Lagmental Vicfred

The Gradient Is Metric-Dual to the Differential by Vicfred

tags: mathematics, Riemannian geometry, geodesics, connections

The metric converts the one-form df into the vector field grad f. The point is to make the formal expression readable enough to audit line by line.

Notation

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 \((g_{ij})\).

$$ g(\operatorname{grad}f,X)=df(X) $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \operatorname{grad}f=g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial}{\partial x^i} $$

Stress the formula

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ g_{ij}=\begin{pmatrix}E&F\\F&G\end{pmatrix}\Longrightarrow(g^{ij})=\frac1{EG-F^2}\begin{pmatrix}G&-F\\-F&E\end{pmatrix} $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad g(\operatorname{grad}f,X)=df(X),\\[5pt] \mathsf{C}\;&:\quad \operatorname{grad}f=g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial}{\partial x^i}. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{grad}f=g^{ij}\frac{\partial f}{\partial x^j}\frac{\partial}{\partial x^i} \end{gathered}} $$

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 Sun 29 September 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.