Real-analytic manifold

This article describes an additional structure on a differential manifold
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Definition

Definition in terms of atlases

A real-analytic manifold is the following data:

• A topological manifold $M$
• An atlas of coordinate charts on $M$ (in other words, an open cover of $M$ with homeomorphisms from each member of the open cover, to an open set in $\mathbb {R} ^{n}$)

Satisfying the following condition: the transition function between any two coordinate charts is a real-analytic map with real-analytic inverse.

Upto the following equivalence:

Two atlases on a topological space define the same real-analytic structure if the transition functions between the corresponding coordinate charts are all real-analytic isomorphisms (viz., real-analytic maps with real-analytic inverses).

Definition in terms of sheaves

A real-analytic manifold is the following data:

• A topological manifold $M$
• A subsheaf of the sheaf of continuous functions with real values on $M$, which plays the role of a sheaf of real-analytic functions

satisfying the following condition: every point in $M$ is contained in an open set homeomorphic to an open set in $\mathbb {R} ^{n}$, such that the sheaf restricted to that open set, corresponds to the sheaf of real-analytic functions on the open set in $\mathbb {R} ^{n}$.