Diffeomorphism from $T M$ to $M \times \mathbb{R}^2$.

Question

Let $M$ be a two-dimensional manifold. Suppose there are two vector fields on $M$, $X(p) = (p,V (p))$ and $Y (p) = (p,W (p))$, such that $$\text{span}\{V (p),W (p)\} = T_p M$$ for all $p \in M$. Construct a diffeomorphism from $T M$ to $M \times \mathbb{R}^2$.

Here is what I know so far:

Since $\text{span}\{V (p),W (p)\}$ span $T_p M$, then there is a diffeomorphism. Since

• $T_pM=\cup_{p\in M} T_p M$

• $M\times \mathbb{R}^2=\cup_{p\in M} \{p \} \times \mathbb{R}^2$

We need to find an isomorphism from $\mathbb{R}^2$ to $T_p M$ for all $p \in M$. But I still cannot find it. Any help would be appreciated. Thank you.

Answer 1

For all $p\in M$ and $v\in T_pM$, since $\{X(p),Y(p)\}$ is a basis of $T_pM$, there exist unique real coefficients $a_p(v),b_p(v)$ such that $v=a_p(v)X(p)+b_p(v)Y(p)$. Now, it can be seen that $a,b\colon TM\to\mathbf{R}$ are smooth, since $X$ and $Y$ are, then $$TM\ni(p,v)\mapsto(p,a_p(v),b_p(v))\in M\times\mathbf{R}^2,$$ is the required diffeomorphism, whose inverse is given by $(p,x,y)\mapsto(p,xX(p)+yY(p))$.