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In John M. Lee book Introductions to smooth manifolds Proposition 8.11 is left as exercise. Can anyone give me hints and suggestions to prove it?

In particular, i want to show:

Let $M$ be a smooth $n$-manifold with or without boundary and let $(X_1,\dots,X_k)$ be a linearly independent $k$-tuple of smooth vector fields on an open subset $U$ of $M$, with $1\le k< n$. Then for each $p\in U$ there exist smooth vector fields $X_{k+1},\dots,X_n$ in a neighborhood $V$ of $p$ such that $(X_1,\dots,X_n)$ is a smooth local frame for $M$ on $U \cap V$.

I can easily see that since $X_1|_p,\dots,X_k|_p$ are vectors linearly independent in $T_pM$ then there exist $v_{k+1},\dots,v_n$ in $T_pM$ such that $\{X_1|_p,\dots,X_k|_p,v_{k+1},\dots,v_n\}$ is a basis for $T_pM$.

I also know that I can extend each $v_i$ to a smooth global vector field on $M$, say $X_i$ (with $i>k$). But then, with this argument, it may be that $(X_1,\dots,X_n)$ is not a local frame for $M$ on $U$. I can only say that $(X_1|_p,\dots,X_n|_p)$ is a basis for $T_pM$.

EDIT Read this below as a (too long for a comment) comment for Sou's answer.

Thank you very much Sou! Please let me add some details to see if I have fully understood. Let be $(W,(x^i))$ a smooth chart for $M$ in $p$, then we have $X_i|_q=X_i^j(q)\frac{\partial}{\partial x^j}|_q$ for each $q$ in $U \cap W$ and each $i\le k$, and we have $v_i=v_i^j\frac{\partial}{\partial x^j}|_p\in T_pM$ for each $k<i\le n$.

Now I define $X_{k+i}|_q=v_i^j\frac{\partial}{\partial x^j}|_q$ for each $q\in W$, and these are smooth vector fields on $W$.

Now the map $G:U \cap W\to M(n\times n, \mathbb{R}), \quad q\mapsto (X_i^j(q))$ is smooth and $G^{-1}(GL(n,\mathbb{R}))$ is the neighborhood of $p$ on which $(X_1,\dots,X_n)$ is a smooth local frame, right?

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    $\begingroup$ Have you tried working in local coordinates around $p$, i.e., working in $\Bbb R^n$, and then going back to a neighborhood of $p$ in $M$? $\endgroup$ Jan 22, 2019 at 18:26
  • $\begingroup$ Is there some result for $\mathbb{R}^n$ which may help? $\endgroup$
    – Minato
    Jan 23, 2019 at 6:57
  • $\begingroup$ You may want to look here to compare your work. The accepted answer there by Diane is trustable since he is Geometer btw. $\endgroup$ Jan 25, 2019 at 8:46
  • $\begingroup$ Thank- you Sou, you are a very kind person :) $\endgroup$
    – Minato
    Jan 27, 2019 at 13:38

1 Answer 1

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$\textbf{Hint: }$ After you choose vectors $v_{k+1},\dots,v_n$, you have linearly independent vectors on $T_pM$. Extend $\{v_{k+1},\dots,v_n\}$ around a neighbourhood of $p$, say to constant local vector fields $X_{k+1},\dots,X_n$. Since $\{X_1|_p,\dots,X_k|_p,X_{k+1}|_p,\dots,X_n|_p\}$ linearly independent, the matrix $[X_i^j(p)]$ is invertible. Use continuity of determinant function to show that there is a smaller neighbourhood of $p$ such that $\{X_1,\dots,X_n\}$ is linearly independent there.

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  • $\begingroup$ I have edited following your argument. Am I right with my details? P.S. Could you please give also a look at this my question math.stackexchange.com/questions/3083066/… ? It seems no one wants to answer. Thank you. $\endgroup$
    – Minato
    Jan 24, 2019 at 13:05
  • $\begingroup$ @yili Yes, more or less. $\endgroup$ Apr 5, 2021 at 13:21
  • $\begingroup$ thank you $\ \ $ $\endgroup$
    – yi li
    Apr 5, 2021 at 13:28

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