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  "body": "\n## Statement\n\nLet $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] of dimension $m$.\nWrite $\\ip{\\cdot}{\\cdot}$ for the Euclidean inner product on $\\R^n$.\nAn *isometric embedding* of $(M, g)$ into $\\R^n$ is an embedding $f \\colon M \\to \\R^n$ with $\\ip{df_p(v)}{df_p(w)} = g_p(v, w)$ for all $p \\in M$ and $v, w \\in T_pM$.\n\n1. ($C^1$ theorem, Nash and Kuiper.) Let $n \\ge m + 1$ and let $f \\colon M \\to \\R^n$ be a smooth embedding that is short, that is $\\abs{df_p(v)}^2 \u003c g_p(v, v)$ for all $p$ and all nonzero $v \\in T_pM$. Then for every $\\varepsilon \u003e 0$ there is a $C^1$ embedding $\\tilde f \\colon M \\to \\R^n$ that is isometric for $g$ and satisfies $\\abs{\\tilde f(p) - f(p)} \u003c \\varepsilon$ for all $p \\in M$.\n2. ($C^k$ theorem, Nash.) Let $3 \\le k \\le \\infty$ and let $g$ be of class $C^k$. Then $(M, g)$ has an isometric embedding of class $C^k$ into $\\R^n$. Here $n = m(3m + 11)/2$ if $M$ is compact, and $n = m(m + 1)(3m + 11)/2$ otherwise.\n\n## Remarks\n\nPart 2 is proved by solving the nonlinear system $f^*\\ip{\\cdot}{\\cdot} = g$ with an iteration that combines Newton's method with smoothing operators, the Nash–Moser technique.\nPart 1 is proved by convex integration: one adds many small fast oscillations that increase lengths in a controlled way.\nAn example of part 1: the flat square torus $\\R^2 / \\Z^2$ has a $C^1$ isometric embedding into $\\R^3$.\nIt has no $C^2$ isometric embedding there, because a compact surface in $\\R^3$ has a point of positive [[Gaussian curvature|SQVQ5DQT6W]], while the flat torus has curvature $0$.\nSo the low codimension in part 1 really needs the low regularity $C^1$.\nNash proved part 1 with $n \\ge m + 2$, and Kuiper extended it to $n \\ge m + 1$.\n\n## Explanations\n\n### L1 human\n\nEvery abstract curved space described by a Riemannian metric can be realised as a surface inside some ordinary Euclidean space, with lengths measured in the usual way.\nSo the abstract definition of a Riemannian manifold gives nothing new compared with curved submanifolds of $\\R^n$.\nIf one allows the surface to be only once differentiable, very little room is needed, and strange crumpled embeddings appear.\n\n### L2 human\n\nThe [[Whitney embedding theorem|2GSRBDT96M]] embeds a manifold in $\\R^{2m}$ but ignores the metric; Nash's theorems make the embedding respect the metric at the cost of more dimensions or less smoothness.\nThe $C^1$ theorem is an early instance of the h-principle: a short embedding can be deformed into an isometric one by arbitrarily small moves.\nThe bounds on $n$ in part 2 have been lowered by later work, but the existence statement is what is used in practice.\n",
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