| Speaker: Carlos Enrique Olmos Affiliation: Universidad Nacional de Cordoba Time: 9am Friday, 15 November, 2019 Location: 303e.257 |
| The main purpose of this lecture is to introduce to techniques in homogeneous geometry based in geometric constructions and Killing fields rather than the usual Lie algebra point of view. In particular, Killing fields are regarded as parallel sections, with respect to the so-called Kostant (twisted) connection in the canonical bundle. This point of view will be applied to address a classical problem in Riemannian geometry that goes back to Chern-Kuiper in the 50’s: the study of the nullity distribution of the curvature tensor. For homogeneous Riemannian manifolds nothing was known about this problem, not even a non-trivial example. In a recent work with A.J. Di Scala and F. Vittone we developed a general theory for homo- geneous spaces M= G/H in relation with the nullity. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e. where the nullity distribution is not parallel. More- over, there are irreducible examples of conullity k = 3, the smallest possible, in any dimension. We also proved that H is trivial and G is solvable if k = 3 (and it is an open problem whether G must be always solvable) The leaves of the nullity of M are closed (this involves a delicate argument). This im- plies that M is a Euclidean affine bundle over the quotient by the leaves of the nullity. Moreover, the perpendicular distribution of the nullity defines a metric connection on this bundle with transitive holonomy or, equivalently, a completely non-integrable distribution (this is not in general true for an arbitrary autoparallel and flat invariant distribution). There are some gen- eral obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent. The new results explained in this lecture are part of a joint article with Antonio J. Di Scala and Francisco Vittone. |