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Let $S$ be a regular submanifold of a manifold $M$, meaning a subset of $M$ such that for all $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ of $p$ in the maximal atlas of $N$ such that $U\cap S$ is defined by the vanishing of $n-k$ coordinate functions. Let $i\colon S\to M$ be the inclusion map. Then $i$ is a homeomorphism of $S$ onto its image and the differential $i_*\colon T_pS \to T_pM$ is an injective homomorphism of vector spaces.

If I have a vector field $X$ on $M$, I can "restrict" it to $S$ by defining, for each point $p\in S$, $$ X_{p\in S} = i_*^{-1}(X_{p\in M}), $$ where $X_{p\in S}$ is the tangent vector in $T_pS$ corresponding to the tangent vector $X_{p\in M}$ in $T_pM$.

If $X$ is smooth on $M$, that means it is a smooth section of the tangent bundle $TM$, i.e., it is a smooth map from $M$ to $TM$. If this is the case, does it follow that $X$ is smooth on $S$, i.e., that the restriction of $X$ to $S$ is also a smooth map from $S$ to $TS$?

I think this should be true and I should be able to show it taking a series of compositions of smooth functions. For example, is $i_*^{-1}$ smooth? I know that $i$ is continuous since $S$ has the subspace topology from $M$, but is $i$ a diffeomorphism? I'm a bit confused on this.

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  • $\begingroup$ a sub differential manifold of some differential manifold. if you define that a smooth vector field becomes locally smooth after applying the charts of (smooth!) diffeomorphism, then obviously yes. $\endgroup$ – reuns Apr 6 '16 at 19:19
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    $\begingroup$ A vector field on $X$ may not be tangent to $S$. Given a smooth vector field $V$ on $X$, $V\circ i$ will be a smooth section of the bundle $TX|_S$, but it may not be a section of $TS$. $\endgroup$ – Michael Albanese Apr 6 '16 at 19:20

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