Gudlaugur Thorbergsson

Chapter 10 – A Survey on Isoparametric Hypersurfaces and Their Generalizations

This chapter discusses isoparametric hypersurfaces and their generalizations. The chapter presents a survey of isoparametric hypersurfaces, discusses Dupin hypersurfaces, and describes isoparametric submanifolds in ambient spaces that are finite dimensional Euclidean spaces or infinite dimensional Hilbert spaces. The taut submanifolds in Riemannian manifolds are described. The classification of isoparametric hypersurfaces in spheres with three principal curvatures is related to the various characterizations of the standard embeddings of the projective planes. The chapter discusses isoparametric hypersurfaces in spheres with four different principal curvatures, all of which are assumed to have the same multiplicity. The Clifford examples of Ferus, Karcher, and Munzner together with the homogeneous hypersurfaces are all known examples of isoparametric hypersurfaces in spheres. A difference between isoparametric and Dupin hypersurfaces is that although the parallel hypersurfaces of the Dupin ones are also Dupin, they do not foliate the ambient space as the isoparametric ones do. The hypersurface of a sphere is isoparametric if and only if it is equifocal.

Polar and coisotropic actions on Kähler manifolds

The main result of the paper is that a polar action on a compact irreducible homogeneous Kahler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.

On the Geometry of the Orbits of Hermann Actions

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint U-orbits from flows in the soliton hierarchy associated to a compact Lie group U. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.

Cycles of Bott–Samelson Type for Taut Representations

Bott and Samelson constructed cycles which are concrete representativesof a basis for the Z2-homology of the orbits ofvariationally complete representations of compact Lie groups (theseinclude isotropy representations of symmetric spaces; in this case theorbits are the so-called generalized real flag manifolds). Then theyused these cycles to show that the orbits of those representations aretaut submanifolds. We adapt the construction of Bott and Samelson to theorbits of three representations which are not variationally complete. Inthis case, it also follows that the orbits are taut.

Sextactic points on a simple closed curve

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

A global theory of flexes of periodic functions

For a real valued periodic smooth function u on R,

Polar actions on rank-one symmetric spaces

n\ge 0

Curvature explosion in quotients and applications

, one defines the osculating polynomial

Singular Riemannian Foliations and Isoparametric Submanifolds

\phi_s