first order operators in Riemannian geometry


On a pseudo-Riemannian manifoldMathworldPlanetmath M, and in Euclidean space in particular, one can express the gradient operator, the divergence operator, and the curl operator (which makes sense only if M is 3-dimensional) in terms of the exterior derivativeMathworldPlanetmath. Let 𝒞(M) denote the ring of smooth functions on M; let 𝒳(M) denote the 𝒞(M)-module of smooth vector fields, and let Ω1(M) denote the 𝒞(M)-module of smooth 1-forms. The contractionPlanetmathPlanetmath with the metric tensor g and its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath g-1, respectively, defines the 𝒞(M)-module isomorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

:𝒳(M)Ω1(M),:Ω1(M)𝒳(M).

In local coordinates, this isomorphisms is expressed as

(xi)=jgijdxj,(dxj)=igijxi.

or as the lowering of an index. To wit, for V𝒳(M), we have

V =i=1nVixi,
(V)j =i=1ngijVi,j=1,,n.

The gradient operator, which in tensor notation is expressed as

(gradf)i=gijfxj,f𝒞(M),

can now be defined as

gradf=(df),f𝒞(M).

Another natural structureMathworldPlanetmath on an n-dimensional Riemannian manifold is the volume formMathworldPlanetmath, ωΩn(M), defined by

ω=detgijdx1dxn.

Multiplication by the volume form defines a natural isomorphism between functions and n-forms:

ffω,f𝒞(M).

Contraction with the volume form defines a natural isomorphism between vector fields and (n-1)-forms:

XXω,X𝒳(M),

or equivalently

xi(-1)i+1detgijdx1dxi^dxn,

where dxi^ indicates an omitted factor. The divergence operator, which in tensor notation is expressed as

divX=iXi,X𝒳(M)

can be defined in a coordinate-free way by the following relationMathworldPlanetmathPlanetmath:

(divX)ω=d(Xω),X𝒳(M).

Finally, on a 3-dimensional manifold we may define the curl operator in a coordinate-free fashion by means of the following relation:

(curlX)ω=d(X),X𝒳(M).
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