Straightest, not shortest: the geodesic equation
In the last guide you earned the Levi-Civita connection nabla — the unique metric-compatible, torsion-free way to differentiate one vector field along another. Now spend it on a single object and watch the whole of Riemannian geodesy unfold. A curve gamma: I -> M has a velocity gamma'(t), a vector field living along it. Ask the most naive question imaginable: when does gamma fail to turn? A straight line in R^n is the curve whose velocity never changes, gamma'' = 0. On a manifold there is no ambient space to take that second derivative in, but the connection supplies the covariant version: the curve is a geodesic when nabla_{gamma'} gamma' = 0, that is, when it carries its own velocity parallel to itself. A geodesic is the straightest curve a curved space permits — it does not accelerate as felt from inside the manifold.
Write it in a chart and it becomes a concrete system of ODEs, which is exactly where its power comes from. With coordinates x^i and the Christoffel symbols Gamma^k_ij you computed last time, the condition nabla_{gamma'} gamma' = 0 unpacks into the geodesic equation: for each k, the second coordinate-derivative of the path plus a quadratic correction in the velocities vanishes. The Christoffel terms are the entire price of curvature — set them to zero and you are left with straight lines, x^k(t) = a^k t + b^k. Because this is a second-order ODE with smooth coefficients, the Picard-Lindelof theorem hands you exactly one geodesic for each initial point p and initial velocity v in T_p M, at least for small time. That existence-and-uniqueness fact is the seed everything else in this guide grows from.
geodesic equation (one for each k = 1, ..., n):
d^2 x^k / dt^2 + Gamma^k_ij (x(t)) * (dx^i/dt)(dx^j/dt) = 0
flat case Gamma = 0 => x^k(t) = a^k t + b^k (straight lines)
initial data ( p, v ), v in T_p M => unique geodesic gamma_v(t)
homogeneity gamma_{c v}(t) = gamma_v(c t) (rescale speed = rescale time)Why "straightest" also means "locally shortest"
Calling these curves geodesics suggests they are short, but we defined them purely by straightness — no length appeared. The reconciliation is one of the most satisfying calculations in the subject, and it runs through the calculus of variations. Fix two points and consider all curves between them; the length functional is awkward because it is invariant under reparametrization, so geometers minimize the cleaner energy functional E(gamma) = (1/2) integral of |gamma'(t)|^2 dt instead. Its critical points are exactly the constant-speed curves that also minimize length, which sidesteps the reparametrization degeneracy.
Now compute the first variation: nudge gamma by a variation field, differentiate the energy, integrate by parts using metric-compatibility of nabla, and the boundary terms drop when the endpoints are held fixed. What survives is an integral of <nabla_{gamma'} gamma', variation field> — so the energy is stationary for every allowed nudge precisely when nabla_{gamma'} gamma' = 0. The straightest curves and the energy-critical curves are the same curves; this is why a single equation deserves the name geodesic. The honest caveat is the word locally: a geodesic is a critical point, hence shortest only among nearby competitors and only over short enough stretches. A great-circle arc on S^2 is a geodesic the whole way around, but past the antipode it is the longer of the two arcs between its endpoints — straight everywhere, shortest only at first.
The exponential map: a tangent space spread over the manifold
Here is the device that turns the geodesic ODE into geometry you can hold. Fix p in M. For each tangent vector v in T_p M there is, for small time, a unique geodesic gamma_v with gamma_v(0) = p and gamma_v'(0) = v. The homogeneity gamma_{cv}(t) = gamma_v(ct) lets you trade speed for time, so you can always run a geodesic to time 1 by shrinking v. Define the exponential map exp_p(v) = gamma_v(1): follow the geodesic that leaves p in the direction v, walk for unit time, and report where you land. It is a map from (a neighborhood of 0 in) the flat vector space T_p M into the curved manifold M, and it is the single most important bridge between linear algebra and Riemannian geometry.
Two facts make exp_p the right tool rather than just a definition. First, its differential at the origin is the identity: d(exp_p)_0 = id on T_p M, because the geodesic through p in direction v leaves with exactly velocity v. By the inverse function theorem, exp_p is therefore a diffeomorphism from a small ball around 0 in T_p M onto a neighborhood of p in M — a normal neighborhood. Second, the radial lines t -> tv through the origin of T_p M map to the very geodesics through p, so straight lines in the tangent space become geodesics in the manifold. The exponential map is the precise sense in which the flat tangent space is the manifold's best first-order model: it is the curved space's chart drawn on the flat blackboard of T_p M.
Normal coordinates and the Gauss lemma
Feed the exponential map an orthonormal basis e_1, ..., e_n of T_p M and you manufacture the most useful coordinate system in the subject. Identify T_p M with R^n via that basis, then the inverse of exp_p assigns to each nearby point q the coordinates of the unique small vector v with exp_p(v) = q. These are geodesic normal coordinates centered at p. In them the geometry is as Euclidean as it can possibly be at a single point: the metric components satisfy g_ij(p) = delta_ij, and crucially all first derivatives of the metric vanish at p, which forces every Christoffel symbol Gamma^k_ij(p) = 0. Geodesics through p are literally straight coordinate rays. The manifold has been straightened to first order around p.
Do not over-read this. Normal coordinates kill the metric's first derivatives at p, but they cannot kill the second derivatives — and that obstruction is not a failure but the entire point. What survives at second order is exactly the curvature; in normal coordinates g_ij(x) = delta_ij - (1/3) R_ikjl x^k x^l + (higher order), so the quadratic deviation of the metric from the flat delta_ij reads off the Riemann curvature tensor directly. This is the cleanest statement of what curvature measures: it is the unremovable second-order discrepancy between your manifold and Euclidean space, the precise obstruction to making g constant. The next guide opens exactly here.
How far can you go? Completeness and Hopf-Rinow
All of the above was local: exp_p is a diffeomorphism only on a small ball, and geodesics were guaranteed only for small time. The natural worry is whether a geodesic can run forever or whether it falls off an edge in finite time — think of the flat plane with the origin deleted, where a geodesic aimed at the puncture simply stops existing. A manifold is geodesically complete when every geodesic extends to all of t in R, equivalently when exp_p is defined on the whole tangent space T_p M for every p. This is geodesic completeness, and it is the hypothesis that lets local geometry become global.
The Hopf-Rinow theorem ties the loose ends together and is the capstone of this guide. For a connected Riemannian manifold it declares the following conditions equivalent: M is geodesically complete; M is complete as a metric space under the distance function d(p,q) = infimum of lengths of curves from p to q; and closed bounded subsets of M are compact. Any one of these gives a powerful bonus: between any two points there exists an actual minimizing geodesic realizing the distance d(p,q). So completeness upgrades the local Gauss-lemma guarantee — shortest paths exist not just nearby but globally.