If $f: E \to M$ is a smooth covering map between manifolds, is it true that the differential (or the push-forward) $df_p : T_p E \to T_{f(p)}M$ is invertible for every $p\in E$?
I think this should be true since every smooth covering map is a local diffeomorphism, hence both a smooth immersion and submersion.
Is my reasoning correct? Thanks in advance.