Reference for real analytic manifolds

Question

I'm trying to find a reference for some introduction to real analytic manifolds. I'm especially interested in the fact, that the set of regular points of an analytic function $F \colon M \to \mathbb{R}^n$ is open dense on a real analytic manifold $M$. By a regular point, I mean that the differential $d_xF \colon T_xM \to \mathbb{R}^n$ has maximal rank in $M$.

Answer 1

Let $\overline R$ be the closure of the set of points where $dF_x$ has full rank. If this is not everything, pick a point $x \in \partial \overline R$. Pick a real analytic chart around this point $x$; because the partial derivatives of real analytic functions are real analytic, $dF$ is real analytic. Supposing for notational convenience that $\dim M =n$ (though there is no reason to make this assumption), $\det(dF)$ is also an analytic function. Because it is nonzero somewhere, it must be nonzero on a dense open set (in this chart) - otherwise there will be a point for which $\det(dF)$ vanishes in a small open set around it, and hence $dF$ vanishes everywhere by the identity theorem, contradicting the choice $x \in \partial \overline R$.

Note that there was no point here where we needed the codomain to be $\Bbb R^n$ as opposed to some other analytic manifold.