Hopf-Rinow
Let \((M,g)\) be a connected smooth Riemannian manifold without boundary. Write \(\nabla\) for its Levi–Civita connection and \(T_pM\) for the tangent space at \(p\). The coordinate conventions are those in tangent spaces and derivations.
The geodesic equation gives a solution for a short time from any initial position and velocity. Two global questions remain: can that solution continue indefinitely, and can two points be joined by a geodesic that realizes their distance? The Hopf–Rinow theorem relates these questions to completeness of the distance metric.
Theorem 1. On a compact, connected Riemannian manifold \(M\), any \(p, q \in M\) can be joined by a minimizing geodesic. Furthermore, for any \(p \in M\), \(\exp_p\) is defined on all of \(T_p M\), and any geodesic may be extended indefinitely in all directions.
It turns out we can relax the requirement on \(M\) in Theorem 1 substantially. We require \(M\) to be geodesically complete.
Definition 2. A Riemannian manifold \(M\) is said to be geodesically complete if for all \(p \in M\), \(\exp_p\) is well-defined on all of \(T_p M\). Equivalently, the geodesic \(\gamma(t)\) with \(\gamma(0) = p\) is well-defined for all \(t \in \mathbb R\).
The theorem compares three ways of controlling escape: convergence of Cauchy sequences, compactness of closed bounded sets, and existence of geodesics for all time. The minimizing-geodesic conclusion is what connects the metric and differential-equation descriptions.
Theorem 3 (Hopf-Rinow). Let \(M\) be a connected Riemannian manifold without boundary, equipped with its Riemannian distance. The following are equivalent:
\(M\) is complete as a metric space.
Closed, bounded subsets of \(M\) are compact.
There exists \(p \in M\) such that \(\exp_p\) is defined on the entirety of \(T_p M\).
\(M\) is geodesically complete.
Furthermore, any one of 3 through 3 implies
Any two points \(p,q \in M\) can be joined by a minimizing geodesic.
Two local facts used below follow from the radial-angular orthogonality established in normal coordinates. First, in a sufficiently small normal ball about \(x\), a radial segment to \(\exp_x(v)\) has length \(|v|\) and minimizes length. Indeed, for a competing curve written in polar coordinates \((r(t),\theta(t))\), its speed is at least \(|r'(t)|\). Integrating from the centre to radius \(|v|\) gives the lower bound \(|v|\). If a competing curve leaves a larger normal ball first, it already has at least the larger radius in length and cannot improve this bound. Limits from positive radius handle any passage through the centre. This also identifies a sufficiently small metric closed ball with the exponential image of a compact tangent-space ball.
Second, a length-minimizing pair of unit-speed geodesic segments cannot have a corner. To see this directly, use normal coordinates at the join and let the incoming and outgoing velocities be \(u\) and \(v\), with \(|u|=|v|=1\). Points at parameter distances \(\varepsilon\) on either side have coordinates \(-\varepsilon u\) and \(\varepsilon v\). Their coordinate straight segment has length \(\varepsilon|u+v|+o(\varepsilon)\), since the metric tends uniformly to its Euclidean value on the shrinking coordinate ball. If \(u\ne v\), then \(|u+v|<2\), and this is shorter than the broken segment of length \(2\varepsilon\). Thus minimization forces \(u=v\); ODE uniqueness then joins the two segments into one geodesic.
Proof. We first prove the main geometric step under condition (c): if \(\exp_p\) is defined on all of \(T_pM\), every \(q\in M\) can be joined to this \(p\) by a minimizing geodesic. The case \(p=q\) is immediate. Otherwise put \(r=d(p,q)>0\), and choose \(0<\rho<r\) so small that the closed ball of radius \(\rho\) lies in a normal neighbourhood of \(p\). Its boundary sphere is compact. Choose \(p_0\) on that sphere minimizing \(d(p_0,q)\).
Every path from \(p\) to \(q\) crosses this sphere. Its length is therefore at least \(\rho+d(p_0,q)\). Taking the infimum over paths and using the reverse triangle inequality gives \(r=\rho+d(p_0,q)\). Let \(v\) be the unit initial velocity of the radial geodesic from \(p\) to \(p_0\), and continue it using (c):
Define
This set is closed, and contains \(\rho\). For \(t\in T\), the triangle inequality gives
Thus \(d(p,c(t))=t\), so the initial segment of \(c\) is minimizing. Let \(t_0=\max T\). We claim \(t_0=r\).
If \(t_0<r\), choose \(0<\delta<r-t_0\) with the closed \(\delta\)-ball about \(c(t_0)\) inside a normal neighbourhood. The same compact-sphere argument gives a point \(y\) on its boundary such that
Join \(c(t_0)\) to \(y\) by its radial unit-speed geodesic. Concatenating it with \(c|_{[0,t_0]}\) gives a path of length \(t_0+\delta\), while the triangle inequality gives \(d(p,y)\ge r-d(y,q)=t_0+\delta\). The concatenation is therefore minimizing. A minimizing broken geodesic has no corner: its small segment around a proposed corner lies in a convex normal neighbourhood, where it must equal the unique minimizing geodesic. Its velocities agree at the join. Uniqueness of the geodesic equation now implies \(y=c(t_0+\delta)\), so \(t_0+\delta\in T\), a contradiction. Hence \(t_0=r\), \(c(r)=q\), and \(c|_{[0,r]}\) is minimizing.
We now establish the equivalences. Under (c), the preceding argument implies, for every \(R>0\),
One inclusion follows from the length of a radial geodesic; the other follows from the minimizing geodesic just constructed. The tangent-space ball is compact, so its continuous image is compact. Every closed bounded subset of \(M\) is a closed subset of such a ball. This proves (c) implies (b).
Condition (b) implies (a): a Cauchy sequence is bounded, hence lies in a compact closed ball. It has a convergent subsequence, and the Cauchy property makes the entire sequence converge to the same limit.
For (a) implies (d), suppose a unit-speed geodesic \(c:(a,b)\to M\) has a finite maximal right endpoint \(b\). The estimate \(d(c(t),c(u))\leq|t-u|\) shows that \(c(t)\) is Cauchy as \(t\uparrow b\), hence converges to \(x\) by completeness. Choose a compact coordinate neighbourhood \(K\) of \(x\) whose interior contains \(x\); eventually \(c(t)\in K\). On \(K\), positive definiteness gives a uniform lower metric bound \(g_y(v,v)\geq c_0|v|^2\), so unit tangent vectors have bounded coordinate components. They form a closed subset of a compact base times a bounded closed vector ball, hence a compact subset of \(TM\).
The geodesic vector field is smooth. Local ODE existence gives, around each point of this compact set of initial data, a neighbourhood and a common short existence interval. A finite subcover gives a single \(\varepsilon>0\) valid for all these data. Starting at any \(t>b-\varepsilon/2\) now gives a solution through \(t+\varepsilon>b\); it agrees with \(c\) on the overlap by uniqueness, contradicting maximality. Reverse time to treat the left endpoint. Constant geodesics already extend for all time, and nonconstant constant-speed geodesics are affine rescalings of unit-speed ones.
Finally (d) implies (c) directly. Under (d), the first argument applies at every base point, giving the minimizing-geodesic conclusion for every pair. A compact connected Riemannian manifold is complete as a metric space, so Theorem 1 follows as well. ◻
Reading: Carmo (1992), Riemannian geometry.