The skew-torsion holonomy theorem and naturally reductive spaces

TL;DR

This paper proves a new geometric holonomy theorem for skew 1-forms, establishing the uniqueness of the canonical connection in certain naturally reductive spaces and extending classification results without relying on algebraic classifications.

Contribution

It introduces a geometric proof of a holonomy theorem for skew 1-forms and applies it to classify the isometry groups of naturally reductive spaces, generalizing previous results.

Findings

Proved a Simons-type holonomy theorem using geometric methods.

Established the uniqueness of the canonical connection in certain naturally reductive spaces.

Extended classification of isometry groups for naturally reductive spaces.

Abstract

We prove a Simons-type holonomy theorem for totally skew 1-forms with values in a Lie algebra of linear isometries. The only transitive case, for this theorem, is the full orthogonal group. We only use geometric methods and we do not use any classification (not even that of transitive isometric actions on the sphere or the list of rank one symmetric spaces). This result was independently proved, by using an algebraic approach, by Paul-Andy Nagy. We apply this theorem to prove that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. From this it follows easily how to obtain the full isometry group of a naturally reductive space. This generalizes known classification results of Onishchick, for normal homogeneous spaces with simple group of isometries, and Shankar, for homogeneous spaces of positive curvature. This also answers a question posed by J. Wolf and Wang-Ziller. Namely, to explain why the presentation group of an isotropy irreducible space, strongly or not, cannot be enlarged (unless for spheres, or for compact simple Lie groups with a bi-invariant metric).