I want to find a diffeomorphism F : $\mathbb B^n$ → $\mathbb B^n$ such that F(p) = q for all p,q$\in$ $\mathbb B^n$. Here $\mathbb B^n$ is the just unit open ball in $\mathbb R^n$.
My idea is let r(t)=(1-t)p+tq, considering p and q as vector. Then r(t) is an integral curve for a vector field in the line segment between p,q. Of course, at every point, the corresponding vector is same. Then use extension lemma to extend this vector field into a vector field V in $\mathbb B^n$, whose support is in $\mathbb B^n$ also. Then V is a complete vector field, and it generates a global flow $\theta _t()$, which is a diffeomorphism.
My only question is can I claim that $\theta_t(p)$=q? If I can then I am done. Also, is $\theta_t(p)$ is a extension of r(t)? If it is, do I need to prove it?