Let $M$ a smooth manifold,$ \ p \in M$, $x\in \mathbb{R}^{n}$ and a linear map $L:T_{p}M\to\mathbb{R}^{n}.$ Show that exist some differentiable

Question

Let $M$ a smooth manifold,$ \ p \in M$, $x\in \mathbb{R}^{n}$ and a linear map $L:T_{p}M\to\mathbb{R}^{n}.$ Show that exist some differentiable application $f:M^{m}\to\mathbb{R}^{n}$ tal que $f(p)=x$ and $df(p)=L.$

I'm thinking we should use Urysohn's lemma or Tietz's lemma, but I can't. Could someone give me a hint?

Answer 1

Take a local chart around $p$, so a differential mapping $\varphi:U\rightarrow\mathbb R^m$ that induces an isomorphism $$d\varphi_p:T_pM\xrightarrow{\cong}T_{\varphi(p)}\mathbb R^m.$$

Consider then the linear morphism $\tilde L:=L\circ d\varphi_p^{-1}$ and set $$f=\tilde L\circ\varphi.$$

Indeed, you get by the chain rule $$df_p=d\tilde L_{\varphi(p)}\circ d\varphi_p=\tilde L\circ d\varphi_p=L.$$

We used the fact that the differential of any linear morphism $u:\mathbb R^p\rightarrow\mathbb R^q$ at any point is itself.

Hint for the finishing touch: to get the property $f(p)=x$ we could deform the solution, by composing $\tilde L$ with an appropiate translation, since its differential is the identity. If you need details, I will provide them if you ask me in the comments.