= Boundary obstruction for commuting volume-preserving vector fields
Suppose a nonempty compact $(n+1)$-manifold with a <volume form> has $n$ pointwise independent <volume-preserving vector fields> that commute and are tangent to its boundary. If restriction $H^1_{\rm dR}(M)\to H^1_{\rm dR}(\partial M)$ is injective, the boundary must have at least two connected components.
Indeed $\eta=\iota_{X_1}\cdots\iota_{X_n}\omega$ is nowhere zero. <Cartan's magic formula> and commutation show $d\eta=0$, while boundary tangency gives $j^*\eta=0$. Injectivity in <de Rham cohomology> makes $\eta=df$. On a connected boundary $f$ is constant; a maximum or minimum differing from that boundary value would be an interior <critical point>, contradicting $df\ne0$. Empty boundary is also impossible.
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