If an $N$-dimensional compact manifold has $N$ commuting vector fields, does this mean the manifold is actually a torus?
2026-04-28 16:29:08.1777393748
$N$ commuting vector fields on an $N$-dimensional compact manifold
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Yes, this is true (if the manifold is assumed connected; otherwise you could have a disjoint union of tori). This is an exercise on page 116 of Differentiable Manifolds: A First Course by Lawrence Conlon. A proof can be found in the paper Integrably parallelizable manifolds by Vagn Lundsgaard Hansen (Lemma 2), which is in free access. The author doesn't define "rank" there, but it's understood as the maximal number of linearly independent commuting vector fields.