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

$<R(X,Y)Y,X>$

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 $\Pi$, denoted as $K(\Pi )$.

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