Homogeneous Riemannian manifolds with non-trivial nullity

Abstract

We develop a general theory for irreducible homogeneous spaces $M=G/H$, in relation to the nullity $\nu$ of their curvature tensor. 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. Moreover, we construct irreducible examples of conullity $k=3$, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that $H$ is trivial and $G$ is solvable if $k=3$. Another of our main results is that the leaves of the nullity are closed (we used a rather delicate argument). This implies that $M$ is a Euclidean affine bundle over the quotient by the leaves of $\nu$. Moreover, we prove that ${\nu }^{\perp }$ defines a metric connection on this bundle with transitive holonomy or, equivalently, ${\nu }^{\perp }$ is completely non-integrable (this is not in general true for an arbitrary autoparallel and flat invariant distribution). We also found some general 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.