# 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

## Contents

## Definition

### Given data

A Riemannian manifold viz a differential manifold equipped with a Riemannian metric .

### Definition part

Let be a tangent plane to at a point . Then, the sectional curvature of at is defined as follows: take two linearly independent vectors and in , and calculate:

viz the inner product of and with respect to .

Divide this by the square of the area of the parallelogram formed by and . This ratio defines the sectional curvature of , denoted as .

Here is the more explicit formula:

Note that the denominator cannot vanish because and 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

- Riemannian manifold with positively lower-bounded curvature
- Positively curved Riemannian manifold
- Quasi-positively curved Riemannian manifold
- Nonnegatively curved Riemannian manifold
- Negatively curved Riemannian manifold
- Constant-curvature metric
- Flat metric

## Facts

### Sectional curvature determines Riemann curvature tensor

*This is analogous to the fact that Ricci curvature determines the Ricci curvature tensor*

The following facts are true about the Riemann curvature tensor:

- The Riemann curvature tensor can be viewed as a -tensor:

- The above map is antisymmetric in (
*For full proof, refer: Antisymmetry of Riemann curvature tensor*and antisymmetric in and . Further it is symmetric in the pairs and . Hence, it is a symmetric bilinear form on .

Thus, given the sectional curvature for all tangent planes, we can back-calculate the Riemann curvature tensor from it by the usual polarization trick. The idea is to view the Riemann curvature tensor as a symmetric bilinear form on , and the sectional curvature as the values that this form takes on pairs of (special kinds of) *unit vectors* in .