A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between the coordinate charts are C^∞ smooth. A manifold with a smooth structure is called a smooth manifold (or differentiable manifold). A smooth structure is used to define differentiability for real-valued functions on a manifold. This extends to a notion of when a map between two differentiable manifolds is smooth, and naturally to the definition of a diffeomorphism. In addition, the smooth structure is used to define manifold tangent vectors, the collection of which is the tangent bundle.
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