A local diffeomorphism is a differentiable map that restricts near every point to a diffeomorphism onto an open subset of its target. The local inverse functions need not combine into a single global inverse function: different parts of the domain of a function can map onto the same output region.
For continuously differentiable maps between open subsets of , being a local diffeomorphism is equivalent to having an invertible derivative at every point. The inverse function theorem proves one direction. For the other, the chain rule for each local inverse function gives two inverse linear maps. The periodic angle example under the inverse function theorem has this property everywhere, even though it has infinitely many inputs for every point of the punctured plane. The noninjective map with constant Jacobian determinant gives another useful distinction: even a constant nonzero Jacobian determinant need not give global injectivity.
Every local diffeomorphism is an open map. To see this, take an open subset of the source and intersect it with the neighbourhoods where the map is a diffeomorphism. Each local image is open, and their union is the full image. Thus nearby outputs remain available locally, despite the possible global ambiguity.
The displayed smooth map from to the punctured plane has Jacobian determinant one: in polar coordinates its radius is and its angle is , so the determinant is . Nevertheless for every integer . 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.
An open map sends every open subset of its source to an open subset of its target. This is an image property; continuity instead concerns preimages of open sets. A local diffeomorphism has both properties, but openness alone does not promise unique recovery of an input. In the periodic coordinate map under the inverse function theorem, open patches give open output regions while multiple angle values still give the same output.
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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).