Inmathematics, asmooth structure on amanifold allows for an unambiguous notion ofsmooth function. In particular, a smooth structure allowsmathematical analysis to be performed on the manifold.^[1]

A smooth structure on a manifold${\displaystyle M}$  is a collection of smoothly equivalent smooth atlases. Here, asmooth atlas for a topological manifold${\displaystyle M}$  is anatlas for${\displaystyle M}$  such that eachtransition function is asmooth map, and two smooth atlases for${\displaystyle M}$  aresmoothly equivalent provided theirunion is again a smooth atlas for${\displaystyle M.}$  This gives a naturalequivalence relation on the set of smooth atlases.

Asmooth manifold is a topological manifold${\displaystyle M}$  together with a smooth structure on${\displaystyle M.}$ 

By taking the union of allatlases belonging to a smooth structure, we obtain amaximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. Thus, we may regard a smooth structure as a maximal smooth atlas and vice versa.

In general, computations with the maximal atlas of a manifold are rather unwieldy. For most applications, it suffices to choose a smaller atlas. For example, if the manifold iscompact, then one can find an atlas with only finitely many charts.

If${\displaystyle \mu }$  and${\displaystyle \nu }$  are two maximal atlases on${\displaystyle M}$  the two smooth structures associated to${\displaystyle \mu }$  and${\displaystyle \nu }$  are said to be equivalent if there is adiffeomorphism${\displaystyle f:M\to M}$  such that${\displaystyle \mu \circ f=\nu .}$ ^{[citation needed]}

John Milnor showed in 1956 that the 7-dimensional sphere admits a smooth structure that is not equivalent to the standard smooth structure. A sphere equipped with a nonstandard smooth structure is called anexotic sphere.

TheE8 manifold is an example of atopological manifold that does not admit a smooth structure. This essentially demonstrates thatRokhlin's theorem holds only for smooth structures, and not topological manifolds in general.

The smoothness requirements on the transition functions can be weakened, so that the transition maps are only required to be${\displaystyle k}$ -times continuously differentiable; or strengthened, so that the transition maps are required to be real-analytic. Accordingly, this gives a${\displaystyle C^k}$  or(real-)analytic structure on the manifold rather than a smooth one. Similarly, acomplex structure can be defined by requiring the transition maps to be holomorphic.

• Smooth frame – Generalization of an ordered basis of a vector spacePages displaying short descriptions of redirect targets
• Atlas (topology) – Set of charts that describes a manifold