first order operators in Riemannian geometry
On a pseudo-Riemannian manifold , and in Euclidean space in
particular, one can express the gradient operator, the divergence
operator, and the curl operator (which makes sense only if is
3-dimensional) in terms of the exterior derivative
. Let denote
the 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 is
the volume form
, , defined by
Multiplication by the volume form defines a natural isomorphism between functions and -forms:
Contraction with the volume form defines a natural isomorphism between vector fields and -forms:
or equivalently
where indicates an omitted factor. The divergence operator, 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 curl operator in a coordinate-free fashion by means of the following relation:
Title | first order operators in Riemannian geometry |
---|---|
Canonical name | FirstOrderOperatorsInRiemannianGeometry |
Date of creation | 2013-03-22 15:28:18 |
Last modified on | 2013-03-22 15:28:18 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 7 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 70G45 |
Classification | msc 53B20 |
Related topic | Gradient |
Related topic | Curl |
Related topic | Divergence |
Related topic | LeibnizNotationForVectorFields |