Smooth Structure

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 smooth maps. 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 the notion of a smooth map between two differentiable manifolds, 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.

Two smooth structures are considered equivalent if there is a homeomorphism of the manifold which pulls back one atlas to an atlas compatible with the other one, i.e., a diffeomorphism. For instance, any two smooth structures on the circle are equivalent, as can be seen by integration.

It is surprising that some manifolds admit more than one smooth structure. The first such example was an exotic sphere of , the seven-dimensional hypersphere, constructed by Milnor (1956) as an -fiber bundle over using quaternions. Work of Casson, Freedman, and Donaldson led to the discovery that four-dimensional Euclidean space has smooth structures that are distinct from the standard structure (Gompf 1983, Kirby 2011). These are called exotic R⁴s, and some of their techniques involve Donaldson theory.

Another approach to smooth structures is through sheaf theory. A coordinate chart on an -dimensional manifold is a homeomorphism to an open set in , and its component functions give local coordinates. A smooth atlas determines a sheaf by assigning to each open set the real-valued functions that are smooth functions in every coordinate chart.

Conversely, a subsheaf of real-valued continuous functions determines a smooth structure if every point belongs to an open set with a homeomorphism to an open set such that the following condition holds. For every open set and every real-valued continuous function on , iff is a smooth function on . These homeomorphisms are coordinate charts whose transition functions are smooth maps (Balzin 2021, Definitions 2.18 and 2.20).