Homogeneous Riemannian manifolds with non-trivial nullity

TL;DR

This paper develops a comprehensive theory for irreducible homogeneous spaces with non-trivial nullity in their curvature tensor, constructing examples and analyzing their geometric properties, including the structure of nullity distributions and holonomy.

Contribution

It introduces invariant distributions related to nullity, constructs new examples with minimal conullity, and proves structural properties like closed leaves and conditions on the acting group.

Findings

Existence of an order-two transvection with null Jacobi operator.

Construction of irreducible examples with conullity 3.

Nullity leaves are closed, leading to a Euclidean affine bundle structure.

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.