The nullity of homogeneous Riemannian manifolds

CARLOS OLMOS

Maynooth

Conferencia; Irish Geometry Conference 2017; 2017

Maynooth University, Department of Mathematics and Statistics

The nullity distribution of the curvature tensor of a Riemannian space was defined by Chern andKuiper in 1952. This distribution turns out to be autoparallel around the points where the dimension is locally constant. Nevertheless, nothing was known about the nullity distribution in homogeneous spaces. In this talk we will mainly refer to some recent results obtained jointly with Antonio J. Di Scala and Francisco Vittone, that will motivate the presentation of some interesting points of view in homogeneous geometry.Let M=G/H be a locally irreducible homogeneous Riemannian manifold. We prove that if M is either compact, or G is semisimple or G is two step nilpontent, or dim(M)= 3, then the distribution of nullity is trivial.We will present also a general structure theory for homogenous manifolds with non-trivial nullitythat predictsthe existence of a transvection (i.e. a Killing field which is parallel at some point) with null Jacobi operator and not in the nullity. With the aid of this result we are able to find a one parameter family irreducible homogeneous spaces of dimension 4 with non-trivial distribution ofnullity (as far as we know these are the first known examples)By making use of the above mentioned structure theorem, we also show that any homogeneous space with a transitive semisimple subgroup of isometries, has trivial nullity distribution.