Lagmental Vicfred

Jacobi Fields Describe Infinitesimal Families of Geodesics by Vicfred

Last updated: Mon 22 June 2026

tags: mathematics, curvature, Gauss--Bonnet, Jacobi fields

Variation through geodesics produces a vector field satisfying a second-order curvature equation. I will separate the object being defined from the consequence being claimed.

Notation

Curvature measures the failure of covariant derivatives to commute. The Riemann tensor \(R(X,Y)Z\) contracts to Ricci curvature and restricts to sectional curvature \(K(\sigma)\).

$$ J(t)=\left.\frac{\partial}{\partial s}\right|_{s=0}\gamma_s(t) $$

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.

$$ \frac{D^2J}{dt^2}+R(J,\dot\gamma)\dot\gamma=0 $$

Stress the formula

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

$$ K\equiv\kappa,\quad J\perp\dot\gamma\Longrightarrow J''+\kappa J=0\Longrightarrow\begin{cases}J=A\sin(\sqrt\kappa t)+B\cos(\sqrt\kappa t),&\kappa>0,\\J=At+B,&\kappa=0.\end{cases} $$

Interpretation

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad J(t)=\left.\frac{\partial}{\partial s}\right|_{s=0}\gamma_s(t),\\[5pt] \mathsf{C}\;&:\quad \frac{D^2J}{dt^2}+R(J,\dot\gamma)\dot\gamma=0. \end{aligned} $$

Limit of the argument

Sign conventions for \(R\) vary by author. A sphere may receive the opposite tensor sign unless the convention is stated.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \frac{D^2J}{dt^2}+R(J,\dot\gamma)\dot\gamma=0 \end{gathered}} $$

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 Sun 24 May 2026. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.