Definition:Geodesic

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 $\nabla$).

This article is complete as far as it goes, but it could do with expansion. In particular: add an "informal" definition to the effect that a geodesic is an arc on a surface between two points which is the shortest curve between those two points, as this brings it into the context that is generally understood by the non-mathematical reader I think an elementary definition for the geodesic on a surface $S \subset \R^3$ is wished. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Results about geodesics can be found here.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 4$: Connections. Covariant Derivatives Along Curves