Lagmental Vicfred

Gauss--Bonnet Turns Total Curvature into Euler Characteristic by Vicfred

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

The integral of Gaussian curvature over a closed oriented surface is topological. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

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 \(K(\sigma)\).

$$ \Sigma\ \text{closed and oriented} $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ \int_\Sigma K\,dA=2\pi\chi(\Sigma) $$

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.

$$ \Sigma_g:\qquad\int_{\Sigma_g}K\,dA=2\pi(2-2g)=4\pi(1-g) $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \Sigma\ \text{closed and oriented},\\[5pt] \mathsf{C}\;&:\quad \int_\Sigma K\,dA=2\pi\chi(\Sigma). \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_\Sigma K\,dA=2\pi\chi(\Sigma) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Tue 21 March 2017. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.