Local diffeomorphism

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

Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be differentiable manifolds. A function [math]\displaystyle{ f : X \to Y }[/math] is a local diffeomorphism, if for each point [math]\displaystyle{ x \in X }[/math] there exists an open set [math]\displaystyle{ U }[/math] containing [math]\displaystyle{ x, }[/math] such that [math]\displaystyle{ f(U) }[/math] is open in [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ f\vert_U : U \to f(U) }[/math] is a diffeomorphism.

A local diffeomorphism is a special case of an immersion [math]\displaystyle{ f : X \to Y, }[/math] where the image [math]\displaystyle{ f(U) }[/math] of [math]\displaystyle{ U }[/math] under [math]\displaystyle{ f }[/math] locally has the differentiable structure of a submanifold of [math]\displaystyle{ Y. }[/math] Then [math]\displaystyle{ f(U) }[/math] and [math]\displaystyle{ X }[/math] may have a lower dimension than [math]\displaystyle{ Y. }[/math]

Characterizations

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 [math]\displaystyle{ f : M \to N }[/math] is a local diffeomorphism if and only if the derivative [math]\displaystyle{ D f_p : T_p M \to T_{f(p)} N }[/math] is a linear isomorphism for all points [math]\displaystyle{ p \in M. }[/math] This implies that [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] must have the same dimension.

A map [math]\displaystyle{ f : X \to Y }[/math] between two connected manifolds of equal dimension ([math]\displaystyle{ \operatorname{dim} X = \operatorname{dim} Y }[/math]) 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

For instance, even though all manifolds look locally the same (as [math]\displaystyle{ \R^n }[/math] for some [math]\displaystyle{ n }[/math]) 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 [math]\displaystyle{ \R^4 }[/math] that make [math]\displaystyle{ \R^4 }[/math] 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

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 [math]\displaystyle{ n. }[/math]

Examples

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.

See also

• Diffeomorphism – Isomorphism of smooth manifolds; a smooth bijection with a smooth inverse
• Homeomorphism – Mapping which preserves all topological properties of a given space
• Invariance of domain – Theorem in topology about homeomorphic subsets of Euclidean space
• Physics:Large diffeomorphism – Class of diffeomorphism
• Local homeomorphism – Mathematical function revertible near each point
• Physics:Spacetime symmetries

References

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