---
tag: PAG39H901E
kind: statement
role: theorem
name: Grauert's theorem on real analytic manifolds
msc: [32C09, "32E10", 32C05]
uses: [Z7H98E1NHP, HMCD0JBFST, JSGWAN2RJV, VKFW27WXEZ]
about: [Z7H98E1NHP]
sources:
  - url: https://en.wikipedia.org/wiki/Stein_manifold
---

## Statement

Let $M$ be a [[real analytic manifold|Z7H98E1NHP]] of dimension $n$ that is Hausdorff and second countable.
Then:

1. $M$ is a closed real analytic [[submanifold|VKFW27WXEZ]] of some [[Stein manifold|HMCD0JBFST]] $X$ of complex dimension $n$. Moreover $M$ is totally real in $X$: $T_pM \cap i\,T_pM = 0$ for $p \in M$.
2. Let $Y$ be a [[complex manifold|JSGWAN2RJV]] that contains $M$ as a closed totally real real analytic submanifold of real dimension $n$. Then $M$ has a basis of open neighbourhoods in $Y$ that are Stein manifolds.
3. $M$ admits a real analytic embedding onto a closed real analytic submanifold of $\R^N$ for some $N$.

## 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.

## Explanations

### L1 human

A real analytic manifold is a space that looks locally like $\R^n$, glued by maps given by convergent power series.
The theorem says that any such space is a real slice of a very well behaved complex manifold, in the way $\R^n$ is the real slice of $\C^n$.
As a consequence, every such space can be drawn inside some $\R^N$ by maps given by power series.

### L2 human

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.