Helmholtz decomposition of a vector field on surface Does it make sense to do Helmholtz decomposition of a vector field defined on a surface or on a manifold? I am mostly interested in the surface case. I was trying to find a reference for this and found only a handful of them mostly from electromagnetics literature. e.g.  
Scharstein, Robert W. "Helmholtz decomposition of surface electric current in electromagnetic scattering problems." System Theory, 1991. Proceedings., Twenty-Third Southeastern Symposium on. IEEE.
On the related note, is there a well defined curl operator for a (tangential) vector field defined on a surface?
 A: The presence of a curl term in the Helmholtz decomposition is very much a three-dimensional phenomenon. While it is reasonable to define the curl of a vector field tangent to a surface, what you get is either a field normal to the sphere or a scalar field, so it's not going to help us decompose tangent vector fields.
The Hodge decomposition tells us that any vector field on a surface can be decomposed as $$ X = \nabla f + J  \nabla g + \sum_{i=1}^m c_iY_i$$ where $f,g$ are functions, $J$ is rotation by $\pi /2$ in the tangent plane, $c_i$ are some real coefficients determined by $X$ and the $Y_i$ are some finite collection of curl-free, divergence-free vector fields (independent of $X$) which describe the relevant features of the topology of the surface. 
Very roughly, there is one field $Y_i$ going around each independent "loop" in the surface - so a plane or a sphere would have none, an infinite cylinder would have one and a torus would have two. If you're interested in some search terms to learn the details: the $Y_i$ are vector fields dual to a basis of the space of harmonic one-forms, which provide representatives of every cohomology class.
In three-dimensional manifolds you get the more familiar
$$ X = \nabla f + \operatorname{curl} V +  \sum_{i=1}^m c_iY_i. $$
In higher dimensions, scalar fields and vector fields no longer suffice to describe the whole de Rham complex, so it's necessary to switch to talking about differential forms.
