Definition
Smooth structure
A choice of mutually compatible smooth coordinate charts on a topological manifold, represented by their maximal smooth atlas.
Definition
Let be a topological manifold. A smooth structure on is a maximal smooth atlas on . Equivalently, it is an equivalence class of smooth atlases, where two atlases are equivalent when they are compatible.
A smooth manifold is a topological manifold together with a chosen smooth structure.
Specifying a smooth structure
It is unnecessary to write down every chart in the maximal atlas. Any smooth atlas determines a unique smooth structure by adjoining every chart smoothly compatible with it. For example, the identity chart on generates its standard smooth structure.
Dependence on the choice
The topology of does not always determine its smooth structure up to diffeomorphism. Distinct smooth structures on the same underlying topological manifold can therefore produce non-diffeomorphic smooth manifolds.
References
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: smooth structures and smooth manifolds.