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:

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

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

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.

For a pseudo-Riemannian manifold

Further information: Sectional curvature for a pseudo-Riemannian manifold

We can also define the sectional curvature of a pseudo-Riemannian manifold. The same definition works.

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