Two meanings of 'geodesic' that you must keep apart
From the curvature rung you already have the geodesic equation nabla_{c'} c' = 0: a curve whose acceleration, measured by the Levi-Civita connection, vanishes. That is the locally straightest notion — a geodesic does not turn, it just coasts. But the word 'geodesic' carries a second, totally different connotation: globally shortest, the curve of least length joining two points. These two meanings are NOT the same, and the entire drama of this guide is figuring out exactly when, and how far, they coincide. Confusing them is the single most common beginner error in global Riemannian geometry, so we pin both down before anything else.
A picture makes the gap vivid. On the round sphere S^2 a great-circle arc is a geodesic in BOTH senses as long as it is short: the equator from 0 to a point a quarter-turn away is the unique shortest path. But continue past the far pole, three-quarters of the way around, and you still have a perfectly good geodesic — nabla_{c'} c' = 0 holds at every point — yet it is now LONGER than going the other way. It stopped being a minimizing geodesic the moment you passed the antipode, even though it never stopped being a geodesic. Locally straightest is a pointwise, differential condition; globally shortest is a competition against all other curves at once.
Why shortest curves obey the geodesic equation: the first variation
Why should a shortest path have zero acceleration at all? The honest answer is the calculus of variations, and you compute it directly. Fix endpoints p and q, take a curve c joining them, and wiggle it through a smooth family c_s of curves with the same endpoints, where c_0 = c. The first variation formula differentiates the length L(c_s) at s = 0 and, after integration by parts, produces an integral of the covariant acceleration against the variation field V: d/ds L(c_s) at s=0 equals minus the integral of g(V, nabla_{c'} c') along the curve (boundary terms drop because the endpoints are fixed). The bending of c is paired against how you push it.
Now the logic closes itself. If c is shortest then L(c_s) is minimized at s = 0, so the derivative must vanish for EVERY allowed variation field V. The only way the integral of g(V, nabla_{c'} c') can be zero for all V is for the other factor to vanish identically: nabla_{c'} c' = 0. So a length-minimizer is automatically a solution of the geodesic equation — provided you parametrize by arc length, because length is reparametrization-invariant and that freedom must be spent first. This is precisely the same Euler-Lagrange reasoning that gives straight lines in the plane; curvature only changes what 'straight' computes to.
- Fix endpoints p and q and pick a curve c joining them, parametrized by arc length so |c'| = 1; this spends the reparametrization freedom up front.
- Wiggle c through a smooth family c_s with c_0 = c and the same endpoints, recording the infinitesimal push by the variation field V = d/ds c_s at s = 0, which vanishes at both ends.
- Differentiate L(c_s) at s = 0 and integrate by parts: the result is d/ds L(c_s) = minus the integral of g(V, nabla_{c'} c') dt, with boundary terms gone because V vanishes at the endpoints.
- If c is shortest the derivative is zero for EVERY admissible V, and the only field orthogonal to all V is the zero field — forcing nabla_{c'} c' = 0, the geodesic equation.
Length versus energy: a cleaner functional with the same minimizers
There is an annoyance hiding in the length functional: it is blind to reparametrization. Trace the same arc fast or slow, in fits and starts, and L is identical, so length has a whole infinite-dimensional symmetry that makes its critical points non-isolated and its variational analysis sticky. The standard fix is to minimize the energy functional E(c) = (1/2) times the integral of |c'|^2 instead. Energy is NOT reparametrization-invariant — speeding up genuinely costs more energy — and that is exactly the feature we want, because it pins down a preferred parametrization.
The two are linked by one inequality. Cauchy-Schwarz gives L(c)^2 <= 2(b-a) E(c), with equality exactly when the speed |c'| is constant. So among all reparametrizations of a fixed arc, energy is smallest precisely for the constant-speed one, and there it equals length up to the harmless factor. The payoff: minimizing energy automatically delivers a constant-speed minimizing geodesic — you get the geodesic equation AND the arc-length-proportional parametrization in one variational problem, with no reparametrization slack to fight. Every serious proof in this rung, including Hopf-Rinow below and the index form in Guide 2, runs on energy, not length.
Four faces of completeness — and one word that ties them
Now the central concept. A Riemannian manifold (M, g) becomes a metric space the moment you define d(p, q) = the infimum of lengths of curves from p to q — the Riemannian distance. With a genuine distance in hand we can ask the most basic analytic question: are Cauchy sequences guaranteed to converge? That is metric completeness, and it can fail spectacularly. Take the plane and delete the origin: a straight path heading toward the puncture is Cauchy but converges to nothing inside the space. The manifold has a 'missing point', a hole that no chart sees but every traveler feels.
There is a second, geometrically flavored notion: geodesic completeness, which asks whether every geodesic can be extended for all time. Equivalently, via the exponential map exp_p you met in the curvature rung, it asks that exp_p be defined on the WHOLE tangent space T_p M, so you can shoot a geodesic in any direction and never run off the edge. The punctured plane fails this too: a geodesic aimed at the missing origin reaches it in finite time and simply stops, with nowhere to continue. Geodesic completeness and metric completeness feel like different demands — one about Cauchy sequences, one about extending curves — and a priori there is no reason they should agree.
The Hopf-Rinow theorem: completeness, all at once
Here is the theorem that organizes the whole subject. For a connected Riemannian manifold (M, g), the Hopf-Rinow theorem declares the following four conditions EQUIVALENT, plus a bonus consequence. Read the equivalence as the deep content: four definitions of 'no holes', forged in different worlds — metric, geodesic, topological — turn out to be one and the same. That is not obvious, and proving it is the real work; the slogan 'complete = complete' hides a genuine theorem.
Hopf-Rinow. For (M, g) connected Riemannian, these are equivalent:
(1) (M, d) is complete as a metric space (Cauchy sequences converge)
(2) M is geodesically complete (exp_p defined on all of T_p M, some p)
(3) exp_p is defined on all of T_p M for EVERY p
(4) every closed and bounded subset of M is compact (Heine-Borel holds)
==> BONUS: any two points p, q are joined by a MINIMIZING geodesic
(a geodesic c with L(c) = d(p, q))The crown is the bonus clause: in a complete manifold, any two points are joined by a minimizing geodesic — a geodesic whose length equals the distance d(p, q), realizing the infimum rather than merely approaching it. This is exactly the existence statement that the first-variation argument could not give you: variation told you a minimizer, IF it exists, must be a geodesic; Hopf-Rinow supplies the 'if it exists'. The proof of the bonus is beautiful and concrete — from p, shoot the geodesic toward q at the right initial direction and ride it; a continuity-and-connectedness argument shows the set of times where it stays distance-minimizing is open, closed, and nonempty, so it reaches q minimizing all the way. Completeness is the hypothesis that keeps the geodesic from falling off an edge mid-journey.
Hypotheses that bite, and where the rung heads next
Hold the hypotheses up to the light, because every one of them is load-bearing. Drop completeness and the punctured plane kills both the equivalence and the bonus: it is geodesically incomplete, metrically incomplete, NOT Heine-Borel (the unit disk minus center is closed and bounded but not compact), and there are pairs of points with no minimizing geodesic at all. Compactness, by contrast, is a free pass: every COMPACT Riemannian manifold is automatically complete, because a compact metric space is complete — so the sphere, the torus, and every closed surface satisfy Hopf-Rinow without you lifting a finger. That is why global theorems so often start 'let M be a compact (or complete) Riemannian manifold'.
Two clarifications worth burning in. First, Hopf-Rinow guarantees a minimizing geodesic EXISTS between any two points, but never that it is unique: on the sphere, antipodal points are joined by infinitely many minimizing great-circle arcs. Uniqueness, and the precise radius out to which exp_p stays a minimizing diffeomorphism, is the story of conjugate points and the cut locus in Guide 2 — the cut locus is exactly where geodesics stop minimizing. Second, the theorem says nothing about curvature; it is pure completeness. Curvature enters only when we start CONTROLLING geodesics, which is the entire remaining program of this rung.
- Guide 2 differentiates families of geodesics to get Jacobi fields, defines conjugate points (where nearby geodesics refocus), and builds the index form of the second variation — the machinery that decides when a geodesic stops minimizing.
- Guide 3 turns a positive lower curvature bound into the Bonnet-Myers theorem (the manifold must be compact with finite fundamental group) and Synge's theorem, plus the Bochner technique.
- Guide 4 takes nonpositive curvature instead and proves Cartan-Hadamard: a complete simply-connected manifold of nonpositive curvature is diffeomorphic to R^n via the exponential map.
- Guide 5 reaches the comparison summit — Rauch and Toponogov comparison, the sphere theorem, and the splitting theorem — where curvature bounds become sharp geometric and topological conclusions.