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  "body": "\n## Summary\n\nA real analytic manifold is a [[topological manifold|QQQ9B1DK4P]] with an [[atlas|J7P32JQTVE]] whose transition maps are real [[analytic functions|081WKEDK6C]].\nEvery real analytic manifold is a [[smooth manifold|F45VQ47GPV]], but the analytic structure is more rigid.\nAn analytic function on a connected manifold is determined by its values on any nonempty open set.\nSpheres, projective spaces, Lie groups and real algebraic manifolds are real analytic.\nEvery second countable real analytic manifold has a complexification and can be embedded analytically in some $\\R^N$.\n\n## Explanations\n\n### L1 human\n\nA manifold is a space that looks locally like $\\R^n$, and its coordinate changes say how to pass between local pictures.\nIn a real analytic manifold these coordinate changes are given locally by convergent power series.\nFor example, on the circle the angle coordinate on two overlapping arcs differs by a constant on each piece of the overlap, which is analytic.\nFunctions that are analytic on such a manifold behave like polynomials: if one vanishes on an open set of a connected manifold, it vanishes everywhere.\n",
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