In vector analysis/calculus, where we operated in $R^3$, the curl of a vector field is just the cross product of the del operator(or just the nabla, $\nabla$) with that vector field, $\vec{\nabla} \times \vec{v}$.
Now, on a smooth three-dimensional manifold, $M$, we define the affine connection as a mapping having three properties that come from the properties that the del operator from vector calculus satisfies. Namely:
$1) \nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ$
$2) \nabla_X(Y+Z)=\nabla_XY+\nabla_XZ$
$3) \nabla_X(fY)=X(f)Y+f\nabla_XY$
for $X, Y, Z\in \mathfrak{X}(M)$ and $f, g$ are $C^{\infty}$ on $M$.
So, extending the concept of the curl of a vector field from vector calculus, how is the curl of a vector field defined on smooth manifolds and what changes in the geometric interpretation of it in relation with its vector calculus analog?