---
tag: Z7H98E1NHP
kind: concept
name: Real analytic manifold
msc: [32C05, 58A07]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_manifold
---

## Summary

A real analytic manifold is a [[topological manifold|QQQ9B1DK4P]] with an [[atlas|J7P32JQTVE]] whose transition maps are real [[analytic functions|081WKEDK6C]].
Every real analytic manifold is a [[smooth manifold|F45VQ47GPV]], but the analytic structure is more rigid.
An analytic function on a connected manifold is determined by its values on any nonempty open set.
Spheres, projective spaces, Lie groups and real algebraic manifolds are real analytic.
Every second countable real analytic manifold has a complexification and can be embedded analytically in some $\R^N$.

## Explanations

### L1 human

A manifold is a space that looks locally like $\R^n$, and its coordinate changes say how to pass between local pictures.
In a real analytic manifold these coordinate changes are given locally by convergent power series.
For example, on the circle the angle coordinate on two overlapping arcs differs by a constant on each piece of the overlap, which is analytic.
Functions that are analytic on such a manifold behave like polynomials: if one vanishes on an open set of a connected manifold, it vanishes everywhere.