Local diffeomorphism

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

Formal definition

Let X and Y be differentiable manifolds. A function,

$f:X\to Y\,$

is a local diffeomorphism, if for each point x in X, there exists an open set U containing x, such that

$f(U)\,$

is open in Y and

$f|_{U}:U\to f(U)\,$

is a diffeomorphism.

Discussion

For instance, even though all manifolds look locally the same (as Rⁿ for some n) 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 R that make R into a differentiable manifold, but both structures are not locally diffeomorphic (see below). Note also that 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 local diffeomorphism from the 2-sphere to Euclidean two-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

Every local diffeomorphism is also a local homeomorphism and therefore an 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 evenly covered by the map.

According to the inverse function theorem, a smooth map f : M → N is a local diffeomorphism if and only if the derivative Df_p : T_pM → T_f(p)N is a linear isomorphism for all points p in M. Note that this implies that M and N must have the same dimension.

Local flow diffeomorphisms

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, MR 2428390 .

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