Lagmental Vicfred

Geodesics Have Zero Covariant Acceleration by Vicfred

tags: mathematics, Riemannian geometry, geodesics, connections

A geodesic parallel-transports its own velocity and locally extremizes length. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The mathematical object

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 \((g_{ij})\).

$$ \nabla_{\dot\gamma}\dot\gamma=0 $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \ddot\gamma^k+\Gamma^k_{ij}(\gamma)\dot\gamma^i\dot\gamma^j=0 $$

One explicit computation

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \frac{d^2\gamma^k}{dt^2}+\sum_{i,j=1}^{n}\Gamma^k_{ij}(\gamma(t))\frac{d\gamma^i}{dt}\frac{d\gamma^j}{dt}=0\qquad(k=1,\ldots,n) $$

Why the identity matters

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad \nabla_{\dot\gamma}\dot\gamma=0,\\[5pt] \mathsf{C}\;&:\quad \ddot\gamma^k+\Gamma^k_{ij}(\gamma)\dot\gamma^i\dot\gamma^j=0. \end{aligned} $$

Where it can fail

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \ddot\gamma^k+\Gamma^k_{ij}(\gamma)\dot\gamma^i\dot\gamma^j=0 \end{gathered}} $$

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 Mon 27 July 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.