# Hint for proof of proposition 10.15, lines a) and c) in John M. Lee's book "An Introduction to Smooth Manifolds"

I have been thinking about how to prove proposition 10.15, lines a) and c) in John Lee's book "An Introduction to Smooth Manifolds" (I leave a screenshot of the proposition below), but I can't figure out how to construct the remaining sections in a). Each section can be written locally as a linear combination of the local canonical smooth frame $$t_{i}: V \rightarrow E$$ such that, using a local trivialization $$\Phi : \pi ^{-1}(V) \to V\times \mathbb{R}^k$$, we get $$\Phi\circ t_{i} : V \to V \times \mathbb{R}^k$$ is given by $$\Phi \circ t_{i} (p) = (p,e_{i})$$. So, $$\sigma _{j} = \sum_{i=1}^{k} \lambda_{i,j} t_{i}$$ for $$1\leq j \leq m \lt k$$, where the $$\lambda_{i,j}$$'s are smooth as real valued functions on $$V$$. So the statement that we can extend the linearly independent set $$\{\sigma_{j}\}_{j=1,...,m}$$ to a local frame is equivalent to stating that the matrix valued function $$[\lambda_{i,j}]_{i = 1,...,k}^{j = 1,...,m}$$, which is a matrix with maximal rank at each point, can be "extended" to a smooth map $$M: W \subset V \to GL(k,\mathbb{R})$$.

Having said this, I would appreciate it if you could give me a hint for the idea of the proofs of a) and c) :).

Proposition 10.15 (Completion of Local Frame for Vector Bundles). Suppose $$\pi:E\longrightarrow M$$ is a smooth vector bundle of rank $$k$$.

(a) If $$(\sigma_1,\ldots,\sigma_m)$$ is a linearly independent $$m$$-tuple of smooth local sections of $$E$$ over an open subset $$U\subseteq M$$, with $$1\leq m, then for each $$p\in U$$ there exist smooth sections $$\sigma_{m+1},\ldots,\sigma_k$$ defined on some neighborhood $$V$$ of $$p$$ such that $$(\sigma_1,\ldots,\sigma_k)$$ is a smooth local frame for $$E$$ over $$U\cap V$$.

(b) If $$(v_1,\ldots,v_m)$$ is a linearly independent $$m$$-tuple of elements of $$E_p$$ for some $$p\in M$$, with $$1\leq m, then there exists a smooth local frame $$(\sigma_i)$$ for $$E$$ over some neighborhood of $$p$$ such that $$\sigma_i(p)=v_i$$ for $$i =1,\ldots,m$$.

(c) If $$A\subseteq M$$ is a closed subset and $$(\tau_1,\ldots,\tau_k)$$ is a linearly independent $$k$$-tuple of sections of $$E|_A$$ that are smooth in the sense described in Lemma 10.12, then there exists a smooth local frame $$(\sigma_1,\ldots,\sigma_k)$$ for $$E$$ over some neighborhood of $$A$$ such that $$\sigma_i|_A=\tau_i$$ for $$i=1,\ldots,k$$.

Exercise 10.16. Prove the preceding proposition.

You almost got it. Continuing what you said, at point $$p$$ we can extend the matrix $$\begin{bmatrix} \lambda_1^1(p) & \cdots & \lambda_m^1(p)\\ \vdots & \vdots& \vdots\\ \lambda_1^k(p) & \cdots & \lambda_m^k(p) \end{bmatrix}$$ to a $$GL_k(\mathbb{R})$$ matrix by extending the linear independent vectors $$v_1(p):=\begin{bmatrix} \lambda_1^1(p) \\ \vdots \\ \lambda_1^k(p) \end{bmatrix},\dots,v_m(p):=\begin{bmatrix} \lambda_m^1(p) \\ \vdots \\ \lambda_m^k(p) \end{bmatrix}$$ to a basis of $$\mathbb{R}^n$$. Let $$v_1(p),\dots v_m(p),v_{m+1},\dots,v_k$$ be that basis. You have to show that the map $$F:U \to \mathbb{R}$$ given by $$p \mapsto \det\left(\begin{bmatrix} v_1(p) & \cdots & v_m(p)& v_{m+1} & \cdots & v_k \end{bmatrix}\right)$$ is smooth. Use the fact that $$F^{-1}(0)$$ is closed to conclude.
• Does not the vectors $v_{m+1}, ..., v_{k}$ depend on the point $p$? Jul 22 at 17:05