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

be differentiable manifolds. A function is a local diffeomorphism, there exists an open set such that the image is a diffeomorphism.

A local diffeomorphism is a special case of an immersion is an embedded submanifold.

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 is a local diffeomorphism if and only if the derivative is a linear isomorphism have the same 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 is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.

Here is an alternative argument for the case of an immersion: 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

is a stronger condition than "to be locally diffeomorphic." Indeed, although locally defined 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, one can impose two different differentiable structures into a differentiable manifold, but both structures are not locally diffeomorphic (see \mathbb{R}^4).

As another example, there can be no local 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, and the 2-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

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 by the map.

Local flow diffeomorphisms

See also

Notes

References

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