Metric Foliations of Homogeneous Three-Spheres

TL;DR

This paper investigates the relationship between homogenous and metric foliations on three-spheres, establishing a characterization of naturally reductive three-spheres based on their foliations.

Contribution

It proves that a homogenous three-sphere is naturally reductive if and only if all its metric foliations are homogenous, linking geometric properties to foliation structures.

Findings

Homogenous three-spheres are naturally reductive iff all metric foliations are homogenous.

Clarifies the relationship between homogeneity and metric properties in foliations.

Provides a characterization of naturally reductive three-spheres based on foliation properties.

Abstract

A smooth foliation of a Riemannian manifold is metric when its leaves are locally equidistant and is homogenous when its leaves are locally orbits of a Lie group acting by isometries. Homogenous foliations are metric foliations, but metric foliations need not be homogenous foliations. We prove that a homogenous three-sphere is naturally reductive if and only if all of its metric foliations are homogenous.