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.
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.
- 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
This section is empty. You can help by adding to it. (July 2010)
- Invariance of domain – Theorem in topology about homeomorphic subsets of Euclidean space
- Local homeomorphism – Continuous open map that, around every point in its domain, has a neighborhood on which it restricts to a homomorphism
- Spacetime symmetries