Grauert's theorem on real analytic manifolds

Remarks

The idea of the proof is as follows. A complexification of $M$ exists by gluing the holomorphic extensions of the real analytic transition maps of $M$. Near $M$, the square of a suitable distance to $M$ gives a strictly plurisubharmonic function whose sublevel sets shrink to $M$. Grauert's solution of the Levi problem for manifolds shows that these sublevel sets are Stein manifolds. A Stein manifold $X$ embeds holomorphically and properly in $\C^N$, and restricting such an embedding to $M$ and taking real and imaginary parts gives the real analytic embedding into $\R^{2N}$. Example: the circle $M = \R/\Z$ sits in the cylinder $X = \C/\Z$ as the image of $\R$, and thin annuli $\set{\abs{\operatorname{Im} z} < \varepsilon}$ are Stein neighbourhoods of it. The countability hypothesis is needed for an embedding into a Euclidean space, since every subset of $\R^N$ is second countable.

Going deeper

For a student

The result is the real analytic analogue of the Whitney embedding theorem for smooth manifolds. Its proof shows how several complex variables helps with real problems: real analytic functions on $M$ are restrictions of holomorphic functions on a Stein neighbourhood. Cartan's theorems A and B on that neighbourhood then give many global real analytic functions on $M$. The Stein neighbourhoods are often called Grauert tubes.