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  "body": "\n## Definition\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] with curvature $R(X, Y)Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z$ (see [[Riemann curvature tensor|JA29ETCM02]]).\nLet $p \\in M$ and let $\\sigma \\subseteq T_pM$ be a $2$-dimensional linear subspace, spanned by $u, v$.\nThe *sectional curvature* of $\\sigma$ is\n$$K(\\sigma) = K(u, v) = \\frac{g\\bigl(R(u, v)v, u\\bigr)}{g(u, u) \\, g(v, v) - g(u, v)^2}.$$\n\n## Remarks\n\nThe value does not depend on the choice of basis $u, v$ of $\\sigma$, because numerator and denominator both change by the square of the determinant of the change of basis.\n\nExample: on the unit sphere $S^n$, $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so for orthonormal $u, v$ the numerator is $g(g(v, v)u - g(u, v)v, u) = 1$ and $K(\\sigma) = 1$ for every plane.\n\nNon-example: $K(u, v)$ is not defined when $u$ and $v$ are linearly dependent, since then they span no plane and the denominator is $0$.\n\nFor a surface, the only plane at $p$ is $T_pM$, and $K$ is the [[Gaussian curvature|SQVQ5DQT6W]].\nA manifold has constant sectional curvature $k$ if and only if $R(X, Y)Z = k\\bigl(g(Y, Z)X - g(X, Z)Y\\bigr)$.\nWith the opposite sign convention for $R$, the numerator is written $g(R(u, v)u, v)$.\n",
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