Lagmental Vicfred

The Laplace--Beltrami Operator Includes the Volume Density by Vicfred

tags: mathematics, Riemannian geometry, geodesics, connections

Divergence of the metric gradient gives the intrinsic Laplacian on a Riemannian manifold. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Definitions first

A Riemannian metric \(g_p:T_pM\times T_pM\to\mathbf R\) varies smoothly and assigns lengths and angles. In coordinates it is a positive-definite matrix \((g_{ij})\).

$$ \Delta_gf=\operatorname{div}_g(\operatorname{grad}_gf) $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \Delta_gf=\frac1{\sqrt{|g|}}\partial_i\!\left(\sqrt{|g|}\,g^{ij}\partial_jf\right) $$

A small case in full

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ g=dr^2+r^2d\theta^2\Longrightarrow\Delta f=\frac1r\frac{\partial}{\partial r}\!\left(r\frac{\partial f}{\partial r}\right)+\frac1{r^2}\frac{\partial^2f}{\partial\theta^2} $$

The reusable statement

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \Delta_gf=\operatorname{div}_g(\operatorname{grad}_gf),\\[5pt] \mathsf{C}\;&:\quad \Delta_gf=\frac1{\sqrt{|g|}}\partial_i\!\left(\sqrt{|g|}\,g^{ij}\partial_jf\right). \end{aligned} $$

A nearby false statement

Christoffel symbols depend on coordinates even though the Levi--Civita connection and geodesic equation are intrinsic.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \Delta_gf=\frac1{\sqrt{|g|}}\partial_i\!\left(\sqrt{|g|}\,g^{ij}\partial_jf\right) \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 20 August 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.