Local diffeomorphism

Let $X$ and $Y$ be differentiable manifolds. A function $f:X\; \backslash to\; Y$ is a local diffeomorphism if, for each point $x\; \backslash in\; X$, there exists an open set $U$ containing $x$ such that the image $f(U)$ is open in $Y$ and $$f\backslash vert\_U\; :\; U\; \backslash to\; f(U)$$ is a diffeomorphism.

A local diffeomorphism is a special case of an immersion $f\; :\; X\; \backslash to\; Y$. In this case, for each $x\; \backslash in\; X$, there exists an open set $U$ containing $x$ such that the image $f(U)$ is an embedded submanifold, and $f|\_U:U\; \backslash to\; f(U)$ is a diffeomorphism. Here $X$ and $f(U)$ have the same dimension, which may be less than the dimension of $Y$.Lee, Introduction to smooth manifolds, Proposition 5.22

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 $f:X\; \backslash to\; Y$ is a local diffeomorphism if and only if the derivative $D\; f\_x\; :\; T\_x\; X\; \backslash to\; T\_\{f(x)\}\; Y$ is a linear isomorphism for all points $x\; \backslash in\; X$. This implies that $X$ and $Y$ have the same dimension.Lee, Introduction to smooth manifolds, Proposition 4.8

It follows that a map $f\; :\; X\; \backslash to\; Y$ between two manifolds of equal dimension ($\backslash operatorname\{dim\}\; X\; =\; \backslash operatorname\{dim\}\; Y$) 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 $x\; \backslash in\; X$, both $T\_xX$ and $T\_\{f(x)\}Y$ have the same dimension, thus $Df\_x$ is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective.Axler, Linear algebra done right, Theorem 3.21

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.

All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if $X$ and $Y$ have the same dimension, and $x\; \backslash in\; X$ and $y\backslash in\; Y$, then there exist open neighbourhoods $U$ of $x$ and $V$ of $y$ and a diffeomorphism $f:U\; \backslash to\; V$. However, this map $f$ need not extend to a smooth map defined on all of $X$, let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism $f:X\; \backslash to\; Y$ 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 $\backslash R^4$ that each make $\backslash R^4$ into a differentiable manifold, but both structures are not locally diffeomorphic (see Exotic $\backslash mathbb\{R\}^4$).{{Citation needed|date=June 2024|reason=they are not diffeomorphic, but existence of a local diffeomorphism is weaker condition}}

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.

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 $n.$

• 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 {{em|evenly covered}} by the map.

{{See also|Flow (mathematics)|}}

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• {{annotated link|Diffeomorphism}}
• {{annotated link|Homeomorphism}}
• {{annotated link|Invariance of domain}}
• {{annotated link|Large diffeomorphism}}
• {{annotated link|Local homeomorphism}}
• {{annotated link|Spacetime symmetries}}

{{reflist}}

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|mr=2954043

|year=2013

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}}

• {{Citation

|last1=Axler

|first1=Sheldon

|title=Linear algebra done right

|series=Undergraduate Texts in Mathematics

|edition=Fourth

|publisher=Springer, Cham

|year=2024

|isbn=978-3-031-41026-0

|mr=4696768

|doi=10.1007/978-3-031-41026-0

|doi-access=free

}}

{{Manifolds}}

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Category:Theory of continuous functions

Category:Diffeomorphisms

Category:Functions and mappings

Category:Inverse functions