Local diffeomorphism


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,
is a local diffeomorphism, if for each point x in X, there exists an open set U containing x, such that
is open in Y and
is a diffeomorphism.
A local diffeomorphism is a special case of an immersion f from X to Y, where the image f of U under f locally has the differentiable structure of a submanifold of Y. Then f and X may have a lower dimension than Y.

Discussion

For instance, even though all manifolds look locally the same 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. Although local diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these diffeomorphisms to ensure that the domain is the entire 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