I am trying to do an exercise that is as follows :
Let $M$ and $N$ be manifolds. The set of maps $f\in C_S^1(M,N)$ such that $f:M\rightarrow N$ is a covering space with the extra restriction that instead of local homeomorphisms we have local diffeomorphisms ,is an open set in $C_S^r(M,N)$.
Now my first idea to try and prove this would be to use the fact that if we have a local diffeomorphism $f: U\rightarrow V$ then the set of diffeomorphisms is open in $C_S^r(U,V)$, but then I realized I have the problem that $C_S^r(U,V)$ is not open in $C_S^r(M,N)$, so I needed to try another approach . I have been thinking about this but I haven't gotten anywhere useful. Any help with this is aprecciated, Thanks in advance.