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Symmetry and shape

Final Activity Report Summary - SYMMETRY AND SHAPE (Symmetry and shape)

Our project is devoted to the study of subsets of spaces with a high degree of symmetry. In the context which we are working in (Riemannian geometry) it is understood that an object has a high degree of symmetry if it is invariant by the action of certain group of isometries (which are the transformations of Riemannian manifolds that leave this geometric structure invariant). Such an object is called a homogeneous submanifold. The main purpose of this project has been to classify these spaces, calculate their shape and determine to what extent this shape is characteristic of them.

The main result of our project, however, is concerned with the first question above. In doing so we moved to the study of more general spaces, the so-called symmetric spaces of noncompact type (many of these questions have already been addressed in the compact case). These spaces are very important in Riemannian geometry and their study goes back to the work by É. Cartan. Among all isometric actions that can be studied on a manifold, the so called hyperpolar actions have been of particular relevance over the last few years.

This question is in general very difficult to handle, so we devoted our attention to a particular situation: the case when such a hyperpolar action induces a foliation on the manifold, roughly speaking, when all the homogeneous submanifolds induced by the action (the orbits of the action) have the same dimension. The main result obtained during this fellowship is the classification of all homogeneous submanifolds arising as the orbits of hyperpolar actions inducing a foliation on a symmetric space of non-compact type (J. Berndt, J. C. Díaz-Ramos and H. Tamaru, preprint arXiv:math.DG/08073517).