Geodesics: The Straightest Possible Paths on a Curved Surface

The precise statement: two equivalent definitions

Fix a Riemannian manifold (M, g) with its unique Levi-Civita connection ∇ — the connection that is torsion-free and preserves the metric. A smooth curve γ: I → M is a geodesic if its velocity is parallel along it:

• Connection form: ∇_γ′ γ′ = 0. The intrinsic acceleration vanishes.
• Coordinate form: in local coordinates (x¹,…,xⁿ), writing γ(t) = (x¹(t),…,xⁿ(t)),
  ẍᵏ + Γᵏ_ij ẋⁱ ẋʲ = 0, for k = 1,…,n,
  summing over repeated indices i, j (Einstein convention). The Γᵏ_ij are the Christoffel symbols ½ gᵏˡ(∂ᵢg_jl + ∂ⱼg_il − ∂ₗg_ij).

This is a second-order ODE, quadratic in velocity. Note a subtlety of terminology: a geodesic is parametrized at constant speed (|γ′| is automatically constant because ∇ preserves g). A curve that merely traces a geodesic's image but with varying speed is a pregeodesic and satisfies ∇_γ′γ′ = f(t) γ′ instead.

The picture: no self-turning, ants that never steer

Imagine an ant walking on a surface, forbidden to turn its own body — it may only go 'straight ahead,' letting the surface bend under it. The path it traces is a geodesic. The full acceleration vector of the curve may be large (a great circle on a sphere is genuinely curving in ℝ³), but all of that acceleration points normal to the surface; the component tangent to the surface — the part the ant could feel as a steer left or right — is zero. That tangential part is the geodesic curvature κ_g, and geodesics are exactly the curves with κ_g ≡ 0.

A second picture: parallel transport. Slide the velocity vector along the curve keeping it 'as parallel as the geometry allows.' A geodesic is a curve that transports its own velocity onto itself — it is self-parallel. This is why geodesics are the natural notion of 'unaccelerated' motion, and why in general relativity free-falling particles trace geodesics of spacetime.

The mechanism: variation of energy and the exponential map

Where does the geodesic equation come from? From a variational principle. Define the energy of a curve E(γ) = ½ ∫ₐᵇ g(γ′, γ′) dt. Take a variation γ_s with fixed endpoints and differentiate. The first variation formula gives
(d/ds)|₀ E(γ_s) = − ∫ₐᵇ g(V, ∇_γ′γ′) dt,
where V is the variation field. This vanishes for all V iff ∇_γ′γ′ = 0. So geodesics are precisely the critical points of energy — the Euler–Lagrange equation of the length/energy functional.

Existence and uniqueness are then pure ODE theory: given a point p and a velocity v ∈ T_pM, Picard–Lindelöf yields a unique geodesic γ_v with γ_v(0) = p, γ_v′(0) = v, at least for small time. Collecting these defines the exponential map exp_p(v) = γ_v(1), a diffeomorphism from a neighborhood of 0 ∈ T_pM onto a neighborhood of p. It converts the linear space T_pM into the curved manifold, and is the engine behind normal coordinates and the local minimizing property (via the Gauss lemma).

Worked example: great circles on the sphere

Take the unit sphere S² with coordinates (θ, φ), θ = colatitude, φ = longitude, and metric ds² = dθ² + sin²θ dφ². The nonzero Christoffel symbols are Γ^θ_φφ = −sinθ cosθ and Γ^φ_θφ = Γ^φ_φθ = cotθ. The geodesic equations become:

• θ̈ − sinθ cosθ φ̇² = 0,
• φ̈ + 2 cotθ θ̇ φ̇ = 0.

Try the equator θ ≡ π/2, φ = t. Then θ̇ = θ̈ = 0 and cotθ = 0, so both equations are satisfied identically — the equator is a geodesic. By the rotational symmetry of the sphere (isometries carry geodesics to geodesics), every great circle is a geodesic. And conversely: solving the system, one finds every geodesic lies in a plane through the origin, i.e. is a great circle. New York (≈40.7°N) to Madrid (≈40.4°N) both sit near the same latitude, yet the great-circle route bows north — the flat map lies, the geodesic tells the truth.

Why completeness matters: Hopf–Rinow and its failure

Local existence is cheap; global minimization is not, and this is where hypotheses bite. The Hopf–Rinow theorem (Heinz Hopf and Willi Rinow, 1931) says: for a connected Riemannian manifold M, the following are equivalent — (i) M is complete as a metric space; (ii) M is geodesically complete (every geodesic extends to all of ℝ); (iii) closed bounded sets are compact. And any of these implies the punchline: every pair of points is joined by a minimizing geodesic.

Drop completeness and the conclusion fails cleanly. Take the punctured plane ℝ² ∖ {0}. The points (−1, 0) and (1, 0) 'want' to be connected by the segment through the origin — but the origin is gone. No geodesic (straight segment) realizes the infimum distance 2; every admissible path detours and is strictly longer. So the distance is achieved by no geodesic. This is not pathology — an open ball, or any manifold with a puncture or edge, behaves the same way. Completeness is exactly the hypothesis that rules out 'running off the edge in finite length.'

Significance: what geodesics unlock

Geodesics are the connective tissue of geometry and physics.

• General relativity: Einstein's equivalence principle says free-falling bodies follow geodesics of curved spacetime; planetary orbits and light-bending are geodesic phenomena. The famous timelike/null geodesics replace Newton's straight-line inertia.
• Curvature and comparison: how nearby geodesics spread or focus is governed by the Jacobi equation J″ + R(J, γ′)γ′ = 0; positive curvature makes them reconverge (conjugate points), driving Bonnet–Myers, Rauch, and Toponogov comparison theorems and the whole sphere-theorem industry.
• Topology: closed geodesics detect the fundamental group; the Cartan–Hadamard theorem shows a complete simply-connected manifold of nonpositive curvature is diffeomorphic to ℝⁿ via exp_p.
• Applied: shortest air/sea routes, optimal control, geodesic domes, shape analysis, and manifold learning where 'geodesic distance' captures intrinsic structure a straight-line distance misses.

Geodesics as 'straight lines' across geometries: what plays the role of a line, and how it behaves.

Space	Geodesics are	Two points joined by a minimizer?	Behavior of nearby geodesics
Euclidean ℝⁿ (flat)	Straight lines	Yes, unique	Stay parallel — separation grows linearly
Sphere S² (curvature +1)	Great circles	Yes, but not unique (antipodes: ∞ many)	Converge and refocus (conjugate points)
Hyperbolic plane ℍ² (curvature −1)	Diameters & circular arcs ⊥ boundary	Yes, unique	Diverge exponentially
Flat cylinder	Helices, circles, vertical lines	Not always unique (wrap-around)	Parallel locally; global self-intersections
Punctured plane ℝ²∖{0}	Straight lines missing the origin	No — path through 0 has no minimizer	Incomplete: geodesics run off to the hole