Symmetry and shape

According to Felix Klein, geometry is the study of those properties of a space that are invariant under the action of a group of transformations. In Riemannian geometry the group of transformations of the space is called the isometry group, although the space can sometimes have additional properties arising from mathematical or physical requirements that also need to be preserved by the transformation group. This is the case of a submanifold, whose geometry is influenced by the structure of the ambient space.

Intuitively, symmetry is the correspondence of shape at each point of a space. An important problem in geometry and in several physical sciences is to determine the symmetries of a space in terms of its shape. Submanifolds that are invariant under the isometries of the ambient manifold are called extrinsically homogeneous, or just homogeneous when there is no possibility of confusion. The main idea of my research project is to classify homogeneous submanifolds of a given Riemannian manifold, study the geometric properties of homogeneous submanifolds, and determine whether their shape is characteristic of them.