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.
Best Answer
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.