In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|_U : U → f(U) is a diffeomorphism. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Note that:

• Every local diffeomorphism is also a local homeomorphism and therefore an open map.
• A diffeomorphism is just a bijective local diffeomorphism.

According to the inverse function theorem, a smooth map f : M → N is a local diffeomorphism if and only if the derivative Df_p : T_pM → T_f(p)N is a linear isomorphism for all points p in M. Note that this implies that M and N must have the same dimension. In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ...