Another word for a (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its "functional structure" (Bredon 1995) and so differs from a topological manifold because the notion of differentiability exists on it. Every smooth manifold is a topological manifold, but not necessarily vice versa. (The first nonsmooth topological manifold occurs in four dimensions.) Milnor (1956) showed that a seven-dimensional hypersphere can be made into a smooth manifold in 28 ways.

Bredon, G. E. Topology & Geometry. New York: Springer-Verlag, p. 69, 1995.

Milnor, J. "On Manifolds Homeomorphic to the 7-Sphere." Ann. Math. 64, 399-405, 1956.

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Weisstein, Eric W. "Smooth Manifold." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SmoothManifold.html