Submanifolds and holonomy

CARLOS OLMOS

Auckland

Seminario; Seminar of Geomety and Analysis, University of Auckland; 2019

Department of Mathematics, Univesity of Auckland, NZ.

Symmetric spaces play a central role in Riemannian geometry. On the one hand, this family of spacesis so large that some of the results that are true for symmetric spaces can be generalized for arbitraryRiemannian manifolds. On the other hand, this family is so particular, that the geometry of symmetricspaces is very rich and non-generic. Some of the more beautiful and important results in Riemanniangeometry are theorems that under mild non-generic assumptions imply symmetry. This is the case ofthe 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, bythe orbits of s-representations (i.e., isotropy representations of semisimple symmetric spaces. or equivalentlytheir holonomy representations). Some remarkable results in subman- ifold geometry aretheorems that imply that a given non-generic submanifold is an orbit of an s-representation. This is thecase of the well-known theo- rem of Thorbergsson about the homogeneity of isoparametric submanifoldsof 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 onsubmanifold geometry be- ing the main ingredients Riemannian or normal holonomy. This suggests thatthere would be a subtle relation between this two contexts. This is ex- plained, in part, by the fact thatusing submamifold geometry, with normal holonomy ingredients one can prove, geometrically, theBerger holonomy theorem , or the Simons (algebraic) holonomy theorem . We will sketch the proof ofsuch results.