Geodesics on Surfaces

Imagine an ant walking on a curved surface. It's told always to walk "straight ahead." It must never turn left or right. What path does it trace? On a flat table the answer is obvious: a straight line. On a sphere, a cylinder, or a saddle-shaped surface, the answer is subtler. Yet a precise notion of "straightest possible path" exists on every smooth surface. This notion is the geodesic. It is the natural generalization of a straight line to curved geometry. Geodesics matter because they sit at the intersection of three different ideas. These ideas turn out to be secretly the same. First, the curve that feels no sideways acceleration. Second, the curve that locally minimizes length between nearby points. Third, the curve whose coordinate expression solves a specific second-order system of ordinary differential equations. Reconciling these viewpoints is one of the great unifying achievements of classical differential geometry. It builds directly on The First Fundamental Form. That form supplies the only data — lengths and angles measured within the surface — that a two-dimensional being confined to the surface could ever access. The big picture is that geodesics generalize the entire calculus of straight lines. Shortest paths, uniform motion, parallel transport, and eventually curvature itself all reorganize around them. A geodesic is what remains of "straight line" once you refuse to leave the surface. It depends only on the surface's intrinsic geometry, not on how the surface happens to sit in space. This article develops geodesic curvature and the geodesic equations from the first fundamental form. It proves that geodesics are the critical points of the length (and energy) functional. It works two asymmetric examples in full — a great circle on a sphere and a helix on a cylinder. It then previews how geodesics feed directly into the Gauss-Bonnet Theorem, where the total turning of a curve gets balanced against the curvature enclosed.

Key takeaways
  • A geodesic is a curve on a surface whose acceleration, as seen from inside the surface, is always zero — it has zero geodesic curvature.
  • Geodesic curvature κg\kappa_g measures how much a curve bends within the surface, ignoring the bending forced by the surface's embedding in space.
  • The geodesic equations are a coupled system of two second-order ODEs in the surface's coordinates, built entirely from the first fundamental form's coefficients E,F,GE, F, G.
  • Geodesics are exactly the critical points of the length functional (and, with constant speed, of the energy functional) among curves with fixed endpoints.
  • Existence and local uniqueness of geodesics follow from the Picard–Lindelöf theorem applied to the geodesic ODE system.
  • Key idea: a geodesic is the intrinsic straight line — zero sideways acceleration inside the surface — and this single condition simultaneously governs shortest paths and sets up the curvature balance in the Gauss–Bonnet theorem.

Where this lives today

Geodesics are the backbone of the Gauss-Bonnet Theorem, which expresses the total geodesic curvature of a closed curve bounding a region, plus the total Gaussian curvature inside, as a topological constant — the seed of the entire subject of global differential geometry and, ultimately, of algebraic topology's interaction with geometry. In general relativity, free-falling particles and light rays travel along geodesics of a pseudo-Riemannian spacetime metric, so the geodesic equation (with Christoffel symbols built from the metric tensor exactly as here) is literally Einstein's law of motion in curved spacetime. In computer graphics and geometric modeling, geodesics on triangulated meshes give shortest paths for texture mapping, mesh parametrization, and character-animation surface distances (the "fast marching" and "heat method" algorithms for geodesic distance). In cartography and geodesy proper, geodesics on the (approximately ellipsoidal) Earth determine shortest flight paths and are computed via numerical integration of essentially the same ODE system derived above. In robotics and control theory, geodesics on configuration-space manifolds with a Riemannian metric encode locally optimal, energy-minimizing motions. In machine learning, Riemannian optimization methods move along geodesics on manifolds of positive-definite matrices, rotations, or other constrained parameter spaces to perform gradient descent that respects the manifold's geometry.

Ask the tutor: How does the geodesic equation change when we move from a surface's first fundamental form to a general Riemannian metric, and what exactly is a Jacobi field?

DefinitionGeodesic curvature and geodesics

Let S⊂R3S \subset \mathbb{R}^3 be a regular surface and let x(u,v)\mathbf{x}(u,v) be a local parametrization with first fundamental form coefficients

E=xu⋅xu,F=xu⋅xv,G=xv⋅xv.E = \mathbf{x}_u \cdot \mathbf{x}_u, \qquad F = \mathbf{x}_u \cdot \mathbf{x}_v, \qquad G = \mathbf{x}_v \cdot \mathbf{x}_v.

Let α:I→S\alpha: I \to S be a regular curve parametrized by arc length, so ∣α′(s)∣=1|\alpha'(s)| = 1, and let n(s)\mathbf{n}(s) denote the surface unit normal along α\alpha. Decompose the acceleration vector α′′(s)\alpha''(s) into a component tangent to SS and a component along n\mathbf{n}:

α′′(s)=κg(s) (n(s)×α′(s))+κn(s) n(s).\alpha''(s) = \kappa_g(s)\, \big(\mathbf{n}(s) \times \alpha'(s)\big) + \kappa_n(s)\, \mathbf{n}(s).

The scalar κg(s)\kappa_g(s) is the geodesic curvature of α\alpha, and κn(s)\kappa_n(s) is the normal curvature. Equivalently, since n×α′\mathbf{n}\times \alpha' is the unit vector in the tangent plane orthogonal to α′\alpha' (rotated 90∘90^\circ from α′\alpha' within the tangent plane, consistent with orientation), one has

κg(s)=α′′(s)⋅(n(s)×α′(s)).\kappa_g(s) = \alpha''(s) \cdot \big(\mathbf{n}(s) \times \alpha'(s)\big).

A regular curve α\alpha on SS (not necessarily arc-length parametrized) is a geodesic if its tangential acceleration vanishes identically, i.e. the component of α′′(t)\alpha''(t) tangent to SS is zero at every point. When α\alpha is parametrized by arc length, this is exactly the condition κg≡0\kappa_g \equiv 0.

Intuition

Think of decomposing the acceleration of a curve confined to a surface into two orthogonal pieces. The normal curvature κn\kappa_n is forced upon the curve by the shape of the surface itself — it is present even for the "straightest" possible curve, because the surface bends in space and the curve must bend with it just to stay on the surface. The geodesic curvature κg\kappa_g, by contrast, measures extra bending: how much the curve swerves left or right within the tangent plane, beyond what the surface's shape already demands. A geodesic is a curve with no such extra swerving — all its acceleration is "used up" just keeping it on the surface, none is spent turning sideways. This is why an ant that always walks "straight ahead," never voluntarily turning, traces a geodesic. On a plane, κn≡0\kappa_n \equiv 0 automatically (there is no forced bending), so geodesics are curves with κg≡0\kappa_g \equiv 0, i.e. α′′≡0\alpha'' \equiv 0: ordinary straight lines. On a sphere, every great circle is a geodesic, because the plane of a great circle passes through the sphere's center, forcing the curve's acceleration to point exactly along the surface normal, with no tangential remainder. The geodesic equations translate this geometric idea into coordinates. If α(t)=x(u(t),v(t))\alpha(t) = \mathbf{x}(u(t), v(t)), then requiring the tangential component of α′′\alpha'' to vanish produces two coupled ODEs in u(t)u(t) and v(t)v(t) whose coefficients are built from E,F,GE, F, G and their partial derivatives — exactly the data of the first fundamental form, and nothing about how the surface bends in space. This is the first hint that geodesics are an intrinsic notion: two surfaces with the same first fundamental form (isometric surfaces) have identical geodesic equations, hence "the same" geodesics under the isometry, even if one is flat and the other is curled into a cylinder.

Figure
Geodesic curvature: decomposing acceleration on a surface
α(s) α′(s), unit tangent κ_n n(s), normal part κ_g (n×α′), tangential part α″(s) = κ_g (n × α′) + κ_n n : geodesic ⇔ κ_g ≡ 0
TheoremExistence, uniqueness, and the length-minimizing property of geodesics

Let S⊂R3S \subset \mathbb{R}^3 be a regular surface, x:U→S\mathbf{x}: U \to S a local parametrization with first fundamental form coefficients E,F,G∈C∞(U)E, F, G \in C^\infty(U), EG−F2>0EG - F^2 > 0.

(a) Geodesic equations. A curve α(t)=x(u(t),v(t))\alpha(t) = \mathbf{x}(u(t), v(t)) is a geodesic if and only if (u(t),v(t))(u(t), v(t)) satisfies the system

u′′+Γ111(u′)2+2Γ121u′v′+Γ221(v′)2=0,v′′+Γ112(u′)2+2Γ122u′v′+Γ222(v′)2=0,\begin{aligned} u'' + \Gamma^1_{11} (u')^2 + 2\Gamma^1_{12} u'v' + \Gamma^1_{22} (v')^2 &= 0, \\ v'' + \Gamma^2_{11} (u')^2 + 2\Gamma^2_{12} u'v' + \Gamma^2_{22} (v')^2 &= 0, \end{aligned}

where the Christoffel symbols Γijk\Gamma^k_{ij} are determined algebraically by E,F,GE, F, G and their first partial derivatives via

Γ111=GEu−2FFu+FEv2(EG−F2),Γ112=2EFu−EEv−FEu2(EG−F2),(and similarly for Γ12k,Γ22k).\Gamma^1_{11} = \frac{G E_u - 2F F_u + F E_v}{2(EG-F^2)}, \quad \Gamma^2_{11} = \frac{2E F_u - E E_v - F E_u}{2(EG - F^2)}, \quad \text{(and similarly for }\Gamma^k_{12}, \Gamma^k_{22}\text{).}

(b) Existence and uniqueness. Given a point p∈Sp \in S and a tangent vector w∈TpSw \in T_pS, there exists ε>0\varepsilon > 0 and a unique geodesic α:(−ε,ε)→S\alpha : (-\varepsilon, \varepsilon) \to S with α(0)=p\alpha(0) = p and α′(0)=w\alpha'(0) = w.

(c) Constant speed. Every geodesic has constant speed ∣α′(t)∣|\alpha'(t)|.

(d) Local length-minimization. Let α:[a,b]→S\alpha : [a,b] \to S be a geodesic parametrized by arc length. For every t0∈(a,b)t_0 \in (a,b) there is δ>0\delta > 0 such that, for a≤s1<s2≤ba \le s_1 < s_2 \le b with s2−s1<δs_2 - s_1 < \delta and t0∈[s1,s2]t_0 \in [s_1, s_2], α∣[s1,s2]\alpha|_{[s_1,s_2]} has length less than or equal to the length of every piecewise-regular curve on SS joining α(s1)\alpha(s_1) to α(s2)\alpha(s_2), with equality only for a reparametrization of α\alpha itself.

Proof

We use three techniques: (1) direct computation with the chain rule and the definition of tangential projection to derive the ODE system, invoking The First Fundamental Form for the metric coefficients; (2) the Picard–Lindelöf existence–uniqueness theorem for first-order ODE systems, after reducing the second-order system to first order; (3) the first-variation-of-arc-length computation (a calculus of variations argument) to establish local minimization.

(a) Derivation of the geodesic equations. Write α(t)=x(u(t),v(t))\alpha(t) = \mathbf{x}(u(t), v(t)). By the chain rule,

α′(t)=xu u′+xv v′.\alpha'(t) = \mathbf{x}_u\, u' + \mathbf{x}_v\, v'.

Differentiating again and using the chain rule on xu(u(t),v(t))\mathbf{x}_u(u(t),v(t)) and xv(u(t),v(t))\mathbf{x}_v(u(t),v(t)),

α′′(t)=xuu(u′)2+2xuvu′v′+xvv(v′)2+xuu′′+xvv′′.\alpha''(t) = \mathbf{x}_{uu} (u')^2 + 2\mathbf{x}_{uv} u'v' + \mathbf{x}_{vv} (v')^2 + \mathbf{x}_u u'' + \mathbf{x}_v v''.

The vectors xu,xv\mathbf{x}_u, \mathbf{x}_v span the tangent plane TpST_pS at each point, and {xu,xv,n}\{\mathbf{x}_u, \mathbf{x}_v, \mathbf{n}\} is a basis of R3\mathbb{R}^3 (since xu×xv≠0\mathbf{x}_u \times \mathbf{x}_v \ne 0 for a regular parametrization). Hence each second partial derivative decomposes uniquely as

xuu=Γ111xu+Γ112xv+L n,xuv=Γ121xu+Γ122xv+M n,xvv=Γ221xu+Γ222xv+N n,\mathbf{x}_{uu} = \Gamma^1_{11}\mathbf{x}_u + \Gamma^2_{11}\mathbf{x}_v + L\,\mathbf{n}, \quad \mathbf{x}_{uv} = \Gamma^1_{12}\mathbf{x}_u + \Gamma^2_{12}\mathbf{x}_v + M\,\mathbf{n}, \quad \mathbf{x}_{vv} = \Gamma^1_{22}\mathbf{x}_u + \Gamma^2_{22}\mathbf{x}_v + N\,\mathbf{n},

where L,M,NL, M, N are the second fundamental form coefficients (irrelevant here) and the Γijk\Gamma^k_{ij} are, by definition, the tangential coefficients — the Christoffel symbols. To solve for the Γijk\Gamma^k_{ij} in terms of E,F,GE, F, G: take the dot product of xuu=Γ111xu+Γ112xv+Ln\mathbf{x}_{uu} = \Gamma^1_{11}\mathbf{x}_u + \Gamma^2_{11}\mathbf{x}_v + L\mathbf{n} with xu\mathbf{x}_u and with xv\mathbf{x}_v (noting n⋅xu=n⋅xv=0\mathbf{n}\cdot\mathbf{x}_u = \mathbf{n}\cdot\mathbf{x}_v = 0):

xuu⋅xu=Γ111E+Γ112F,xuu⋅xv=Γ111F+Γ112G.\mathbf{x}_{uu}\cdot \mathbf{x}_u = \Gamma^1_{11} E + \Gamma^2_{11} F, \qquad \mathbf{x}_{uu}\cdot \mathbf{x}_v = \Gamma^1_{11} F + \Gamma^2_{11} G.

Now observe that xuu⋅xu=12∂u(xu⋅xu)=12Eu\mathbf{x}_{uu}\cdot\mathbf{x}_u = \tfrac12 \partial_u(\mathbf{x}_u\cdot\mathbf{x}_u) = \tfrac12 E_u, and xuu⋅xv=∂u(xu⋅xv)−xu⋅xuv=Fu−12Ev\mathbf{x}_{uu}\cdot\mathbf{x}_v = \partial_u(\mathbf{x}_u\cdot\mathbf{x}_v) - \mathbf{x}_u\cdot\mathbf{x}_{uv} = F_u - \tfrac12 E_v (using xu⋅xuv=12∂v(xu⋅xu)=12Ev\mathbf{x}_u\cdot\mathbf{x}_{uv} = \tfrac12\partial_v(\mathbf{x}_u\cdot\mathbf{x}_u) = \tfrac12 E_v). This gives the linear system

E Γ111+F Γ112=12Eu,F Γ111+G Γ112=Fu−12Ev.E\,\Gamma^1_{11} + F\,\Gamma^2_{11} = \tfrac12 E_u, \qquad F\,\Gamma^1_{11} + G\,\Gamma^2_{11} = F_u - \tfrac12 E_v.

Since EG−F2>0EG - F^2 > 0 (regularity), this 2×22\times 2 system is invertible, and solving by Cramer's rule yields exactly the formulas stated for Γ111,Γ112\Gamma^1_{11}, \Gamma^2_{11}. The formulas for Γ12k\Gamma^k_{12} and Γ22k\Gamma^k_{22} follow identically from dotting xuv\mathbf{x}_{uv} and xvv\mathbf{x}_{vv} with xu,xv\mathbf{x}_u, \mathbf{x}_v and using the analogous identities xuv⋅xu=12Ev\mathbf{x}_{uv}\cdot \mathbf{x}_u = \tfrac12 E_v, xuv⋅xv=12Gu\mathbf{x}_{uv}\cdot\mathbf{x}_v = \tfrac12 G_u, xvv⋅xu=Fv−12Gu\mathbf{x}_{vv}\cdot\mathbf{x}_u = F_v - \tfrac12 G_u, xvv⋅xv=12Gv\mathbf{x}_{vv}\cdot\mathbf{x}_v = \tfrac12 G_v. This proves the Γijk\Gamma^k_{ij} depend only on E,F,GE, F, G and their first partials — an intrinsic fact. Substituting the decomposition of xuu,xuv,xvv\mathbf{x}_{uu}, \mathbf{x}_{uv}, \mathbf{x}_{vv} back into the expression for α′′(t)\alpha''(t) and collecting tangential terms, the tangential component of α′′\alpha'' is

(u′′+Γ111(u′)2+2Γ121u′v′+Γ221(v′)2)xu+(v′′+Γ112(u′)2+2Γ122u′v′+Γ222(v′)2)xv.\Big(u'' + \Gamma^1_{11}(u')^2 + 2\Gamma^1_{12}u'v' + \Gamma^1_{22}(v')^2\Big)\mathbf{x}_u + \Big(v'' + \Gamma^2_{11}(u')^2 + 2\Gamma^2_{12}u'v' + \Gamma^2_{22}(v')^2\Big)\mathbf{x}_v.

Since xu,xv\mathbf{x}_u, \mathbf{x}_v are linearly independent, this vanishes identically if and only if both coefficients vanish, which is precisely the stated geodesic equation system. This proves (a).

(b) Existence and uniqueness. Introduce p=u′p = u', q=v′q = v'. The geodesic system becomes the first-order system in (u,v,p,q)(u, v, p, q):

u′=p,v′=q,p′=−Γ111p2−2Γ121pq−Γ221q2,q′=−Γ112p2−2Γ122pq−Γ222q2.u' = p, \quad v' = q, \quad p' = -\Gamma^1_{11}p^2 - 2\Gamma^1_{12}pq - \Gamma^1_{22}q^2, \quad q' = -\Gamma^2_{11}p^2 - 2\Gamma^2_{12}pq - \Gamma^2_{22}q^2.

The right-hand side is a C∞C^\infty (in particular, locally Lipschitz) function of (u,v,p,q)(u,v,p,q) on the open set where (u,v)∈U(u,v) \in U, since the Γijk\Gamma^k_{ij} are smooth functions of E,F,GE, F, G and their derivatives, which are themselves smooth by the regularity of x\mathbf{x}. By the Picard–Lindelöf theorem, for any initial data (u0,v0,p0,q0)(u_0, v_0, p_0, q_0) there exists ε>0\varepsilon > 0 and a unique C2C^2 solution (u(t),v(t))(u(t), v(t)) on (−ε,ε)(-\varepsilon, \varepsilon) with u(0)=u0u(0) = u_0, v(0)=v0v(0) = v_0, u′(0)=p0u'(0) = p_0, v′(0)=q0v'(0) = q_0. Given p∈Sp \in S with coordinates (u0,v0)(u_0,v_0) and w=a xu+b xv∈TpSw = a\,\mathbf{x}_u + b\,\mathbf{x}_v \in T_pS, take (p0,q0)=(a,b)(p_0, q_0) = (a,b); the resulting solution α(t)=x(u(t),v(t))\alpha(t) = \mathbf{x}(u(t),v(t)) is, by part (a), the unique geodesic with α(0)=p\alpha(0) = p, α′(0)=w\alpha'(0) = w. This proves (b).

(c) Constant speed. Let α\alpha be a geodesic, so α′′(t)\alpha''(t) is everywhere normal to SS, i.e. α′′(t)=λ(t)n(t)\alpha''(t) = \lambda(t)\mathbf{n}(t) for some scalar function λ\lambda. Then

ddt∣α′(t)∣2=2 α′(t)⋅α′′(t)=2λ(t) α′(t)⋅n(t)=0,\frac{d}{dt}|\alpha'(t)|^2 = 2\,\alpha'(t)\cdot \alpha''(t) = 2\lambda(t)\,\alpha'(t)\cdot \mathbf{n}(t) = 0,

since α′(t)∈Tα(t)S\alpha'(t) \in T_{\alpha(t)}S is orthogonal to n(t)\mathbf{n}(t). Hence ∣α′(t)∣2|\alpha'(t)|^2 is constant, so ∣α′(t)∣|\alpha'(t)| is constant. This proves (c).

(d) Local length-minimization. This is a first-variation argument. Fix a geodesic α:[a,b]→S\alpha: [a,b] \to S parametrized by arc length, so ∣α′∣≡1|\alpha'| \equiv 1 by (c). Fix t0∈(a,b)t_0 \in (a,b); by continuity of x\mathbf{x} and its inverse there is a coordinate neighborhood x(U)\mathbf{x}(U) around α(t0)\alpha(t_0) and δ>0\delta > 0 such that α([t0−δ,t0+δ])⊂x(U)\alpha([t_0-\delta, t_0+\delta]) \subset \mathbf{x}(U) and, shrinking δ\delta if necessary (using existence of geodesic normal coordinates, or more elementarily by continuity of solutions to the geodesic ODE on compact parameter sets), such that α∣[s1,s2]\alpha|_{[s_1,s_2]} for s1,s2∈[t0−δ,t0+δ]s_1, s_2 \in [t_0-\delta,t_0+\delta] is the unique geodesic segment joining its endpoints within x(U)\mathbf{x}(U). We show any piecewise-regular curve β:[s1,s2]→S\beta : [s_1,s_2]\to S joining α(s1)\alpha(s_1) to α(s2)\alpha(s_2) within this neighborhood has length L(β)≥L(α∣[s1,s2])=s2−s1L(\beta) \ge L(\alpha|_{[s_1,s_2]}) = s_2 - s_1, with equality iff β\beta is a reparametrization of α\alpha. Consider the energy functional for a curve γ:[s1,s2]→S\gamma : [s_1,s_2] \to S,

J(γ)=∫s1s2∣γ′(t)∣2 dt.J(\gamma) = \int_{s_1}^{s_2} |\gamma'(t)|^2\, dt.

By the Cauchy–Schwarz inequality applied to the functions 11 and ∣γ′(t)∣|\gamma'(t)| on [s1,s2][s_1,s_2],

L(γ)2=(∫s1s2∣γ′(t)∣ dt)2≤(s2−s1)∫s1s2∣γ′(t)∣2 dt=(s2−s1)J(γ),L(\gamma)^2 = \left(\int_{s_1}^{s_2} |\gamma'(t)|\,dt\right)^2 \le (s_2 - s_1)\int_{s_1}^{s_2}|\gamma'(t)|^2\,dt = (s_2-s_1) J(\gamma),

with equality if and only if ∣γ′(t)∣|\gamma'(t)| is constant on [s1,s2][s_1,s_2]. Thus it suffices to show α\alpha minimizes JJ among curves with the same endpoints and same parameter interval, and that any JJ-minimizer among constant-speed curves must itself have κg≡0\kappa_g \equiv 0; combined with Cauchy–Schwarz this yields the length-minimization with equality forcing reparametrization of a geodesic. To show α\alpha is a critical point of JJ (and in fact the minimizer, by the local uniqueness from part (b) together with a standard second-variation / Jacobi field argument establishing that sufficiently short geodesic segments have no conjugate points, hence are strict local minima — this is the content of the Jacobi field theory, which we invoke here as the mechanism making the critical point a genuine minimum rather than merely a critical point): let γr(t)=x(u(t,r),v(t,r))\gamma_r(t) = \mathbf{x}(u(t,r), v(t,r)) for r∈(−η,η)r \in (-\eta,\eta) be a smooth variation of α=γ0\alpha = \gamma_0 with fixed endpoints, u(si,r)=u(si,0)u(s_i,r) = u(s_i,0), v(si,r)=v(si,0)v(s_i,r) = v(s_i,0) for i=1,2i=1,2 and all rr. Then

J(γr)=∫s1s2(Eut2+2Futvt+Gvt2) dt,J(\gamma_r) = \int_{s_1}^{s_2} \Big(E u_t^2 + 2F u_t v_t + G v_t^2\Big)\, dt,

where subscripts denote partial derivatives and E,F,GE,F,G are evaluated along (u(t,r),v(t,r))(u(t,r),v(t,r)). Differentiating under the integral sign at r=0r=0 and integrating by parts in tt (the boundary terms vanish since the variation fixes endpoints), a direct computation — using the same identities for ∂E/∂u\partial E/\partial u, etc., as in part (a) — shows

ddr∣r=0J(γr)=−2∫s1s2[(u′′+Γ111(u′)2+2Γ121u′v′+Γ221(v′)2) ξ+(v′′+Γ112(u′)2+2Γ122u′v′+Γ222(v′)2) η]dt,\frac{d}{dr}\Big|_{r=0} J(\gamma_r) = -2\int_{s_1}^{s_2} \Big[\big(u'' + \Gamma^1_{11}(u')^2 + 2\Gamma^1_{12}u'v' + \Gamma^1_{22}(v')^2\big)\, \xi + \big(v'' + \Gamma^2_{11}(u')^2 + 2\Gamma^2_{12}u'v' + \Gamma^2_{22}(v')^2\big)\, \eta \Big] dt,

where ξ=∂u/∂r∣r=0\xi = \partial u/\partial r|_{r=0}, η=∂v/∂r∣r=0\eta = \partial v/\partial r|_{r=0} are the (arbitrary, endpoint-vanishing) components of the variation field. Since α\alpha satisfies the geodesic equations from part (a), both bracketed expressions vanish identically, so ddr∣r=0J(γr)=0\frac{d}{dr}\big|_{r=0}J(\gamma_r) = 0 for every variation with fixed endpoints. Thus α\alpha is a critical point of JJ, and by the fundamental lemma of the calculus of variations, a curve is a critical point of JJ for all fixed-endpoint variations if and only if it satisfies the geodesic equations — so critical points of JJ are exactly the geodesics. Finally, restricting to a sufficiently short parameter interval around t0t_0 (which is possible by the smooth dependence of geodesics on initial conditions and the inverse function theorem applied to the exponential map, guaranteeing no conjugate points arise before length δ\delta), the second variation of JJ along α\alpha is positive definite on variations vanishing at the endpoints, so α\alpha is a strict local minimizer of JJ, hence of LL by the Cauchy–Schwarz argument above. This proves (d). ■\blacksquare

∎

Worked examples

Example 1: A great circle on the unit sphere is a geodesic, verified directly and via the geodesic equations. Parametrize the unit sphere by

x(θ,φ)=(sin⁡θcos⁡φ, sin⁡θsin⁡φ, cos⁡θ),θ∈(0,π), φ∈(0,2π),\mathbf{x}(\theta,\varphi) = (\sin\theta\cos\varphi,\ \sin\theta\sin\varphi,\ \cos\theta), \qquad \theta \in (0,\pi),\ \varphi \in (0,2\pi),

with θ\theta the polar angle from the north pole and φ\varphi the azimuthal angle. A direct computation gives

xθ=(cos⁡θcos⁡φ,cos⁡θsin⁡φ,−sin⁡θ),xφ=(−sin⁡θsin⁡φ,sin⁡θcos⁡φ,0),\mathbf{x}_\theta = (\cos\theta\cos\varphi, \cos\theta\sin\varphi, -\sin\theta), \qquad \mathbf{x}_\varphi = (-\sin\theta\sin\varphi, \sin\theta\cos\varphi, 0),

so

E=xθ⋅xθ=1,F=xθ⋅xφ=0,G=xφ⋅xφ=sin⁡2θ.E = \mathbf{x}_\theta\cdot\mathbf{x}_\theta = 1, \qquad F = \mathbf{x}_\theta\cdot\mathbf{x}_\varphi = 0, \qquad G = \mathbf{x}_\varphi \cdot \mathbf{x}_\varphi = \sin^2\theta.

Take the great circle through the point θ0=π/3\theta_0 = \pi/3, φ0=π/4\varphi_0 = \pi/4 tilted so that it is not the equator and not a meridian — a genuinely asymmetric case. Concretely, consider the great circle cut by the plane through the origin with unit normal

c=(12,0,12),\mathbf{c} = \Big(\tfrac{1}{\sqrt2}, 0, \tfrac{1}{\sqrt2}\Big),

i.e. the set {(x,y,z)∈S2:12x+12z=0}\{(x,y,z) \in S^2 : \tfrac{1}{\sqrt2}x + \tfrac{1}{\sqrt2}z = 0\}. Parametrize it by arc length: pick an orthonormal basis of c⊥\mathbf{c}^\perp, e.g.

e1=(0,1,0),e2=c×e1=(12,0,−12),\mathbf{e}_1 = (0,1,0), \qquad \mathbf{e}_2 = \mathbf{c}\times \mathbf{e}_1 = \Big(\tfrac{1}{\sqrt2}, 0, -\tfrac{1}{\sqrt2}\Big),

and set

α(s)=cos⁡s e1+sin⁡s e2=(sin⁡s2, cos⁡s, −sin⁡s2).\alpha(s) = \cos s\, \mathbf{e}_1 + \sin s\, \mathbf{e}_2 = \Big(\tfrac{\sin s}{\sqrt2},\ \cos s,\ -\tfrac{\sin s}{\sqrt2}\Big).

This is a unit-speed curve (check: α′(s)=(cos⁡s2,−sin⁡s,−cos⁡s2)\alpha'(s) = (\tfrac{\cos s}{\sqrt2}, -\sin s, -\tfrac{\cos s}{\sqrt2}), and ∣α′(s)∣2=12cos⁡2s+sin⁡2s+12cos⁡2s=cos⁡2s+sin⁡2s=1|\alpha'(s)|^2 = \tfrac12\cos^2 s + \sin^2 s + \tfrac12\cos^2 s = \cos^2 s + \sin^2 s = 1, confirmed) lying on S2S^2 (check: ∣α(s)∣2=12sin⁡2s+cos⁡2s+12sin⁡2s=1|\alpha(s)|^2 = \tfrac12\sin^2 s + \cos^2 s + \tfrac12 \sin^2 s = 1, confirmed). Compute the acceleration directly:

α′′(s)=(−sin⁡s2, −cos⁡s, sin⁡s2)=−α(s).\alpha''(s) = \Big(-\tfrac{\sin s}{\sqrt2},\ -\cos s,\ \tfrac{\sin s}{\sqrt2}\Big) = -\alpha(s).

On the unit sphere the outward unit normal at the point α(s)\alpha(s) is n(s)=α(s)\mathbf{n}(s) = \alpha(s) itself. So α′′(s)=−n(s)\alpha''(s) = -\mathbf{n}(s), which is entirely normal to the sphere — its tangential component is exactly zero at every ss. Hence κg≡0\kappa_g \equiv 0 and α\alpha is a geodesic, and moreover κn≡−1\kappa_n \equiv -1 (constant), consistent with a great circle having normal curvature equal in magnitude to the sphere's curvature 11. This matches the abstract theorem: solving the geodesic equations for S2S^2 (using the computed E=1,F=0,G=sin⁡2θE=1, F=0, G=\sin^2\theta) gives the Christoffel symbols

Γ111=0, Γ121=0, Γ221=−sin⁡θcos⁡θ, Γ112=0, Γ122=cot⁡θ, Γ222=0,\Gamma^1_{11}=0,\ \Gamma^1_{12}=0,\ \Gamma^1_{22} = -\sin\theta\cos\theta,\ \Gamma^2_{11}=0,\ \Gamma^2_{12} = \cot\theta,\ \Gamma^2_{22}=0,

and the system θ′′−sin⁡θcos⁡θ (φ′)2=0\theta'' - \sin\theta\cos\theta\,(\varphi')^2 = 0, φ′′+2cot⁡θ θ′φ′=0\varphi'' + 2\cot\theta\,\theta'\varphi' = 0 is satisfied by the coordinate functions θ(s),φ(s)\theta(s), \varphi(s) of our explicit great circle (this can be checked by substituting cos⁡θ(s)=−sin⁡s2\cos\theta(s) = -\frac{\sin s}{\sqrt2}, obtained from the zz-coordinate, and differentiating, though the direct extrinsic computation above already suffices to prove geodesy).

Example 2: The helix on a circular cylinder, and why it is a geodesic despite visibly curving in space. Let the cylinder of radius R=2R = 2 be parametrized by

x(u,v)=(2cos⁡u, 2sin⁡u, v),u∈(0,2π), v∈R.\mathbf{x}(u,v) = (2\cos u,\ 2\sin u,\ v), \qquad u \in (0,2\pi),\ v \in \mathbb{R}.

Then xu=(−2sin⁡u,2cos⁡u,0)\mathbf{x}_u = (-2\sin u, 2\cos u, 0), xv=(0,0,1)\mathbf{x}_v = (0,0,1), giving

E=4,F=0,G=1,E = 4, \qquad F = 0, \qquad G = 1,

constants. Since E,F,GE,F,G are constant, all Christoffel symbols vanish (each involves only first derivatives of E,F,GE,F,G, which are all zero). The geodesic equations reduce to

u′′=0,v′′=0,u'' = 0, \qquad v'' = 0,

so every geodesic on this cylinder has the form u(t)=at+bu(t) = at+b, v(t)=ct+dv(t) = ct+d for constants a,b,c,da,b,c,d — a coordinate-linear curve. Take the asymmetric case a=3,b=π6,c=5,d=−1a = 3, b = \tfrac{\pi}{6}, c = 5, d = -1 (an arbitrary, non-axis-aligned, non-unit-slope helix):

α(t)=x(3t+π6, 5t−1)=(2cos⁡(3t+π6), 2sin⁡(3t+π6), 5t−1).\alpha(t) = \mathbf{x}(3t + \tfrac{\pi}{6},\ 5t - 1) = \big(2\cos(3t+\tfrac\pi6),\ 2\sin(3t + \tfrac\pi6),\ 5t-1\big).

We verify geodesy directly. Compute

α′(t)=(−6sin⁡(3t+π6), 6cos⁡(3t+π6), 5),α′′(t)=(−18cos⁡(3t+π6), −18sin⁡(3t+π6), 0).\alpha'(t) = \big(-6\sin(3t+\tfrac\pi6),\ 6\cos(3t+\tfrac\pi6),\ 5\big), \qquad \alpha''(t) = \big(-18\cos(3t+\tfrac\pi6),\ -18\sin(3t+\tfrac\pi6),\ 0\big).

The cylinder's outward unit normal at α(t)\alpha(t) is n(t)=(cos⁡(3t+π6),sin⁡(3t+π6),0)\mathbf{n}(t) = \big(\cos(3t+\tfrac\pi6), \sin(3t+\tfrac\pi6), 0\big) (the radial direction in the xyxy-plane). Observe

α′′(t)=−18 n(t),\alpha''(t) = -18\,\mathbf{n}(t),

which is purely normal — zero tangential component. Hence α\alpha is a geodesic (its speed ∣α′(t)∣=36+25=61|\alpha'(t)| = \sqrt{36+25} = \sqrt{61} is constant, consistent with part (c) of the theorem, though this is not an arc-length parametrization since 61≠1\sqrt{61}\ne 1; reparametrizing by s=61 ts = \sqrt{61}\,t gives the arc-length geodesic). Geometrically this makes sense: unrolling the cylinder onto the plane (an isometry, since E,F,GE,F,G match the flat metric du2↦4 du2du^2 \mapsto 4\,du^2 up to relabeling, or more directly, the map (u,v)↦(2u,v)(u,v)\mapsto(2u,v) is a local isometry onto the plane with the flat metric) sends the helix to the straight line 2u=6t+π32u = 6t + \tfrac\pi3, v=5t−1v = 5t-1 in the plane, confirming that a helix — which visibly curves in R3\mathbb{R}^3 — is nonetheless intrinsically straight, because the cylinder's intrinsic (first-fundamental-form) geometry is identical to that of a flat sheet of paper rolled up without stretching.

CounterexampleDropping arc-length / constant-speed reduces the equations to nonsense, and dropping regularity breaks existence

The theorem's part (a)–(b) implicitly requires the parametrization x\mathbf{x} to be regular, i.e. EG−F2>0EG - F^2 > 0 everywhere on UU, precisely so the Christoffel symbols are well-defined (the 2×22\times 2 linear system solved in the proof requires EG−F2≠0EG-F^2 \ne 0 to invert) and so Picard–Lindelöf applies to a genuine, non-degenerate ODE system. Consider the "parametrization" of the plane by

x(u,v)=(u3,v,0),(u,v)∈R2.\mathbf{x}(u,v) = (u^3, v, 0), \qquad (u,v) \in \mathbb{R}^2.

Here xu=(3u2,0,0)\mathbf{x}_u = (3u^2, 0, 0), xv=(0,1,0)\mathbf{x}_v = (0,1,0), so E=9u4E = 9u^4, F=0F=0, G=1G=1, and EG−F2=9u4EG - F^2 = 9u^4, which vanishes at u=0u=0: x\mathbf{x} fails to be a regular parametrization along the entire line u=0u=0 (there, xu=0\mathbf{x}_u = 0, so the map is not an immersion). Attempt to run the proof of part (a) at u=0u = 0: the formula

Γ111=GEu−2FFu+FEv2(EG−F2)=1⋅36u3−0+02⋅9u4=2u\Gamma^1_{11} = \frac{GE_u - 2FF_u + FE_v}{2(EG-F^2)} = \frac{1\cdot 36u^3 - 0 + 0}{2\cdot 9u^4} = \frac{2}{u}

blows up as u→0u \to 0. The geodesic ODE system is therefore singular along u=0u=0, Picard–Lindelöf's Lipschitz hypothesis fails there, and indeed existence and uniqueness of geodesics through points with u0=0u_0 = 0 genuinely fail for this ODE in these coordinates: for instance, both the constant curve u(t)≡0u(t) \equiv 0 and (after a short computation solving u′′=−2u(u′)2u'' = -\frac{2}{u}(u')^2 away from u=0u=0, then examining limiting behavior) certain non-constant solutions can be made to agree to first order at u=0u=0 with u′(0)=0u'(0)=0, illustrating the loss of uniqueness that the regularity hypothesis is designed to prevent. Of course the underlying surface here is just the plane R2\mathbb{R}^2 (via (x,y)=(u3,v)(x,y) = (u^3,v) away from u=0u=0), whose true geodesics are ordinary straight lines; the pathology is an artifact of choosing an irregular parametrization, which is exactly why the hypothesis EG−F2>0EG-F^2>0 is non-negotiable in the theorem — it is a statement about the parametrization used to compute the equations, not merely a cosmetic simplification.

Scope, edge cases & common traps

SituationWhat holdsWhat can fail
Compact surface (e.g. sphere)Geodesics extend for all time (Hopf–Rinow, stated beyond this article's scope)Geodesics need not be globally length-minimizing past a certain length (e.g. two antipodal points on S2S^2 are joined by infinitely many minimizing great-circle arcs, and any great circle arc longer than half the circle is not minimizing)
Non-simply-connected or incomplete surfaceLocal existence/uniqueness (Theorem b) still holdsGlobal uniqueness of geodesic between two points fails; a geodesic may not be extendable to all of R\mathbb{R} (e.g. an open disk with the boundary circle removed)
Straight line on a planeGeodesic with κg≡0\kappa_g \equiv 0, κn≡0\kappa_n \equiv 0—
Curve of latitude on sphere (not equator)Has constant κn\kappa_n, but κg≢0\kappa_g \not\equiv 0Not a geodesic — a classic trap, since it looks "circular and symmetric" like the equator
Parametrization singular (EG−F2=0EG-F^2=0)—Christoffel symbols undefined/blow up; existence–uniqueness can fail, as shown above

The mathematician behind it

Origins · 1 minCarl Friedrich Gauss and how this was discovered
CF
Carl Friedrich Gauss
1777–1855 · Göttingen, Kingdom of Hanover

Gauss's 1827 treatise Disquisitiones generales circa superficies curvas ("General Investigations of Curved Surfaces") founded intrinsic differential geometry: the idea that a surface's geometry — including its geodesics — can be studied entirely through measurements made within the surface, via what we now call the first fundamental form, without reference to how the surface sits in space. Gauss developed this theory partly out of his work on geodetic surveying of the Kingdom of Hanover, where "geodesic" originally meant the literal shortest path measured across the curved Earth, and partly out of pure mathematical curiosity about curvature, culminating in his Theorema Egregium (the intrinsic nature of Gaussian curvature), which sets the stage for the Gauss–Bonnet theorem.

History
- 1827: Gauss publishes Disquisitiones generales circa superficies curvas, introducing the first fundamental form and intrinsic surface geometry.
- 1830s–1840s: Ferdinand Minding and others develop the theory of geodesics on surfaces of constant curvature, connecting them to non-Euclidean geometry.
- 1848: Pierre Ossian Bonnet proves the local Gauss–Bonnet formula, relating geodesic curvature of a boundary curve to total Gaussian curvature enclosed.
- 1870s: Elwin Bruno Christoffel introduces the symbols Γijk\Gamma^k_{ij} in the context of quadratic differential forms, later adopted throughout Riemannian geometry.
- 1900s: Tullio Levi-Civita and Gregorio Ricci-Curbastro generalize geodesics to arbitrary Riemannian manifolds via parallel transport and covariant differentiation, the modern language in which the geodesic equation is usually first taught.
Is every locally length-minimizing curve a geodesic, and conversely?

Yes to both, under the right local hypotheses. The theorem shows geodesics are locally length-minimizing (part d), and the converse also holds: if α\alpha is parametrized by arc length and locally minimizes length among all nearby fixed-endpoint curves, then it must be a critical point of the energy functional JJ on short intervals (since local minimizers of length that are constant-speed are automatically local minimizers of energy, by the Cauchy–Schwarz equality analysis in the proof), hence satisfies the geodesic equations by the fundamental lemma of the calculus of variations. The subtlety is "local": a geodesic segment need not be globally minimizing once it grows long enough, as the antipodal great-circle-arc example on the sphere shows.

Why do the Christoffel symbols only depend on the first fundamental form, not on how the surface sits in R3\mathbb{R}^3?

This is proved directly in the derivation: the linear system pinning down Γ111,Γ112\Gamma^1_{11}, \Gamma^2_{11} (and similarly for the other indices) involves only EE, FF, GG, and their first partial derivatives, arising from identities like xuu⋅xu=12Eu\mathbf{x}_{uu}\cdot\mathbf{x}_u = \tfrac12 E_u. No second fundamental form coefficients L,M,NL, M, N (which encode extrinsic bending) appear. This is the algebraic seed of Gauss's Theorema Egregium: because geodesics, hence the Christoffel symbols, are built purely from E,F,GE,F,G, two surfaces with the same first fundamental form under a diffeomorphism (an isometry) have corresponding geodesics, even if one is flat and the other is bent into space, as with the plane and the cylinder.

Do geodesics always exist for all time, or can they "run off the edge" of a surface?

Local existence (Theorem part b) only guarantees a geodesic on some small interval (−ε,ε)(-\varepsilon,\varepsilon). Whether it extends to all of R\mathbb{R} depends on completeness of the surface: on a compact surface without boundary (like a sphere or torus), geodesics extend indefinitely, a fact governed by the Hopf–Rinow theorem (beyond this article's scope). On an incomplete surface, such as an open disk or a surface with a puncture, a geodesic can indeed reach the boundary or the missing point in finite parameter time and fail to extend further, exactly as a straight line in the open unit disk (with the flat metric) exits through the boundary circle in finite time.

Are geodesics unique between two given points?

Not globally, only locally. Locally (Theorem part b), a geodesic is uniquely determined by a point and an initial velocity, which pins down a unique geodesic — but that does not immediately give uniqueness between two fixed points, since different initial velocities at the first point can lead to different geodesics that both pass through the second point later. The clearest counterexample is two antipodal points on a sphere, connected by infinitely many great-circle arcs (every great circle through both poles works), all geodesics, all of the same minimal length π\pi (in the unit sphere). Uniqueness between two points is a delicate global question, generally false without further hypotheses like negative curvature or small enough separation.

How does geodesic curvature relate to the Gauss-Bonnet theorem?

The local Gauss–Bonnet formula states that for a region RR on a surface bounded by a piecewise-smooth curve ∂R\partial R with exterior angles θi\theta_i at the corners,

∫∂Rκg ds+∬RK dA+∑iθi=2π,\int_{\partial R} \kappa_g \, ds + \iint_R K \, dA + \sum_i \theta_i = 2\pi,

where KK is Gaussian curvature. Geodesic curvature is exactly the boundary integrand that "pays for" the enclosed curvature; when ∂R\partial R is itself a union of geodesic arcs, κg≡0\kappa_g \equiv 0 along it, so the formula reduces to a pure statement about enclosed curvature and corner angles — this is precisely how geodesic triangles on a sphere are shown to have angle sums exceeding π\pi, foreshadowing the full Gauss-Bonnet Theorem.

Does a geodesic always have zero normal curvature too?

No — this is a common confusion. A geodesic has κg≡0\kappa_g \equiv 0, but κn\kappa_n is generally nonzero and is entirely determined by the surface's second fundamental form and the curve's direction; it need not vanish. On the unit sphere, every great circle geodesic has constant normal curvature κn=−1\kappa_n = -1 (as computed in Worked Example 1), reflecting the sphere's own curvature, not any property special to geodesics. A curve with both κg≡0\kappa_g \equiv 0 and κn≡0\kappa_n \equiv 0 would have to be a straight line, which only lies on a surface where the surface itself is flat in that direction.

Why does the parametrization need to be regular (EG−F2≠0EG - F^2 \ne 0) for the geodesic equations to make sense?

The Christoffel symbol formulas all have 2(EG−F2)2(EG-F^2) in the denominator, arising from inverting the 2×22\times2 linear system built from xu,xv\mathbf{x}_u, \mathbf{x}_v decompositions. If EG−F2=0EG - F^2 = 0 at some point, xu\mathbf{x}_u and xv\mathbf{x}_v become linearly dependent there (the parametrization is not an immersion), the basis {xu,xv,n}\{\mathbf{x}_u,\mathbf{x}_v,\mathbf{n}\} degenerates, and the Christoffel symbols are undefined or blow up, as demonstrated in the counterexample above with x(u,v)=(u3,v,0)\mathbf{x}(u,v)=(u^3,v,0). This is purely a defect of the chosen coordinates, not of the underlying surface, but it shows the hypothesis is essential for the proof method, even when the surface itself is perfectly smooth.

Before this

Where this leads