Real analytic manifold

Remarks

Example: the sphere $S^n \subseteq \R^{n+1}$ with the two stereographic projections is a real analytic manifold. The transition map is $y \mapsto y / \abs{y}^2$ on $\R^n \setminus \set{0}$, whose components are quotients of polynomials by a nonvanishing polynomial, hence real analytic.

Non-example: on $\R$ take the identity chart and the chart $\psi$ with $\psi(x) = x$ for $x \le 0$ and $\psi(x) = x + e^{-1/x^2}$ for $x > 0$. The map $\psi$ is a homeomorphism of $\R$ with positive derivative, and the transition map $\psi$ is smooth, so this atlas makes $\R$ a smooth manifold. But $\psi$ is not real analytic at $0$: all its derivatives at $0$ agree with those of $x \mapsto x$, while $\psi(x) \neq x$ for $x > 0$. So the atlas is not real analytic.

A real analytic manifold is in particular a smooth manifold, since its maximal real analytic atlas lies in a unique maximal smooth atlas.