Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 19 / 2 / c / Solution

Surjectivity was proved in part (b). Fix $q\in N$ and choose a normal ball $B=B_N(q,\epsilon )$ on which ${\exp }_q$ is a diffeomorphism from the tangent ball. For each $p\in f^{-1}(q)$ define a sectionCompleteness of $M$ guarantees that the source Riemannian exponential map is defined. Preservation of geodesics and uniqueness of their initial-value problem give $f\circ s_p={\mathrm{id}}_B$. Since $df$ is invertible, differentiation of this identity shows that $s_p$ is a local diffeomorphism; it is injective because it is a section. Its image $U_p$ is open, and $f:U_p\to B$ is a diffeomorphism.

These images are disjoint. If $s_p(y)=s_{p^{\prime }}(y)$, lift the unique radial geodesic from $y$ back to $q$ starting at that common point. Both section constructions give this same lift, and uniqueness forces its endpoint to be both $p$ and $p^{\prime }$. Conversely, for any $x\in f^{-1}(B)$, lift that reversed radial geodesic from $f(x)$ to $q$ starting at $x$. Completeness supplies its full finite interval; its endpoint is some $p\in f^{-1}(q)$. Reversing the lift shows $x=s_p(f(x))$. ConsequentlyEvery point has such an evenly covered neighborhood, proving that the complete local isometry is a covering. This argument does not assume a uniform positive source injectivity radius.