A note on the uniqueness of the canonical connection of a naturally reductive space

TL;DR

This paper proves the uniqueness of the canonical connection in certain naturally reductive spaces, extending previous results to non-compact cases and providing specific cases where uniqueness holds or fails.

Contribution

It extends the uniqueness result of the canonical connection to non-compact naturally reductive spaces and computes their isometry groups.

Findings

Canonical connection is unique for hyperbolic spaces of dimension not equal to three.

Canonical connection is unique for spheres with symmetric presentation.

Full isometry group of compact, locally irreducible naturally reductive spaces is computed.

Abstract

We extend the result in J. Reine Angew. Math. 664, 29-53, to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In particular, the canonical connection is unique for the hyperbolic space when the dimension is different from three. We also prove that the canonical connection on the sphere is unique for the symmetric presentation. Finally, we compute the full isometry group (connected component) of a compact and locally irreducible naturally reductive space.