# Differential local diffeomorphism

##### 2020-02-19 08:20Smoothness is a local property. (The term global refers to the whole space X. ) Definition 1. 2. A smooth map f: X Y between subsets of Euclidean space is a dieomorphism if it is a bijection, and if the inverse f1: Y Xis also smooth. Xand Y are dieomorphic if such an fexists.

Diffeomorphism. If U, V are connected open subsets of Rn such that V is simply connected, a differentiable map f: U V is a diffeomorphism if it is proper and if the differential Dfx: Rn Rn is bijective at each point x in U. differential local diffeomorphism General. A local diffeomorphism is a local homeomorphism and so also an open map. By the inverse function theorem.

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3 Answers. For any f (y)f (R), f a local diffeomorphism implies there is U open about y such that f (U) is open in R about f (y). Necessarily f (U)f (R) and so immediately this means f (R) is open. For 2) the image is an interval because f (R) is an open connected set which is an interval from basic real analysis. Of course if f(U) admits a differential manifold structure the answer is yes, but in general I think the situation is more complicated. \endgroup John Mar 19 '13 at 21: 07 \begingroup I think its called Inverse function theorem in english \endgroup trew Mar 19 '13 at 21: 12 Apr 07, 2013 Let f: R R be a local diffeomorphism (diffeomrophism in a neighborhood of each point). How to show that a locally diffeomorphic mapping from R to R is op. Ok, here is what I am thinking. . that since we are dealing with a diffeomorphism we have a bijective differential morphism between manifolds whose inverse is also bijective (a differential local diffeomorphism Stack Exchange network consists of 174 Q& A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share