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Let $f:M\to N$ be a surjective submersion smooth map between two smooth manifolds $M$ and $N$. Consider the differential map $df:TM\to TN$ between the tangent bundles. Given an involutive subbundle $A$ of $TM$. Is $df(A)$ an involutive subbundle of $TN$?

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There is no reason why $df(A)$ should even be a subbundle. For instance, let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}:(x,y)\mapsto x$ be the projection and take $A=\text{Span}(\partial_{y}+x\partial_{x})\subset T\mathbb{R}^{2}$. Then $df(A)$ is zero at $0\in\mathbb{R}$, but one-dimensional at other points $p\in\mathbb{R}$.

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