first order operators in Riemannian geometry

On a pseudo-Riemannian manifold $M$, 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 derivative. Let ${𝒞}^{\mathrm{\infty }}(M)$ denote the ring of smooth functions on $M$; let $𝒳(M)$ denote the ${𝒞}^{\mathrm{\infty }}(M)$-module of smooth vector fields, and let ${\mathrm{\Omega }}^1(M)$ denote the ${𝒞}^{\mathrm{\infty }}(M)$-module of smooth 1-forms. The contraction with the metric tensor $g$ and its inverse $g^{-1}$, respectively, defines the ${𝒞}^{\mathrm{\infty }}(M)$-module isomorphisms

$$\mathrm{♭}:𝒳(M)\to {\mathrm{\Omega }}^1(M),\mathrm{♯}:{\mathrm{\Omega }}^1(M)\to 𝒳(M).$$

In local coordinates, this isomorphisms is expressed as

$${\left(\frac{\partial }{\partial x^i}\right)}^{\mathrm{♭}}=\sum _j{g}_{ij}d{x}^j,{\left(d{x}^j\right)}^{\mathrm{♯}}=\sum _i{g}^{ij}\frac{\partial }{\partial x^i}.$$

or as the lowering of an index. To wit, for $V\in 𝒳(M)$, we have

$V$ $={\displaystyle \sum _{i=1}^n}{V}^i{\displaystyle \frac{\partial }{\partial x^i}},$

${(V^{\mathrm{♭}})}_j$ $={\displaystyle \sum _{i=1}^n}{g}_{ij}{V}^i,j=1,\mathrm{\dots },n.$

The gradient operator, which in tensor notation is expressed as

$${(\mathrm{grad}f)}^i=g^{ij}\frac{\partial f}{\partial x^j},f\in {𝒞}^{\mathrm{\infty }}(M),$$

can now be defined as

$$\mathrm{grad}f={(df)}^{\mathrm{♯}},f\in {𝒞}^{\mathrm{\infty }}(M).$$

Another natural structure on an $n$-dimensional Riemannian manifold is the volume form, $\omega \in {\mathrm{\Omega }}^n(M)$, defined by

$$\omega =\sqrt{det{g}_{ij}}d{x}^1\wedge \mathrm{\dots }\wedge d{x}^n.$$

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

$$f\mapsto f\omega ,f\in {𝒞}^{\mathrm{\infty }}(M).$$

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

$$X\mapsto X\rfloor \omega ,X\in 𝒳(M),$$

or equivalently

$$\frac{\partial }{\partial x^i}\mapsto {(-1)}^{i+1}\sqrt{det{g}_{ij}}d{x}^1\wedge \mathrm{\dots }\wedge \widehat{d{x}^i}\wedge \mathrm{\dots }\wedge d{x}^n,$$

where $\widehat{d{x}^i}$ indicates an omitted factor. The divergence operator, which in tensor notation is expressed as

$$\mathrm{div}X={\nabla }_i{X}^i,X\in 𝒳(M)$$

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

$$(\mathrm{div}X)\omega =d(X\rfloor \omega ),X\in 𝒳(M).$$

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

$$(\mathrm{curl}X)\rfloor \omega =d(X^{\mathrm{♭}}),X\in 𝒳(M).$$

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