| Speaker: Carlos Enrique Olmos Affiliation: National University of Cordoba Time: 9am Friday, 22 November, 2019 Location: 303e.257 |
| Symmetric spaces play a central role in Riemannian geom- etry. On the one hand, this family of spaces is so large that some of the results that are true for symmetric spaces can be generalized for arbitrary Riemannian manifolds. On the other hand, this family is so particular, that the geometry of symmetric spaces is very rich and non-generic. Some of the more beautiful and important results in Riemannian geometry are the- orems that under mild non-generic assumptions imply symmetry. This is the case of the Berger holonomy theorem and the rank rigidity theorem of Ballmann/Burns-Spatzier. A similar role, to that of symmetric spaces in Riemannian geometry, is played, in Euclidean submanifold geometry, by the orbits of s-representations (i.e., isotropy representations of semisimple symmetric spaces. or equiva- lently their holonomy representations). Some remarkable results in subman- ifold geometry are theorems that imply that a given non-generic submanifold is an orbit of an s-representation. This is the case of the well-known theo- rem of Thorbergsson about the homogeneity of isoparametric submanifolds of high codimension. This is also the case of the so-called Rank Rigidity theorem for submanifolds (Di.Scala-O.). The above mentioned results on Riemannian geometry look similar to those on submanifold geometry be- ing the main ingredients Riemannian or normal holonomy. This suggests that there would be a subtle relation between this two contexts. This is ex- plained, in part, by the fact that using submamifold geometry, with normal holonomy ingredients one can prove, geometrically, the Berger holonomy theorem , or the Simons (algebraic) holonomy theorem . We will sketch the proof of such results. 1 |