Manifold
Additional structure

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold.

This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of R n {\displaystyle \mathbb {R} ^{n}} (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms.

The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible.

These notions are made precise in general through the use of pseudogroups.
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