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  "body": "\n## Definition\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]] of dimension $n$.\nA *real analytic atlas* on $M$ is an [[atlas|J7P32JQTVE]] $\\set{(U_i, \\varphi_i)}$ of charts $\\varphi_i \\colon U_i \\to \\R^n$ with the following property.\nFor all $i, j$ the transition map\n$$\\varphi_j \\circ \\varphi_i^{-1} \\colon \\varphi_i(U_i \\cap U_j) \\to \\varphi_j(U_i \\cap U_j)$$\nis real [[analytic|081WKEDK6C]].\nThat is, each component of the transition map is given near each point by a convergent power series.\nA *real analytic manifold* is a topological manifold $M$ together with a maximal real analytic atlas on $M$.\n\n## Remarks\n\nExample: the sphere $S^n \\subseteq \\R^{n+1}$ with the two stereographic projections is a real analytic manifold.\nThe transition map is $y \\mapsto y / \\abs{y}^2$ on $\\R^n \\setminus \\set{0}$, whose components are quotients of polynomials by a nonvanishing polynomial, hence real analytic.\n\nNon-example: on $\\R$ take the identity chart and the chart $\\psi$ with $\\psi(x) = x$ for $x \\le 0$ and $\\psi(x) = x + e^{-1/x^2}$ for $x \u003e 0$.\nThe map $\\psi$ is a homeomorphism of $\\R$ with positive derivative, and the transition map $\\psi$ is smooth, so this atlas makes $\\R$ a [[smooth manifold|F45VQ47GPV]].\nBut $\\psi$ is not real analytic at $0$: all its derivatives at $0$ agree with those of $x \\mapsto x$, while $\\psi(x) \\neq x$ for $x \u003e 0$.\nSo the atlas is not real analytic.\n\nA real analytic manifold is in particular a smooth manifold, since its maximal real analytic atlas lies in a unique maximal smooth atlas.\n",
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