> [!definition] > > Let $X$ be a smooth [[Manifold|manifold]] and $D: \vf(X)^2 \to \vf(X)$ be a [[Connection|connection]]. $D$ is a **covariant derivative** if for any [[Vector Field|vector fields]] $\xi, \eta \in \vf(X)$, > $ > D_\xi \eta - D_\eta \xi = [\xi, \eta] > $ > is their [[Bracket Product|bracket product]].