Geodesic orbit manifolds and Killing fields of constant length

TL;DR

This paper explores the relationship between Killing fields of constant length and the structure of isometry groups on geodesic orbit manifolds, providing new insights and proofs in Riemannian geometry.

Contribution

It establishes that abelian ideals in the isometry algebra have Killing fields of constant length and offers a new proof that nonpositive Ricci curvature geodesic orbit manifolds are symmetric spaces.

Findings

Abelian ideals in the isometry algebra have Killing fields of constant length

New proof that nonpositive Ricci curvature geodesic orbit manifolds are symmetric spaces

Clarifies the structure of isometry groups on geodesic orbit manifolds

Abstract

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We prove the following result: If $\mathfrak{a}$ is an abelian ideal of $\mathfrak{g}$, then every Killing field $X\in \mathfrak{a}$ has constant length. On the ground of this assertion we give a new proof of one result of C. Gordon: Every Riemannian geodesic orbit manifold of nonpositive Ricci curvature is a symmetric space.