Definition:Acceleration of Smooth Curve on Smooth Manifold

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 $D_t$ be the covariant derivative along $\gamma$.

Then $D_t \gamma'$ is called the acceleration of $\gamma$ on $M$.