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

A connected Riemannian manifold is complete when its Riemannian distance is a complete metric space, meaning that every Cauchy sequence converges in the manifold. The Hopf-Rinow theorem says that for a connected finite-dimensional Riemannian manifold, this is equivalent to geodesic completeness, and also to compactness of every closed bounded subset. Under these equivalent conditions any two points are joined by a minimizing geodesic, and the Riemannian exponential map at every point is defined on its entire tangent space.

We use the usual connected, boundaryless-manifold convention for the global statements in Q2–Q5. Connectedness is essential, including for the diameter and group-growth conclusions. Without connectedness, the completeness and covering assertions apply to the target components met by the map; an unrelated incomplete component cannot be controlled by a local isometry into another component.

A local isometry preserves the Levi-Civita connection and hence geodesics, as well as the lengths of lifted curves. We first show that a complete source permits finite-length paths to lift all the way to their endpoints. Begin lifting a piecewise smooth path $c:[0,1]\to N$ from any chosen point above $c(0)$, using local inverse maps. If its lift is defined only up to a maximal $b\le 1$, thenCompleteness gives a limit point in $M$. A local inverse at that point continues the lift, including its endpoint. Thus such a finite obstruction is impossible. Connectedness of $N$ and piecewise smooth paths from $f(p)$ to any target point prove that $f$ is onto; no prior completeness of $N$ is used in this step.

Now lift the initial position and velocity of any geodesic in $N$. The resulting geodesic in $M$ exists for all real time by Hopf-Rinow theorem. Its image extends the original target geodesic by uniqueness of the geodesic initial-value problem. Hence $N$ is geodesically complete, and therefore complete.

The converse is false for a general local isometry, even an onto one. Give the interval $(0,4\pi )$ its ordinary metric and map it to the unit circle by $t\mapsto e^{it}$. This is an onto local isometry to a complete circle, but the sequence $t_j=1/j$ is Cauchy in the interval and has no limit there. It is not a covering map: the identity point of the circle has only one preimage, whereas nearby points have two.

The converse holds for a Riemannian covering. If $f$ is also a covering map and $N$ is complete, every target geodesic defined on $\mathbb{R}$ lifts globally with any specified initial point. A local isometry makes that lift a geodesic. Its initial velocity can be any prescribed source velocity because $df$ is an isomorphism. Thus all source geodesics extend for all time, and $M$ is complete by the Hopf-Rinow theorem.

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.