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

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

A local diffeomorphism is a special case of an immersion  . In this case, for each  , there exists an open set   containing   such that the image   is an embedded submanifold, and   is a diffeomorphism. Here   and   have the same dimension, which may be less than the dimension of  .[1]

Characterizations

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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   have the same dimension.[2]

It follows that a map   between two 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, for any  , both   and   have the same dimension, thus   is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.[3]

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

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All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if   and   have the same dimension, and   and  , then there exist open neighbourhoods   of   and   of   and a diffeomorphism  . However, this map   need not extend to a smooth map defined on all of  , let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism   is a stronger condition than "to be locally diffeomophic." 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 on   that each make   into a differentiable manifold, but both structures are not locally diffeomorphic (see Exotic  ).[citation needed]

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

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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

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Local flow diffeomorphisms

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See also

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Notes

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  1. ^ Lee, Introduction to smooth manifolds, Proposition 5.22
  2. ^ Lee, Introduction to smooth manifolds, Proposition 4.8
  3. ^ Axler, Linear algebra done right, Theorem 3.21

References

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  • 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.
  • Lee, John M. (2013), Introduction to smooth manifolds, Graduate Texts in Mathematics, vol. 218 (Second ed.), New York, NY.: Springer, ISBN 978-1-4419-9981-8, MR 2954043
  • Axler, Sheldon (2024), Linear algebra done right, Undergraduate Texts in Mathematics (Fourth ed.), Springer, Cham, doi:10.1007/978-3-031-41026-0, ISBN 978-3-031-41026-0, MR 4696768