restrictions on the original representation. On the other hand, while most representations do not admit reductions, many geometrically interesting ones do. For instance, it is the case for many representations of low cohomogeneity and for the important class of polar representations. These are precisely the representations admitting reductions to finite groups, the simplest example being that of the Weyl group associated to the adjoint representation of a compact connected Lie group. Another class is built by the representations with non-trivial principal isotropy groups, and, generalizing both previous examples, by representations with non-trivial copolarity.