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Let $M$ be a smooth manifold and $X\colon M \to TM$ a vector field on $M$. I'm having some trouble proving that these assertions are equivalent:

(i) $X$ is smooth.

(ii) for every chart $(U,\varphi) \in \Sigma(M)$, if $X = \sum a_i \partial_i$, then the $a_i$ are smooth.

(iii) for every smooth $f\colon V \subseteq M\to \Bbb R$, the map $X(f)\colon V \to \Bbb R$ is smooth.

I managed to prove that (i) implies (ii) and that (ii) implies (iii). I'm stuck on (iii) implying (i).

My idea is to write $X$ in local coordinates and see somehow that said representation is smooth. Let $(U,\varphi)$ be a chart in $M$ and $(\tilde{U},\tilde{\varphi})$ be the induced chart in $TM$. Writing $\varphi = (x_1,\cdots,x_n)$, we have that $$\begin{align} \tilde{\varphi}\circ X\circ\varphi^{-1}(x) &= \tilde{\varphi}(X(\varphi^{-1}(x))) \\ &= (x, ({\rm d}x_1)_{\varphi^{-1}(x)}(X(\varphi^{-1}(x))),\cdots, ({\rm d}x_n)_{\varphi^{-1}(x)}(X(\varphi^{-1}(x)))) \end{align}$$

I don't know how to say that each $({\rm d}x_i)_{\varphi^{-1}(x)}(X(\varphi^{-1}(x)))$ is smooth using only assertion (iii). Help?


I do realize that proving that (iii) implies (ii) and (ii) implies (i) is easier. But I want to understand this specific implication.

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For all $ \ u \in im( \varphi) \ $ we have

$$(dx_i)_{\varphi^{-1} (u)} \Big( X \big( \varphi^{-1} (u) \big) \Big) = X_{\varphi^{-1} (u)} (x_i) = [X(x_i)] \big( \varphi^{-1} (u) \big) = \{ [X(x_i)] \circ \varphi^{-1} \} (u)$$

Continue from this

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  • $\begingroup$ I got it, thanks a lot! $\endgroup$ – Ivo Terek Jul 17 '16 at 4:21

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