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

 $\flat:\mathcal{X}(M)\to\Omega^{1}(M),\quad\sharp\colon\Omega^{1}(M)\to\mathcal% {X}(M).$

In local coordinates, this isomorphisms is expressed as

 $\left(\frac{\partial}{\partial x^{i}}\right)^{\flat}=\sum_{j}g_{ij}dx^{j},% \quad\left(dx^{j}\right)^{\sharp}=\sum_{i}g^{ij}\frac{\partial}{\partial x^{i}}.$

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

 $\displaystyle V$ $\displaystyle=\sum_{i=1}^{n}V^{i}\frac{\partial}{\partial x^{i}},$ $\displaystyle(V^{\flat})_{j}$ $\displaystyle=\sum_{i=1}^{n}g_{ij}V^{i},\quad j=1,\ldots,n.$

The gradient operator, which in tensor notation is expressed as

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

can now be defined as

 $\operatorname{grad}f=(df)^{\sharp},\quad f\in\mathcal{C}^{\infty}(M).$

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

 $\omega=\sqrt{\det g_{ij}}\,dx^{1}\wedge\ldots\wedge dx^{n}.$

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

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

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

 $X\mapsto X\mathop{\rfloor}\omega,\quad X\in\mathcal{X}(M),$

or equivalently

 $\frac{\partial}{\partial x^{i}}\mapsto(-1)^{i+1}\sqrt{\det g_{ij}}\,dx^{1}% \wedge\ldots\wedge\widehat{dx^{i}}\wedge\ldots\wedge dx^{n},$

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

 $\operatorname{div}X=\nabla_{i}X^{i},\quad X\in\mathcal{X}(M)$

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

 $(\operatorname{div}X)\,\omega=d(X\mathop{\rfloor}\omega),\quad X\in\mathcal{X}% (M).$

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

 $(\operatorname{curl}X)\mathop{\rfloor}\omega=d(X^{\flat}),\quad X\in\mathcal{X% }(M).$
Title first order operators in Riemannian geometry FirstOrderOperatorsInRiemannianGeometry 2013-03-22 15:28:18 2013-03-22 15:28:18 rmilson (146) rmilson (146) 7 rmilson (146) Definition msc 70G45 msc 53B20 Gradient Curl Divergence LeibnizNotationForVectorFields