On Lie algebras associated with a spray

TL;DR

This paper investigates the structure of the Lie algebra of infinitesimal isometries on Riemannian manifolds, revealing it contains at most two specific commutative ideals linked to geometric properties.

Contribution

It introduces a novel analysis of the Lie algebra structure associated with a spray, identifying the origins of its commutative ideals in geometric invariants.

Findings

The Lie algebra has at most two commutative ideals.

One ideal arises from the horizontal nullity space of the Nijenhuis tensor.

The other ideal consists of constant vector fields independent of the metric.

Abstract

The Lie algebra of infinitesimal isometries of a Riemannian manifold contains at most two commutative ideals. One coming from the horizontal nullity space of the Nijenhuis tensor of the canonical connection, the other coming from the constant vectors fields independent of the Riemannian metric.