Lets say I have two (compact) manifolds $U$,$V$ and a diffeomorphism $\psi:U\rightarrow V $.
The shortest way between two points $a$ , $b \in V$ is given by a parametrisation $\gamma :W \rightarrow V $ which I found using Euler-Langrange-Equation. Now I want to have a parametrisation $\eta: W \rightarrow U$ with the same property, that means $\psi(\eta(t))=\gamma(t)$.
Moreover, I need more theory according to lengths on manifolds, I minimized the length using $L=\int \left \|\gamma '(t)) \right \|dt$ but I don't know if this is an elegent way of doing it. The problem which I want to solve is: what is the Length of to points $a,b\in U$ regarding to $V$, without transforming to $V$, finding a parametrisation and integrating over it.