I initially asked this question on [MSE](https://math.stackexchange.com/q/1219052/39599) but I haven't had any luck.




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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.



Is the analytic version of the Whitney Approximation Theorem true?
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.


