Geodesic

Definition

A geodesic is a curve representing the shortest path between two points in a Riemannian Manifold.

Suppose a Riemannian Manifold $(\mathcal{M},g)$. A differentiable curve in $\mathcal{M}$ is defined as a smooth mapping $\gamma :\mathbb{R}\to \mathcal{M}$ from an open interval $({\lambda }_0,{\lambda }_1)\subset \mathbb{R}$ into $\mathcal{M}$. The distance of a smooth curve is given by Arc Length $L(\gamma )={\int }_{{\lambda }_0}^{{\lambda }_1}\sqrt{g_{\gamma (t)}({\gamma }^{\prime }(t),{\gamma }^{\prime }(t))}dt$ The distance between two points $p$ and $q$ on $\mathcal{M}$ is defined as the Infimum of the length taken over all differentiable curves $\gamma :({\lambda }_0,{\lambda }_1)\to \mathcal{M}$ such that $\gamma ({\lambda }_0)=p$ and $\gamma ({\lambda }_1)=q$. $d^{\mathcal{M}(p,q)}={\inf }_{\gamma \in C(p,q)}L(\gamma )$ where $C(p,q)$ is the set of all differentiable curves in $\mathcal{M}$ that join up the points $p$ and $q$.