Local diffeomorphism

A local diffeomorphism restricts near every point to a diffeomorphism onto an open subset. In particular, every local diffeomorphism is an open map.

Table of contents
- Noninjective map with constant Jacobian determinant Local diffeomorphism
- Open map Local diffeomorphism

Noninjective map with constant Jacobian determinant ( $a (s, r) = e^{s} (cos (r e^{- 2 s}), sin (r e^{- 2 s}))$ )

The displayed smooth map from

R^{2}

to the punctured plane has Jacobian determinant one: in polar coordinates its radius is

e^{s}

and its angle is

r e^{- 2 s}

, so the determinant is

e^{2 s} e^{- 2 s} = 1

. Nevertheless

a (0, 2 πk) = (1, 0)

for every integer

k

. Thus a local volume-preserving map need not be injective or have bounded weighted inverse multiplicity. Compactly supported data distributed over many inverse branches disprove a determinant-only dispersion with a nonlinear velocity map estimate.

Open map

An open map sends every open set to an open set.

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Local diffeomorphism by

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A **local diffeomorphism** is a mathematical concept from differential geometry that describes a type of smooth map between two differentiable manifolds (or smooth manifolds).

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