Geodesics — Mathematics Knowledge System

Overview

A geodesic on a Riemannian manifold (M, g) is a curve that locally minimises length — the generalisation of a straight line to curved spaces. Equivalently, geodesics are the curves whose acceleration (in the sense of the Levi-Civita connection) is zero: they are the 'straightest possible' curves. On the sphere, great circles are geodesics. In general relativity, free-falling massive objects travel along geodesics of spacetime. The geodesic equation is a second-order ODE whose coefficients are the Christoffel symbols of the metric.

Intuition

On a flat surface, the shortest path between two points is a straight line. On a curved surface, the shortest path bends to follow the curvature. Imagine stretching a rubber band taut between two points on a sphere: it snaps to a great circle arc. A geodesic is the generalisation of this idea to any Riemannian manifold. Locally it always looks straight (zero acceleration), even if globally it curves to follow the geometry of the space.

Formal Definition

Definition

A geodesic is a smooth curve γ: I → M satisfying the geodesic equation, which says the covariant derivative of the velocity vector along γ is zero. In local coordinates, this becomes a second-order ODE involving the Christoffel symbols Γᵢⱼᵏ of the Levi-Civita connection.

\nabla_{\gamma'} \gamma' = 0

The velocity vector γ' is parallel transported along γ

Geodesic equation (coordinate-free)

\frac{d^2 x^k}{dt^2} + \Gamma^k_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt} = 0

Geodesic equation in local coordinates

\Gamma^k_{ij} = \frac{1}{2} g^{kl}\left( \frac{\partial g_{il}}{\partial x^j} + \frac{\partial g_{jl}}{\partial x^i} - \frac{\partial g_{ij}}{\partial x^l} \right)

Christoffel symbols of the Levi-Civita connection

L(\gamma) = \int_a^b \sqrt{g_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt}}\, dt

Length functional whose critical points are geodesics

Properties

Constant speed

g(\gamma'(t), \gamma'(t)) = \text{const along any geodesic}

Geodesics on S² are great circles

\text{The geodesics of } S^n \text{ with the round metric are great circles: intersections of } S^n \text{ with planes through the origin.}

Geodesics on flat ℝⁿ are straight lines

\Gamma^k_{ij} = 0 \text{ in Cartesian coordinates on } \mathbb{R}^n, \text{ so the geodesic equation is } \ddot{x}^k = 0.

Theorems

Theorem 1: Existence and Uniqueness of Geodesics

\text{For every } p \in M \text{ and every } v \in T_pM, \text{ there exists a unique maximal geodesic } \gamma_v \text{ with } \gamma_v(0) = p, \gamma_v'(0) = v.

Theorem 2: Geodesics are locally length-minimising

\text{Every geodesic segment is locally the shortest curve between its endpoints.}

Theorem 3: Exponential Map

\exp_p: T_pM \to M, \quad \exp_p(v) = \gamma_v(1) \text{ is a local diffeomorphism near } 0.

Worked Examples

1
The flat torus is a quotient of ℝ² by integer translations. The metric is the Euclidean metric.
$g = dx^2 + dy^2$
2
All Christoffel symbols vanish (Euclidean metric), so the geodesic equation is ẍ = 0, ÿ = 0.
$\ddot{x} = 0,\quad \ddot{y} = 0$
3
Solutions are straight lines in ℝ²: (x(t), y(t)) = (x₀ + at, y₀ + bt).
4
On T², these project to curves with slope b/a. If b/a is rational, the geodesic is a closed curve; if irrational, it is dense in T².

✓ Answer

Geodesics on T² are projections of straight lines in ℝ². Those with rational slope are closed; those with irrational slope are everywhere dense in T².

Practice Problems

Mediumfree response

What are the geodesics of the hyperbolic plane H² with metric g = (dx² + dy²)/y²?

Hardproof writing

Prove that the speed |γ'(t)|_g is constant along any geodesic.

Common Mistakes

Common Mistake

All geodesics are globally length-minimising

Geodesics minimise length only locally. On S², the short arc of a great circle minimises length, but the long arc (going the other way around) is also a geodesic yet not a minimiser.

Common Mistake

The Christoffel symbols are tensors

Christoffel symbols Γᵏᵢⱼ transform inhomogeneously under coordinate changes; they are connection coefficients, not tensors.

Quiz

Historical Background

The term 'geodesic' comes from the Greek geodaisia (earth measurement), reflecting the surveyor's problem of finding shortest paths on the Earth. Gauss studied geodesics on surfaces in the 1820s. Riemann generalised this to arbitrary dimensions in 1854. Christoffel introduced the connection coefficients (Christoffel symbols) in 1869 to compute geodesics. The modern variational treatment — geodesics as stationary points of the length functional — unified the subject with the calculus of variations.

1827
Gauss studies geodesics on surfaces in Disquisitiones Generales circa Superficies Curvas
Carl Friedrich Gauss
1854
Riemann generalises geodesics to n-dimensional curved spaces
Bernhard Riemann
1869
Christoffel introduces Christoffel symbols, enabling explicit geodesic equations
Elwin Bruno Christoffel
1915
Einstein uses geodesics of spacetime to model free fall in general relativity
Albert Einstein

Summary

A geodesic is a curve γ satisfying ∇_{γ'} γ' = 0 — its velocity is parallel along itself.
In coordinates, this becomes ẍᵏ + Γᵏᵢⱼ ẋⁱ ẋʲ = 0, a second-order ODE determined by the Christoffel symbols.
Geodesics on S² are great circles; on flat ℝⁿ they are straight lines; on H² they are half-circles and vertical lines.
Speed is constant along geodesics; they locally minimise length but need not do so globally.
The exponential map exp_p: T_pM → M encodes the geodesic flow and is a local diffeomorphism near 0.

References

Bookdo Carmo, M. P. — Riemannian Geometry, Birkhäuser, 1992, Chapter 3
BookLee, J. M. — Riemannian Manifolds: An Introduction to Curvature, Springer, 1997, Chapter 4
WebsiteWikipedia — Geodesic
WebsiteWolfram MathWorld — Geodesic