---
tag: 1X42JAA349
kind: concept
name: Sectional curvature
msc: [53B20, 53C20]
sources:
  - url: https://en.wikipedia.org/wiki/Sectional_curvature
same_as:
  wikidata: Q855622
  wikipedia: Sectional curvature
---

## Summary

The sectional curvature of a [[Riemannian manifold|7CBT4SM83Y]] assigns a number to each $2$-dimensional plane in each tangent space.
It is the [[Gaussian curvature|SQVQ5DQT6W]] at $p$ of the surface swept out by the geodesics from $p$ tangent to that plane.
The sectional curvatures determine the full [[Riemann curvature tensor|JA29ETCM02]], and bounds on them are the hypotheses of the main comparison theorems of global Riemannian geometry.

## Explanations

### L1 human

In two dimensions, curvature at a point is a single number, the Gaussian curvature.
In higher dimensions a point has many surface directions, and the space can be curved differently in each.
Sectional curvature gives the Gaussian curvature for each choice of a $2$-plane of directions.
The round unit sphere has sectional curvature $1$ for every plane, flat space has $0$, and hyperbolic space has $-1$.