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  "body": "\n## Statement\n\nLet $S$ be a complete two-dimensional [[Riemannian manifold|7CBT4SM83Y]] whose [[Gaussian curvature|SQVQ5DQT6W]] is a negative constant.\nThen there is no smooth isometric [[immersion|6RW68JGBHC]] of $S$ into $\\R^3$.\nIn particular, the hyperbolic plane cannot be realized as a smooth surface in $\\R^3$ with its induced metric.\n\n## Remarks\n\nAfter scaling, one may assume $K = -1$; one may also pass to the universal cover, which is the hyperbolic plane.\nFor an immersed surface with $K \u003c 0$, the two asymptotic directions at each point give coordinates in which the angle $\\omega$ between the coordinate curves satisfies the sine-Gordon equation $\\omega_{uv} = \\sin\\omega$.\nCompleteness lets these coordinates cover the whole plane, and the equation then bounds the area of every coordinate quadrilateral by $2\\pi$.\nThis contradicts the fact that the hyperbolic plane has infinite area.\n\nThe [[pseudosphere|44WEBQSKEY]] has constant curvature $-1$ and lies in $\\R^3$, but it is not complete: it ends in a sharp edge.\nThe smoothness hypothesis matters: by the Nash–Kuiper theorem there are $C^1$ isometric embeddings of the hyperbolic plane into $\\R^3$.\nFor constant positive curvature the situation is different, since round spheres are complete and lie in $\\R^3$.\n\n## Explanations\n\n### L1 human\n\nThe hyperbolic plane is a surface that curves like a saddle at every point, by the same amount everywhere, and goes on forever.\nHilbert's theorem says that you cannot build such a surface in ordinary space without creases or edges.\nEvery smooth attempt, like the pseudosphere, must stop at some edge.\nThis is why hyperbolic geometry is usually drawn with models such as the Poincaré disc, which distort distances.\n\n### L2 human\n\nThe theorem explains why the hyperbolic plane is studied as an abstract Riemannian manifold rather than as a surface in space.\nEfimov later showed that a complete surface with curvature bounded above by a negative constant cannot be smoothly immersed in $\\R^3$ either.\nIn contrast, the Nash embedding theorems show that every Riemannian manifold embeds isometrically in some higher-dimensional Euclidean space.\nThe proof is a good example of how the Gauss equation turns a geometric problem into a partial differential equation.\n",
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