Definition:Covariant Derivative along Smooth Curve

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

Let $TM$ be the tangent bundle of $M$.

Let $\nabla$ be a connection in $TM$.

Let $I \subseteq \R$ be a real interval.

Let $\gamma : I \to M$ be a smooth curve.

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

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

Let $D_t : \map {\mathfrak{X}} \gamma \to \map {\mathfrak{X}} \gamma$ be an operator such that $\forall a, b \in \R$, $\forall V, W \in \map {\mathfrak{X}} \gamma$ and $\forall f \in \map {C^\infty} I$ it holds that:


 * $\map {D_t} {a V + b W} = a D_t V + b D_t W$


 * $\map {D_t} {f V} = f' V + f D_t V$

and if $V$ is extendible, then for any extension $\tilde V$:


 * $D_t \map V t = \nabla_{\map {\gamma'} t} \tilde V$

Then $D_t$ is called the covariant derivative along a smooth curve.