Real analytic manifold

Summary

A real analytic manifold is a topological manifold with an atlas whose transition maps are real analytic functions. Every real analytic manifold is a smooth manifold, 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 $\R^N$.