📂Geometry

Geodesics on a Differentiable Manifold

Definition¹

On the manifold $M$, for a curve $\gamma : I \to M$ at point $t_{0} \in I$, if $\dfrac{D}{dt}\left( \dfrac{d \gamma}{d t} \right) = 0$, then $\gamma$ is a geodesic at $t_{0}$. If for all points $t \in I$, $\gamma$ is a geodesic at $t$, then $\gamma$ is called a geodesic.

If $[a,b] \subset I$ and $\gamma : I \to M$ is a geodesic, a contraction mapping $\gamma|_{[a,b]}$ connecting $\gamma (a)$ to $\gamma (b)$ is called the geodesic segment joining $\gamma (a)$ to $\gamma (b)$

Explanation

By abusing the name, the image $\gamma (I)$ of $\gamma$ is also referred to as a geodesic.

The following theorem describes the necessary and sufficient condition for $\gamma$ to be a geodesic, which is consistent with results from differential geometry at $\mathbb{R}^{3}$.

Theorem

The necessary and sufficient condition for $\gamma$ to be a geodesic is as follows:

$$ \gamma \text{ is geodesic.} \iff \dfrac{d^{2} \gamma^{k}}{d t} + \Gamma_{ij}^{k}\dfrac{d \gamma^{i}}{d t} \dfrac{d \gamma^{j}}{d t}\quad \forall k $$