The structure of homogeneous Riemannian manifolds with nullity

TL;DR

This paper investigates the structure of homogeneous Riemannian manifolds with nullity, establishing new algebraic conditions, constructing examples with non-solvable groups, and describing isometry groups in specific cases.

Contribution

It introduces new conditions on Lie algebras related to nullity, constructs examples with non-solvable groups, and characterizes isometry groups for certain nullity cases.

Findings

Existence of transvections in the direction of nullity elements.

Transvections generate an abelian ideal of the isometry algebra.

Examples of homogeneous spaces with non-solvable groups and non-trivial nullity.

Abstract

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 $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ 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 $\tilde{\mathfrak g}$. These results constitute a substantial improvement on the structure theory developed in \cite{DOV}. 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 homogeneouslocally irreducible Riemannian manifold with nullity.