Real analytic manifold with curvature

Question

Suppose $(M,g)$ is a real analytic Riemannian manifold (that is with coordinate charts that are real analytic functions with analytic inverses). Can we conclude that the manifold is always flat? The intuition is that analyticity forces the manifold to be rigid, but it seems strange to me that every real analytic manifold is flat. Are there simple examples of real analytic manifolds which are not flat?