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Let $f:M\rightarrow N$ be a smooth map between two simply connected Riemannian manifolds of the same dimension. It is also given that for every $x\in M$ we have that $Df|_x:T_xM\rightarrow T_{f(x)}N$ is an isometry of inner product spaces. Must $f:M\rightarrow N$ be injective ? I know the answer is yes if geodesic completness is assumed, but what happens if it is not assumed ? Thank you.

Edit: The above version of my question was already answered in the comments thanks to Anton Petrunin. I am now interested in this slightly modified version of my previous question:

Let $f:M\rightarrow N$ be a surjective smooth map between two simply connected Riemannian manifolds of the same dimension. It is also given that for every $x\in M$ we have that $Df|_x:T_xM\rightarrow T_{f(x)}N$ is an isometry of inner product spaces. Must $f:M\rightarrow N$ be injective ? I know the answer is yes if geodesic completness is assumed, but what happens if it is not assumed ? Thank you.

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  • $\begingroup$ Let $N$=plane, $M'\subseteq N$ some annulus, and $M$=an open subset of a double cover of $M'$ which is simply connected and doesn't inject onto $M'$. $\endgroup$
    – Wojowu
    Commented Mar 13, 2023 at 20:08
  • $\begingroup$ @Wojowu double cover of an annulus is not simply-connected. $\endgroup$
    – Anton Petrunin
    Commented Mar 13, 2023 at 20:46
  • $\begingroup$ Fixing problem in Wojowu's suggestion: Consider double cover of a tubular neighborhood of a projective plane embedded into a euclidean space. $\endgroup$
    – Anton Petrunin
    Commented Mar 13, 2023 at 20:49
  • $\begingroup$ @AntonPetrunin Thank you for your comments, I am sorry I still cant wrap my head around your example, can you please write it with more clarification as an answer? $\endgroup$
    – Amr
    Commented Mar 13, 2023 at 20:57
  • $\begingroup$ Use Whitney theorem to embed $\mathbb{R}\mathrm{P}^2$ into $\mathbb{R}^5$, the rest should be clear. $\endgroup$
    – Anton Petrunin
    Commented Mar 13, 2023 at 21:04

1 Answer 1

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Take three copies of the open unit disk in $\mathbb{C}$. Call them $A$, $M$, and $B$, for "attic", "main floor", and "basement" respectively.

Cut slits in them as follows:

  • small slits in $A$ and $M$ along the $y$ axis from $i$ to $i/2$
  • small slit in $M$ and $B$ along the $y$ axis from $-i$ to $-i/2$.

Make stairs from the attic down to the main floor by gluing $A$ to $M$ along the open interval connecting $i$ and $i/2$: attach the right side of the slit in $A$ to the left side of the slit in $M$. Make stairs from the main floor down to the basement by gluing $M$ to $B$ along the open interval connecting $-i$ and $-i/2$: attach the right side of the slit in $M$ to the left side of the slit in $B$.

What results is a non geodesically complete Riemannian manifold that surjects onto the unit disk (another non-complete Riemanninan manifold) in an infinitesimal isometry that is not injective.

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