Problem 8-5 (Completion of Local Frames)

Proof.

(a)

Fix $p\in U$ and let $(v_1,\dots ,v_k)$ be the derivations $(X_1,\dots ,X_k){\text{|}}_p$ at the point $p$. Since $v_1,\dots ,v_k\in T_{p}M$ are $k$ linearly independent elements of an $n$-dimensional vector space, we can find $n-k$ additional vectors $v_{k+1},\dots ,v_n$ such that $v_1,\dots ,v_k,v_{k+1},\dots ,v_n$ form a basis for $T_{p}M$. By Proposition 8.7, we can find smooth global vector fields $X_{k+1}^{(p)},\dots ,X_n^{(p)}\in 𝔛(M)$ on $M$ such that $X_{k+1}^{(p)}{\text{|}}_p=v_{k+1}$, $\text{…}$, $X_n^{(p)}{\text{|}}_p=v_n$. We argue that in a neighborhood $V$ of $p$ which is small enough, the resulting collection of derivations $$X_1{\text{|}}_q,\dots ,X_k{\text{|}}_q,X_{k+1}{\text{|}}_q,\dots ,X_n{\text{|}}_q$$

defines a basis for $T_{q}M$ for all $q\in V$.

By Proposition 8.1, the fact that our collection of vector fields is smooth means that the component functions (stacked together in a single matrix $X$):

$$\left[\begin{array}{cccccc}\hfill X_1^1\hfill & \hfill \text{…}\hfill & \hfill X_k^1\hfill & \hfill X_{k+1}^1\hfill & \hfill \text{…}\hfill & \hfill X_n^1\hfill \\ \hfill ⋮\hfill & \hfill \text{⋱}\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \text{⋱}\hfill & \hfill ⋮\hfill \\ \hfill X_1^n\hfill & \hfill \text{…}\hfill & \hfill X_k^n\hfill & \hfill X_{k+1}^n\hfill & \hfill \text{…}\hfill & \hfill X_n^n\hfill \end{array}\right]$$

are smooth. In particular, this matrix function is continuous. Since the determinant function $\det <!--\; FUNCTION\; APPLICATION\; -->$ is continuous as well, their composition $\det <!--\; FUNCTION\; APPLICATION\; -->\circ X:M\to ℝ$

$$\det <!--\; FUNCTION\; APPLICATION\; -->\left[\begin{array}{cccccc}\hfill \underset{1}{\overset{1}{X}}\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{k}{\overset{1}{X}}\hfill & \hfill \underset{k+1}{\overset{1}{X}}\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{n}{\overset{1}{X}}\hfill \\ \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \text{⋱}\hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \text{⋱}\hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill \\ \hfill \underset{1}{\overset{n}{X}}\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{k}{\overset{n}{X}}\hfill & \hfill \underset{k+1}{\overset{n}{X}}\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{n}{\overset{n}{X}}\hfill \end{array}\right]$$

is continuous too. We know that at $p$, its value is non-zero

$$\det <!--\; FUNCTION\; APPLICATION\; -->\left[\begin{array}{cccccc}\hfill \underset{1}{\overset{1}{X}}(p)\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{k}{\overset{1}{X}}(p)\hfill & \hfill \underset{k+1}{\overset{1}{X}}(p)\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{n}{\overset{1}{X}}(p)\hfill \\ \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \ddots \hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \ddots \hfill & \hfill ⋮<!--\; FUNCTION\; APPLICATION\; -->\hfill \\ \hfill \underset{1}{\overset{n}{X}}(p)\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{k}{\overset{n}{X}}(p)\hfill & \hfill \underset{k+1}{\overset{n}{X}}(p)\hfill & \hfill \dots <!--\; FUNCTION\; APPLICATION\; -->\hfill & \hfill \underset{n}{\overset{n}{X}}(p)\hfill \end{array}\right]\ne 0.$$

By the definition of continuity, we can find a neighborhood $V$ of $p$ on which $\det <!--\; FUNCTION\; APPLICATION\; -->\circ X$ is non-zero. In other words, the columns $(X_1,\dots ,X_n){\text{|}}_q$ form a basis for $T_{q}M$ at each point $q\in V$, i.e., $(X_1,\dots ,X_n)$ is a local frame on $V$.

(b)

By Proposition 8.7, we can find smooth global vector fields $X_1,\dots ,X_k\in 𝔛(M)$ such that $X_1{\text{|}}_p=v_1$, $\text{…}$, $X_k{\text{|}}_p=v_k$. Using a similar argumentation with determinant, one can demonstrate that the collection $X_1,\dots ,X_k$ is linearly independent in some small neighborhood $U$ of $p$. Using part (a), we can extend this linearly independent collection of vector fields to a local frame $X_1,\dots ,X_k,X_{k+1},\dots ,X_n$ in some neighborhood $V\subseteq U$ of $p$.

(c)

By Extension Lemma for Vector Fields (Problem 8-1), for each $1\le j\le n$, there exists a smooth global vector field ${\tilde{X}}_j$ on $M$ such that ${\tilde{X}}_j{\text{|}}_A=X_j$. By a similar argument with determinant, each point $p\in A$ has an open neighborhood $V_p$ on which $({\tilde{X}}_1,\dots ,{\tilde{X}}_n)$ forms a local frame. Taking the union $V:={\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{p\in A}{V}_p$ of all these neighborhoods, we obtain the desired neighborhood of $A$.