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I initially asked this question on MSE but I haven't had any luck.

The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the manifolds are real analytic, is every continuous map between them homotopic to a real analytic map?

I know that the natural generalisation to complex manifolds fails. That is, not every continuous map between complex manifolds is homotopic to a holomorphic map.

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    $\begingroup$ Try the paper: H.L. Royden: The analytic approximation of differentiable mappings, Math. Ann., 139(1960), 171-179. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 22, 2015 at 16:37
  • $\begingroup$ Two comments, which are valid for at least compact manifolds (I'm not very sure about the non-compact case): (1) any smooth manifold admits a refinement of the atlas which is analytic, (2) One possible way to prove your claim is to flow by harmonic map heat flow for a short time. I'm not actually sure if it's proven in the literature but the reference for Ricci flow (the proof should basically be identical, presumably) is Bando "“Real analyticity of solutions of Hamilton’s equation" $\endgroup$
    – Otis Chodosh
    Commented Apr 22, 2015 at 17:57
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    $\begingroup$ The paper: H.Grauert: On Levi's problem and the embedding of real analytic manifolds, Annals of Math. 68 (1958), 460--472. It shows (proposition 5 if I remember right) that real analytic mappings are dense in the Whitney $C^\infty$-topology. From homotopic follows. $\endgroup$
    – Peter Michor
    Commented Apr 22, 2015 at 18:20
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    $\begingroup$ Royden's paper only covers continuous maps between compact manifolds. $\endgroup$
    – Ben McKay
    Commented Aug 21, 2019 at 7:51
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    $\begingroup$ Grauert's paper doesn't really cover maps between manifolds. It only proves that real valued analytic functions are dense in the continuous real valued functions. I think that it should not be difficult to generalize to maps between real analytic manifolds, but I am not aware of a reference where this is done. $\endgroup$
    – Ben McKay
    Commented Aug 21, 2019 at 9:14

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The result is not stated in Grauert's paper. On the other hand, Grauert proves that every real analytic manifold $M$ sits as a real analytic totally real submanifold, and analytic deformation retraction, in a Stein manifold $M_{\mathbb{C}}$. So every continuous map $\phi \colon M \to N$ of real analytic manifolds extends to a continuous map $\phi \colon M_{\mathbb{C}} \to N_{\mathbb{C}}$. Since $M_{\mathbb{C}}$ and $N_{\mathbb{C}}$ are Stein manifolds, Oka's theorem proves that this continuous map is homotopic to a holomorphic map. This then composes with the real analytic inclusion $M \to M_{\mathbb{C}}$ and real analytic deformation retraction $N_{\mathbb{C}} \to N$, so $\phi$ is homotopic to a real analytic map. For me, at least, this is not obviously contained in Grauert's paper (or at least I didn't spot it), although I am sure that Grauert would have seen it as a consequence.

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    $\begingroup$ Thanks for fleshing this out. I could never find a precise statement, so this is perfect. $\endgroup$
    – Michael Albanese
    Commented Sep 25, 2019 at 17:01
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According to this paper by Michael Langenbruch, this was proved by none other than H. Whitney. The paper has lots of references.

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    $\begingroup$ Langenbruch's paper doesn't use the word "manifold", and concerns only analytic functions defined on open subsets of Euclidean space. I don't think that real analytic patching of those approximations is easy, so I don't see how to apply this result on manifolds. $\endgroup$
    – Ben McKay
    Commented Aug 21, 2019 at 7:53
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The book
F. Guaraldo, P. Macrì, A. Tancredi "Topics on Real Analytic Spaces", Advanced Lectures in Mathematics, Braunschweig/Wiesbaden: Friedrich Vieweg & Sohn, pp. X+163 (1986), ISBN:3-528-08963-6 , MR1013362, Zbl 0616.32012.

which is largely devoted to this topic, has a very general and explicit statement (Chapter VII):

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