\begin{definition}[Definition:Curl Operator/Riemannian Manifold]
Let $\struct {M, g}$ be a Riemannian manifold.
Let $TM$ and $T^*M$ be the tangent bundle and the cotangent bundle of $M$ respectively.
Let $\wedge^2 T^*M$ be the subbundle of alternating tensors.
Let $\rd V_g$ be the Riemannian volume form.
Let $\beta : TM \to \wedge^2 T^*M$ be a mapping such that:
:$\map \beta X = X \lrcorner \rd V_g$
where $\lrcorner$ denotes the interior multiplication.
Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields of $M$.
Let $X \in \map {\mathfrak{X}} M$ be a smooth vector field.
Let $\flat$ denote the index lowering.
Then the '''curl operator''' is defined by:
:$\curl X := \beta^{-1} \map d {X^\flat}$
\end{definition}