September 20, 2025

Flat degenerate metrics and Riemannian foliations

Key Points

A counter-example disproves the conjecture that closed manifolds with flat metrics must be finitely covered.
The conjecture originally suggested a link to the structure of fiber bundles over the torus.
The analysis connects the counter-example to the study of transversely flat riemannian foliations.
Implications of this finding broaden the understanding of manifold structures beyond previously held assumptions.

Abstract

Bandyopadhyay, Dacorogna, Matveev and Troyanov Math. Z. 305 (2023), p. 25 conjectured that a closed manifold admitting a flat, non-negative definite metric of constant rank m m should be finitely covered by a fiber bundle over the m 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.

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Cite This Study

Brice Flamencourt (Fri,) studied this question.

synapsesocial.com/papers/68d469c131b076d99fa664e9 https://doi.org/https://doi.org/10.1090/proc/17349

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