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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 $p \colon X \to \mathbb{R}^n$ 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 $X$.

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 $T p \colon TX \to T \mathbb{R}^n$ 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 $\mathbb{R}^n$.

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 $X$ 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 $\mathbb{R}^n$?

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 ?