first order operators in Riemannian geometry
On a pseudo-Riemannian manifold , and in Euclidean space inparticular, one can express the gradient operator, the divergenceoperator, and the curl operator (which makes sense only if is3-dimensional) in terms of the exterior derivative
. Let denotethe ring of smooth functions on ; let denote the-module of smooth vector fields, and let denote the-module of smooth 1-forms. The contraction
with the metric tensor and its inverse
, respectively, defines the-module isomorphisms
In local coordinates, this isomorphisms is expressed as
or as the lowering of an index. To wit, for , we have
The gradient operator, which in tensor notation is expressed as
can now be defined as
Another natural structure on an -dimensional Riemannian manifold isthe volume form
, , defined by
Multiplication bythe volume form defines a natural isomorphism between functions and-forms:
Contraction with thevolume form defines a natural isomorphism between vector fields and-forms:
orequivalently
where indicates an omitted factor. The divergenceoperator, which in tensor notation is expressed as
can be defined in a coordinate-free way by the following relation:
Finally, on a -dimensional manifold we may define the curloperator in a coordinate-free fashion by means of the following relation: