Category:Geodesics

This category contains results about Geodesics.
Definitions specific to this category can be found in Definitions/Geodesics.

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 $\nabla$).