Submanifolds and Berger-type theorems

OLMOS, CARLOS

Rio de Janeiro

Simposio; International Symposium on Differential Geometry "In honor of Marcos Dajczer on his 60th birthday" Abstracts; 2009

IMPA

The normal holonomy of a submanifold of a space form, turns out to be even simpler than Riemannian holonomy. This has interesting consequences not only in submanifold geometry, but also in Riemannian geometry. In fact, the Berger holonomy theorem depends strongly on the fact that the normal holonomy has a very special form. In this talk we would like to draw the attention on some results, similar to that of Berger, in the context of submanifold or Riemannian geometry (that also depend on the special form of the normal holonomy and that can be proven by geometric methods). Finally, we will discuss some applications of the so-called "Skew-torsion holonomy theorem" to naturally reductive spaces, which in particular explains the inextendibility of isotropy irreducible spaces (in the sense of Wolf and Wang-Ziller).