Totally Geodesic Riemannian Foliations on Compact Lie Groups

TL;DR

This paper proves that certain Riemannian submersions with totally geodesic fibers from compact Lie groups are indeed coset foliations, answering a long-standing question in differential geometry.

Contribution

It establishes that Riemannian submersions with totally geodesic fibers from compact Lie groups are coset foliations, even under weaker conditions than previously assumed.

Findings

Affirmative answer to Ranjan's question

Extension to submersions defined on open subsets

Supports the structure of Riemannian foliations on Lie groups

Abstract

In 86, Ranjan questioned whether a submersion $\pi: G\to B$ from a compact simple Lie group with bi-invariant metric is a coset foliation or not, provided the submersion is Riemannian with totally geodesic fibers. Here we answer this question affirmatively, even when the submersion is defined only in an open subset of $G$ (assuming suitable hypothesis).