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  "body": "\n## Summary\n\nThe sectional curvature of a [[Riemannian manifold|7CBT4SM83Y]] assigns a number to each $2$-dimensional plane in each tangent space.\nIt is the [[Gaussian curvature|SQVQ5DQT6W]] at $p$ of the surface swept out by the geodesics from $p$ tangent to that plane.\nThe 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.\n\n## Explanations\n\n### L1 human\n\nIn two dimensions, curvature at a point is a single number, the Gaussian curvature.\nIn higher dimensions a point has many surface directions, and the space can be curved differently in each.\nSectional curvature gives the Gaussian curvature for each choice of a $2$-plane of directions.\nThe round unit sphere has sectional curvature $1$ for every plane, flat space has $0$, and hyperbolic space has $-1$.\n",
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