The structure of the homogeneous Riemannian manifolds with nullity

We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space M=G/H imposes on the Lie algebra g of G and on the Lie algebra g of the full isometry group of M. Namely, we prove that there exists a transvection of M in the direction of any element of the nullity, possibly by enlarging the presentation group G. Moreover, we prove that these transvections generate an abelian ideal of g. These results constitute a substantial improvement on the structure theory developed in our previous paper. In addition we construct examples of homogeneous Riemannian spaces with non-trivial nullity, where G is a non-solvable group, answering a natural open question. Such examples admit (locally homogeneous) compact quotients. In the case of co-nullity 3 we give an explicit description of the isometry group of any homogeneous locally irreducible Riemannian manifold with nullity.