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Assume that $f:\mathbb{R}^N\to\mathbb{R}^N$ is a surjective function and in addition suppose that $f$ is a local diffeomorphism. Take two points in the image of $f$, let's say, $f(x),f(y)$ with $f(x)\neq f(y)$.

Is it possible to find a continuous curve $\alpha :[0,1]\to\mathbb{R}^N$ such that $\alpha(0)=x$, $\alpha(1)=y$ and $f(\alpha (t))=(1-t)f(x)+tf(y)$.

Remark: This question is related to this one

Remark 1: The answer given here by @smiley06, solves this problem, when only one boundary condition is prescribed.

Thank you

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  • $\begingroup$ You're basically asking whether $f$ has to be a covering. For $N\geq2$ it is not true. $\endgroup$ – user8268 Sep 25 '13 at 12:51
  • $\begingroup$ Thank you for your comment @user8268. Could you please give me a example? $\endgroup$ – Tomás Sep 25 '13 at 12:52
  • $\begingroup$ Do you have any example of such a $f$ that is not invertible? I can't think of any. $\endgroup$ – tom Sep 25 '13 at 12:54
  • $\begingroup$ I'll try to draw one $\endgroup$ – user8268 Sep 25 '13 at 12:56
  • $\begingroup$ well and for which $(x,y)$ is $f(x,y)= (0,0)$ ?? $\endgroup$ – tom Sep 25 '13 at 13:01
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For $N=2$; I replace $\mathbb R^2$ with a disk (as it is diffeomorphic); here is a surjective map from a disk to a "disk" (still diffeomorphic to $\mathbb R^2$) and a straight segment that can't be lifted.

really horrible picture

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  • $\begingroup$ Sorry about my dumbness, but could you add some explanation. Thank you $\endgroup$ – Tomás Sep 25 '13 at 13:11
  • $\begingroup$ I know that picture is impossible; try yourself to map a disk to the plane so that the image is a disk, but the map is not 1-1. $\endgroup$ – user8268 Sep 25 '13 at 13:17
  • $\begingroup$ Very nice! I like it. $\endgroup$ – tom Sep 25 '13 at 13:21
  • $\begingroup$ Are all that "loops" necessary? Disclaimer: I am not specialist. $\endgroup$ – Matemáticos Chibchas Sep 25 '13 at 13:41
  • $\begingroup$ Yes the loops are necessary. Look at my animation. You take disk first deform it to the O shape and with one end you fill the hole in the O. $\endgroup$ – tom Sep 25 '13 at 13:59
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All the credit goes to the user8268 but I made a little animation.

enter image description here

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  • $\begingroup$ Unfortunately I don't know how to fill the inside of the disk. $\endgroup$ – tom Sep 25 '13 at 13:52
  • $\begingroup$ Thank you @tom for your kindness. Now I can see, hahah. $\endgroup$ – Tomás Sep 25 '13 at 14:10
  • $\begingroup$ wow! How did you do it? $\endgroup$ – user8268 Sep 25 '13 at 14:11
  • $\begingroup$ I used Autodes Maya, but it is "a little" bit overkill(most of the Avatar scenes were done in it :D). I guess it can be done in something like Flash. $\endgroup$ – tom Sep 25 '13 at 14:25

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