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 $\gamma$ $\nabla$.

Suppose:


 * $\forall t \in I : D_t \gamma' = 0$.

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