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Let $M$ be a smooth manifold of dimension $n>1$. Let $\ell:=\gamma(S^1)$ be a smooth embedded loop in $M$ for some $\gamma:S^1\to M$. Let $\eta_1, \eta_2\in \mathfrak{X}(\ell;TM)$ be two smooth $TM$-valued vector fields along $\ell$.

Question : is it possible to locally extend $\eta_1$ and $\eta_2$ to commuting vector fields in a neighborhood of $\ell$ ?

Edit : If $\eta_1,\eta_2$ are tangent to $\ell$, it seems to be a necessary condition that $\eta_1,\eta_2$ commutes as vector fields on $\ell$.

Noé AC
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    What if $\eta_1,\eta_2$ are tangent to $\ell$ and don't commute there? If you assume $\eta_1$ is transverse to $\ell$ then I think this works: extend $\eta_1$ however you like and then extend $\eta_2$ by flowing it along $\eta_1.$ – Anthony Carapetis Dec 18 '17 at 06:57

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