Definition:Covariant Derivative

Definition
Let $M$ be a smooth manifold with or without boundary.

Let $E$ be a smooth manifold.

Let $\pi : E \to M$ be a smooth vector bundle.

Let $\map \Gamma E$ be the space of smooth sections of $E$.

Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields on $M$.

Let $\map {C^\infty} M$ be the space of smooth real functions on $M$.

Let $\nabla : \map {\mathfrak{X}} M \times \map \Gamma E \to \map \Gamma E$ be the connection, written $\tuple {X, Y} \mapsto \nabla_X Y$ where $X \in \map {\mathfrak{X}} M$, $Y \in \map \Gamma E$, and $\times$ denotes the cartesian product.

Then $\nabla_X Y$ is known as the covariant derivative of $Y$ in the direction $X$.