Definition:Geodesic

Definition

Informal Definition

Let $S$ be a surface.

A geodesic is an arc between two points $A$ and $B$ on $S$ which has the smallest arc length from $A$ and $B$ lying completely in $S$.

Formal Definition

Let $M$ be a smooth manifold with or without boundary.

Let $I \subseteq \R$ be a real interval.

Let $\gamma : I \to M$ be a smooth curve on $M$.

Let $\gamma'$ be the velocity of $\gamma$.

Let $\nabla$ be a connection on $M$.

Let $D_t$ be the covariant derivative along $\gamma$ with respect to $\nabla$.

Suppose:

$\forall t \in I : D_t \gamma' = 0$.

Then $\gamma$ is called the geodesic (with respect to $\nabla$).

Examples

Sphere

A geodesic on the surface $S$ of a sphere between points $A$ and $B$ on $S$ is the part of the great circle on which $A$ and $B$ both lie.

Also see

• Results about geodesics can be found here.