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Flat degenerate metrics and Riemannian foliations

Brice Flamencourt

TL;DR

This work addresses the BDMT conjecture asserting that a closed manifold with a flat non-negative definite metric of constant rank $m$ is finitely covered by a fiber bundle over the $m$-torus $T^m$. The authors construct a concrete counterexample: a compact $5$-manifold obtained as the suspension of the $4$-torus $T^4$ over $S^1$, carrying a flat metric of rank $3$. The metric arises from an explicit cocompact action on $\tilde{M}=\mathbb{R}^4\times\mathbb{R}$ by $\Gamma=\mathbb{Z}^4\rtimes\langle (x,t)\mapsto(Ax,t+1)\rangle$, with $A$ the companion matrix of $P(X)=X^4-2X^3-2X+1$, producing $M=\tilde{M}/\Gamma$ and a flat metric of rank $3$. A group-theoretic obstruction shows that no finite cover of $M$ can fiber over $T^3$, since any finite-index subgroup yields a rank contradiction in the associated commutator subgroup; this links flat metrics to transversely flat Riemannian foliations and clarifies the global structure of such manifolds.

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.

Flat degenerate metrics and Riemannian foliations

TL;DR

This work addresses the BDMT conjecture asserting that a closed manifold with a flat non-negative definite metric of constant rank is finitely covered by a fiber bundle over the -torus . The authors construct a concrete counterexample: a compact -manifold obtained as the suspension of the -torus over , carrying a flat metric of rank . The metric arises from an explicit cocompact action on by , with the companion matrix of , producing and a flat metric of rank . A group-theoretic obstruction shows that no finite cover of can fiber over , since any finite-index subgroup yields a rank contradiction in the associated commutator subgroup; this links flat metrics to transversely flat Riemannian foliations and clarifies the global structure of such manifolds.

Abstract

Bandyopadhyay, Dacorogna, Matveev and Troyanov conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank should be finitely covered by a fiber bundle over the -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.

Paper Structure

This paper contains 2 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. Construction of the counter-example

Key Result

Lemma 2.1

A subgroup of $\Gamma$ is a semi-direct product of the form $L \rtimes K$ where $L$ is a subgroup of $\mathbb{Z}^4$ and $K$ is either the trivial group or the group generated by an element of $\Gamma$ which does not lie in $\mathbb{Z}^4$.

Theorems & Definitions (3)

  • Conjecture 1.1
  • Lemma 2.1
  • proof