Lagmental Vicfred

A Riemannian Metric Converts Velocity into Length by Vicfred

tags: mathematics, Riemannian geometry, geodesics, connections

Integrating the pointwise norm of a curve velocity gives its geometric length. This is a compact note, but the quantifiers and hypotheses stay on the page.

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 \((g_{ij})\).

$$ \|\dot\gamma(t)\|_g=\sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ L_g(\gamma)=\int_a^b\|\dot\gamma(t)\|_g\,dt $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ g=g_{ij}\,dx^i\otimes dx^j,\qquad L_g(\gamma)=\int_a^b\sqrt{\sum_{i,j}g_{ij}(\gamma(t))\dot\gamma^i(t)\dot\gamma^j(t)}\,dt $$

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad \|\dot\gamma(t)\|_g=\sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))},\\[5pt] \mathsf{C}\;&:\quad L_g(\gamma)=\int_a^b\|\dot\gamma(t)\|_g\,dt. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] L_g(\gamma)=\int_a^b\|\dot\gamma(t)\|_g\,dt \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Sat 31 October 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.