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  "body": "\n## Statement\n\nLet $M$ be a [[real analytic manifold|Z7H98E1NHP]] of dimension $n$ that is Hausdorff and second countable.\nThen:\n\n1. $M$ is a closed real analytic [[submanifold|VKFW27WXEZ]] of some [[Stein manifold|HMCD0JBFST]] $X$ of complex dimension $n$. Moreover $M$ is totally real in $X$: $T_pM \\cap i\\,T_pM = 0$ for $p \\in M$.\n2. Let $Y$ be a [[complex manifold|JSGWAN2RJV]] that contains $M$ as a closed totally real real analytic submanifold of real dimension $n$. Then $M$ has a basis of open neighbourhoods in $Y$ that are Stein manifolds.\n3. $M$ admits a real analytic embedding onto a closed real analytic submanifold of $\\R^N$ for some $N$.\n\n## Remarks\n\nThe idea of the proof is as follows.\nA complexification of $M$ exists by gluing the holomorphic extensions of the real analytic transition maps of $M$.\nNear $M$, the square of a suitable distance to $M$ gives a strictly plurisubharmonic function whose sublevel sets shrink to $M$.\nGrauert's solution of the Levi problem for manifolds shows that these sublevel sets are Stein manifolds.\nA Stein manifold $X$ embeds holomorphically and properly in $\\C^N$, and restricting such an embedding to $M$ and taking real and imaginary parts gives the real analytic embedding into $\\R^{2N}$.\nExample: the circle $M = \\R/\\Z$ sits in the cylinder $X = \\C/\\Z$ as the image of $\\R$, and thin annuli $\\set{\\abs{\\operatorname{Im} z} \u003c \\varepsilon}$ are Stein neighbourhoods of it.\nThe countability hypothesis is needed for an embedding into a Euclidean space, since every subset of $\\R^N$ is second countable.\n\n## Explanations\n\n### L1 human\n\nA real analytic manifold is a space that looks locally like $\\R^n$, glued by maps given by convergent power series.\nThe theorem says that any such space is a real slice of a very well behaved complex manifold, in the way $\\R^n$ is the real slice of $\\C^n$.\nAs a consequence, every such space can be drawn inside some $\\R^N$ by maps given by power series.\n\n### L2 human\n\nThe result is the real analytic analogue of the Whitney embedding theorem for smooth manifolds.\nIts proof shows how several complex variables helps with real problems: real analytic functions on $M$ are restrictions of holomorphic functions on a Stein neighbourhood.\nCartan's theorems A and B on that neighbourhood then give many global real analytic functions on $M$.\nThe Stein neighbourhoods are often called Grauert tubes.\n",
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  "name": "Grauert's theorem on real analytic manifolds",
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