Isoparametric foliations and polar actions on complex space forms

Resumo

The notion of symmetry underlies a large number of new ideas and major advances in Science, Engineering and Art. From the mathematical viewpoint, the intuitive idea of symmetry as the balanced correspondence of shape along space translates to the existence of a transformation group acting on such space. The rst natural eld for the study of symmetry is then geometry. Conversely, in his in uential Erlanger Programm, Felix Klein described geometry as the study of those properties of a space that are invariant under a transformation group. Hence, symmetry lies in the very core of geometry. In Riemannian geometry, the natural group to consider is the isometry group, that is, the group of those transformations of the space that preserve distances. The action of a subgroup of the isometry group of a given manifold is called an isometric action. Its cohomogeneity is the lowest codimension of its orbits. Each one of the orbits of such an isometric action is called an (extrinsically) homogeneous submanifold, and the collection of all the orbits is the orbit foliation of the action. The main objects of study in this thesis are certain kinds of submanifolds with a particularly high degree of symmetry. Our ultimate goal is to decide whether the intuitive notion of symmetry is re ected in the mathematical notion of symmetry, namely if the correspondence of shape at di erent parts of the submanifold implies that the submanifold is homogeneous.