Local diffeomorphism

Summary

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.

Formal definition edit

Let   and   be differentiable manifolds. A function   is a local diffeomorphism, if for each point   there exists an open set   containing   such that   is open in   and

 
is a diffeomorphism.

A local diffeomorphism is a special case of an immersion   where the image   of   under   locally has the differentiable structure of a submanifold of   Then   and   may have a lower dimension than  

Characterizations edit

A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.

The inverse function theorem implies that a smooth map   is a local diffeomorphism if and only if the derivative   is a linear isomorphism for all points   This implies that   and   must have the same dimension.

A map   between two connected manifolds of equal dimension ( ) is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding), or equivalently, if and only if it is a smooth submersion. This is because every smooth immersion is a locally injective function while invariance of domain guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.

Discussion edit

For instance, even though all manifolds look locally the same (as   for some  ) in the topological sense, it is natural to ask whether their differentiable structures behave in the same manner locally. For example, one can impose two different differentiable structures on   that make   into a differentiable manifold, but both structures are not locally diffeomorphic (see below). Although local diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire (smooth) manifold. For example, there can be no global diffeomorphism from the 2-sphere to Euclidean 2-space although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, the sphere is compact whereas Euclidean 2-space is not.

Properties edit

If a local diffeomorphism between two manifolds exists then their dimensions must be equal. Every local diffeomorphism is also a local homeomorphism and therefore a locally injective open map. A local diffeomorphism has constant rank of  

Examples edit

A diffeomorphism is a bijective local diffeomorphism. A smooth covering map is a local diffeomorphism such that every point in the target has a neighborhood that is evenly covered by the map.

Local flow diffeomorphisms edit

See also edit

References edit

  • Michor, Peter W. (2008), Topics in differential geometry, Graduate Studies in Mathematics, vol. 93, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2003-2, MR 2428390.