Manifold

A topological space that is locally homeomorphic to Euclidean space of a fixed dimension, equipped when needed with additional structure (smooth, differentiable, Riemannian) to support calculus and geometric constructions.

A topological space that is locally homeomorphic to Euclidean space R^n; often equipped with extra structure (smooth, differentiable, analytic, or topological) specified via atlases of coordinate charts and transition maps of a given regularity class.