Reduction of Homogeneous Riemannian Structures

TL;DR

This paper investigates the reduction of homogeneous Riemannian structures under group actions, analyzing invariant classes, fiber geometry, and applications to almost contact and Hermitian manifolds, including Sasakian structures.

Contribution

It provides new results on the behavior of homogeneous Riemannian tensors under reduction, especially when the acting group is normal or acts freely, and explores applications to specific geometric structures.

Findings

Normal subgroup of symmetries preserves the reducing tensor.

Reduction preserves certain classes of homogeneous tensors depending on fiber geometry.

Reduced structures include homogeneous Kähler tensors in Sasakian cases.

Abstract

The goal of this article is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group $H$ of isometries. In a first result, $H$ is a normal subgroup of the group of symmetries associated to the reducing tensor $\bar{S}$. The situation when $H$ is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fiberings of almost contact manifolds over almost Hermitian manifolds. If the structure is moreover Sasakian, the obtained reduced tensor is homogeneous K\"ahler.