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Local diffeomorphism

From Wikipedia, the free encyclopedia

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 and be differentiable manifolds. A function

is a local diffeomorphism, if for each point there exists an open set containing such that
is open in and
is a diffeomorphism.

A local diffeomorphism is a special case of an immersion where the image of under locally has the differentiable structure of a submanifold of Then and may have a lower dimension than

Discussion

For instance, even though all manifolds look locally the same (as for some ) 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 that make 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

  • Every local diffeomorphism is also a local homeomorphism and therefore an open map.
  • A local diffeomorphism has constant rank of
  • 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 is a local diffeomorphism if and only if the derivative is a linear isomorphism for all points Note that this implies that and must have the same dimension.

Local flow diffeomorphisms

See also

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.
This page was last edited on 11 September 2021, at 15:02
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