Geodesics - Core Definitions

Geodesics are the curves that generalize straight lines to curved spaces. They are characterized as curves with zero acceleration, or equivalently, as locally length-minimizing curves.

Definition and the Geodesic Equation

Definition8.1Geodesic

A smooth curve $\gamma: I \to M$ on a Riemannian manifold $(M, g)$ is a geodesic if its velocity is parallel along itself:

\nabla_{\gamma'(t)} \gamma'(t) = 0,

where $\nabla$ is the Levi-Civita connection. In local coordinates $(x^1, \ldots, x^n)$ , writing $\gamma(t) = (x^1(t), \ldots, x^n(t))$ , the geodesic equation becomes

\ddot{x}^k + \Gamma^k_{ij} \dot{x}^i \dot{x}^j = 0, \quad k = 1, \ldots, n.

RemarkConstant speed

A geodesic has constant speed: $\frac{d}{dt}|\gamma'(t)|^2 = 2g(\nabla_{\gamma'}\gamma', \gamma') = 0$ . So $|\gamma'|$ is constant along any geodesic. A geodesic parametrized with $|\gamma'| = 1$ is a unit-speed geodesic.

The Exponential Map

Definition8.2Exponential map

For $p \in M$ , the exponential map $\exp_p: T_pM \supset U \to M$ is defined by $\exp_p(v) = \gamma_v(1)$ , where $\gamma_v$ is the geodesic with $\gamma_v(0) = p$ and $\gamma_v'(0) = v$ . By ODE existence/uniqueness, $\exp_p$ is defined on some neighborhood $U$ of the origin in $T_pM$ and is a smooth map.

Theorem8.1Local diffeomorphism

The differential $(d\exp_p)_0: T_0(T_pM) \cong T_pM \to T_pM$ is the identity. By the inverse function theorem, $\exp_p$ is a diffeomorphism from a neighborhood of $0 \in T_pM$ to a neighborhood of $p \in M$ .

Definition8.3Normal coordinates and injectivity radius

The coordinates on $M$ near $p$ obtained via $\exp_p$ and an orthonormal basis of $T_pM$ are Riemannian normal coordinates. In these coordinates, $g_{ij}(0) = \delta_{ij}$ , $\Gamma^k_{ij}(0) = 0$ , and geodesics through $p$ are straight lines.

The injectivity radius at $p$ is $\mathrm{inj}(p) = \sup\{r : \exp_p|_{B_r(0)}$ is a diffeomorphism $\}$ .

Geodesics as Variational Curves

Definition8.4Energy functional

The energy of a piecewise smooth curve $\gamma: [a,b] \to M$ is

E(\gamma) = \frac{1}{2}\int_a^b |\gamma'(t)|^2 \, dt.

The length is $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$ . By Cauchy-Schwarz, $L(\gamma)^2 \leq 2(b-a)E(\gamma)$ , with equality if and only if $|\gamma'|$ is constant.

Theorem8.2Geodesics as critical points

A curve $\gamma$ is a geodesic if and only if it is a critical point of the energy functional $E$ (equivalently, a constant-speed critical point of the length functional $L$ ) among curves with fixed endpoints.

ExampleGeodesics on model spaces

$\mathbb{R}^n$ : Straight lines $\gamma(t) = p + tv$ .
$S^n$ : Great circles (intersections of $S^n$ with 2-planes through the origin).
$\mathbb{H}^n$ (upper half-space model): Vertical lines and semicircles orthogonal to the boundary.
Torus $T^2$ (flat): Images of straight lines in $\mathbb{R}^2$ under the projection.

RemarkGeodesics need not minimize

A geodesic is a critical point of the energy (a "stationary path"), but not necessarily a minimizer. On $S^2$ , the long arc of a great circle between two non-antipodal points is a geodesic but not length-minimizing. A geodesic minimizes length only up to the first conjugate point.