The study reveals a counter-example to a conjecture on flat metrics in closed manifolds, suggesting new implications for riemannian foliations.
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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Brice Flamencourt (2025) studied this question.