Sectional curvature

This article defines a notion of curvature for a differential manifold equipped with a Riemannian metric

The equivalent notion for a pseudo-Riemannian manifold is: Sectional curvature for a pseudo-Riemannian manifold

Definition

Given data

A Riemannian manifold $(M,g)$ viz a differential manifold $M$ equipped with a Riemannian metric $g$.

Definition part

Let $\mathrm{\Pi }$ be a tangent plane to $M$ at a point $p\in M$. Then, the sectional curvature of $\mathrm{\Pi }$ at $p$ is defined as follows: take two linearly independent vectors $X$ and $Y$ in $\mathrm{\Pi }$, and calculate:

$g(R(X,Y)Y,X)$

viz the inner product of $R(X,Y)Y$ and $X$ with respect to $g$.

Divide this by the square of the area of the parallelogram formed by $X$ and $Y$. This ratio defines the sectional curvature of $\mathrm{\Pi }$, denoted as $K(\mathrm{\Pi })$.

Here is the more explicit formula:

$\frac{g(R(X,Y)Y,X)}{g(X,X)g(Y,Y)-g(X,Y{)}^2}$

Note that the denominator cannot vanish because $X$ and $Y$ are independent vectors.

Related notions

Related notions of curvature

• Ricci curvature associates a real number to every tangent direction at every point
• Scalar curvature assocaites a real number to every point

Related metric properties

• Riemannian manifold with positively lower-bounded curvature
• Positively curved Riemannian manifold
• Negatively curved Riemannian manifold
• Constant-curvature metric