Flat degenerate metrics and Riemannian foliations

TL;DR

This paper presents a counter-example to a conjecture about flat, non-negative definite metrics on closed manifolds and explores the relationship with transversely flat Riemannian foliations.

Contribution

It provides a counter-example to a conjecture and discusses the connection between flat degenerate metrics and Riemannian foliations.

Findings

Counter-example to the conjecture on flat metrics

Analysis of the link with Riemannian foliations

Insights into the structure of manifolds with degenerate metrics

Abstract

Bandyopadhyay, Dacorogna, Matveev and Troyanov conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank $m$ should be finitely covered by a fiber bundle over the $m$-torus. We give a counter-example to this statement and we discuss the link between this problem and the study of transversely flat Riemannian foliations.