On the affine group of a normal homogeneous manifold

TL;DR

This paper characterizes the affine transformation group of normal homogeneous spaces using geometric methods, completing the understanding of their isometry groups and analyzing fixed points and holonomy properties.

Contribution

It provides a geometric derivation of the affine transformation group for normal homogeneous spaces and explores fixed points and holonomy implications.

Findings

The affine transformation group is explicitly determined for normal homogeneous spaces.

Fixed points of the isotropy group form a torus in these spaces.

Holonomy groups of homogeneous fibrations are Lie groups, contained in affine transformations.

Abstract

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers, in particular this holonomy group is a Lie group (this is a result of Guijarro and Walschap).