Lagmental Vicfred

Sectional Curvature Assigns Curvature to a Two-Plane by Vicfred

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

The Riemann tensor evaluated on an orthonormal basis measures the curvature of a tangent two-plane. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

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 \(K(\sigma)\).

$$ \sigma=\operatorname{span}\{u,v\}\subset T_pM $$

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.

$$ K(\sigma)=\frac{g(R(u,v)v,u)}{g(u,u)g(v,v)-g(u,v)^2} $$

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.

$$ u,v\ \text{orthonormal}\Longrightarrow K(\sigma)=\langle R(u,v)v,u\rangle $$

The reusable statement

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad \sigma=\operatorname{span}\{u,v\}\subset T_pM,\\[5pt] \mathsf{C}\;&:\quad K(\sigma)=\frac{g(R(u,v)v,u)}{g(u,u)g(v,v)-g(u,v)^2}. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] K(\sigma)=\frac{g(R(u,v)v,u)}{g(u,u)g(v,v)-g(u,v)^2} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sun 26 June 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.