Lagmental Vicfred

Gaussian Curvature Is a Determinant of the Shape Operator by Vicfred

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

For a surface in Euclidean three-space, Gaussian curvature is the product of principal curvatures. I want the notation, the mechanism, and the failure mode visible at the same time.

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 \(K(\sigma)\).

$$ S=-dN:T_p\Sigma\to T_p\Sigma $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ K=\det S=\kappa_1\kappa_2 $$

A small case in full

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ I=\begin{pmatrix}E&F\\F&G\end{pmatrix},\quad II=\begin{pmatrix}e&f\\f&g\end{pmatrix}\Longrightarrow K=\frac{eg-f^2}{EG-F^2} $$

The reusable statement

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad S=-dN:T_p\Sigma\to T_p\Sigma,\\[5pt] \mathsf{C}\;&:\quad K=\det S=\kappa_1\kappa_2. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] K=\det S=\kappa_1\kappa_2 \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Tue 03 August 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.