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

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.