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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 29th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexhad need for a stub for [[local diffeomorphism]]

   had need for a stub for local diffeomorphism

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 2nd 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexSuppose a smooth function $p \colon X \to \mathbb{R}^n$ from a _diffeological space_ $X$ to Cartesian space induces at each point an isomorphism on tangent vectors as well as on all higher jets. Then what sensible extra conditions does it take to conclude that $p$ is in fact a local diffeomorphism, i.e. restricts to a diffeomorphism around an open neighbourhood of each point? Here I mean tangents and jets defined by equivalence classes of smooth maps into $X$.

   Suppose a smooth function from a diffeological space to Cartesian space induces at each point an isomorphism on tangent vectors as well as on all higher jets.

   Then what sensible extra conditions does it take to conclude that is in fact a local diffeomorphism, i.e. restricts to a diffeomorphism around an open neighbourhood of each point?

   Here I mean tangents and jets defined by equivalence classes of smooth maps into .

3. • CommentRowNumber3.
   • CommentAuthorigor
   • CommentTimeMay 3rd 2015
   • PermaLink
   Author: igor
   Format: MarkdownItexUnless I'm missing something, but your condition implies that the Jacobian map $T p \colon TX \to T \mathbb{R}^n$ is full rank and even invertible. The inverse function theorem then guarantees that $p$ is a local diffeomorphism about any point $x\in X$ that has an $n$-manifold neighborhood. Is your question then about $X$'s that at some points fail to be $n$-manifold? But then, it seems to me, that essentially by definition there cannot be a local diffeomorphism from any neighborhood of such a point into $\mathbb{R}^n$.

   Unless I’m missing something, but your condition implies that the Jacobian map is full rank and even invertible. The inverse function theorem then guarantees that is a local diffeomorphism about any point that has an -manifold neighborhood. Is your question then about ’s that at some points fail to be -manifold? But then, it seems to me, that essentially by definition there cannot be a local diffeomorphism from any neighborhood of such a point into .

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 5th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexIf you only know that $X$ is a diffeological space and that it has a map $X \to \mathbb{R}^n$ which is an iso on all tangents, what else does it need (if anything) to conclude that $X$ is a manifold sitting by a local diffeomorphism over $\mathbb{R}^n$?

   If you only know that is a diffeological space and that it has a map which is an iso on all tangents, what else does it need (if anything) to conclude that is a manifold sitting by a local diffeomorphism over ?

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeMay 5th 2015
   • (edited May 5th 2015)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWhat happened with [[moneomorphism]] and [[epiomorphism]]? I know that the terms are used rarely, especially outside of Eastern Europe, and especially the second term, but I remember writing about that terminology in nlab and this seemingly completely _vanished_. nLab search and google show no hits at nLab about those. Did I dream about writing it ?

   What happened with moneomorphism and epiomorphism? I know that the terms are used rarely, especially outside of Eastern Europe, and especially the second term, but I remember writing about that terminology in nlab and this seemingly completely vanished. nLab search and google show no hits at nLab about those. Did I dream about writing it ?