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  "body": "\n## Statement\n\nLet $M \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]] with a smooth unit normal field $N$ and [[shape operator|DQCD11X1VM]] $S$.\nLet $I\\!I(X, Y) = \\ip{S X}{Y}$ be its [[second fundamental form|9KHQR16P1B]].\nLet $\\nabla$ be the [[Levi-Civita connection|CAZYGE1T4E]] of the induced metric on $M$, with [[Riemann curvature tensor|JA29ETCM02]] $R(X, Y) Z = \\nabla_X \\nabla_Y Z - \\nabla_Y \\nabla_X Z - \\nabla_{[X, Y]} Z$.\nThen for all tangent vector fields $X, Y, Z, W$ on $M$:\n\n1. (Gauss equation) $\\ip{R(X, Y) Z}{W} = I\\!I(Y, Z) \\, I\\!I(X, W) - I\\!I(X, Z) \\, I\\!I(Y, W)$.\n2. (Codazzi–Mainardi equation) $(\\nabla_X S) Y = (\\nabla_Y S) X$, where $(\\nabla_X S) Y = \\nabla_X (S Y) - S(\\nabla_X Y)$.\n\n## Remarks\n\nLet $D$ be the flat connection of $\\R^3$, given by directional derivatives.\nThe Gauss formula $D_X Y = \\nabla_X Y + I\\!I(X, Y) N$ and the Weingarten formula $D_X N = -S X$ split $D$ into tangent and normal parts.\nSince $D$ has zero curvature, the tangent part of $D_X D_Y Z - D_Y D_X Z - D_{[X, Y]} Z = 0$ gives the Gauss equation, and the normal part gives the Codazzi–Mainardi equation.\nTaking $X, Y$ orthonormal and $Z = Y$, $W = X$ in the Gauss equation gives $\\ip{R(X, Y) Y}{X} = \\det S = K$, the [[Gaussian curvature|SQVQ5DQT6W]].\nExample: for the sphere of radius $r$ with the inner normal, $S = \\frac{1}{r} \\id$, so $(\\nabla_X S) Y = \\frac{1}{r} (\\nabla_X Y - \\nabla_X Y) = 0$ and both sides of the Codazzi equation vanish, while the Gauss equation gives sectional curvature $1/r^2$.\n\n## Explanations\n\n### L1 human\n\nA surface in space has two kinds of data: how lengths are measured on it, and how it bends in space.\nThese cannot be chosen freely.\nThe Gauss–Codazzi equations are the compatibility conditions between them.\n\n### L2 human\n\nThe Gauss equation contains the Theorema Egregium: the left side depends only on the metric, so $K = \\det S$ does too.\nConversely, Bonnet's theorem says that on a simply connected domain, a metric and a symmetric form satisfying these equations come from a surface in $\\R^3$, unique up to rigid motion.\nThis is the surface analogue of the fundamental theorem of space curves.\nThe same equations hold for hypersurfaces of $\\R^{n+1}$, and with extra terms for submanifolds of a general Riemannian manifold.\n",
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