@Echo I think your first semtence in your first comment is not true because we jave effective action

@Echo In your comment is not a possible confusion of effective action with free action?Please read the motivational example again

Any way my previous post below has similar flavour mathoverflow.net/questions/152411/…

Oh Thank you for link to Pestov Inon theorem. I was not aware of this theorem. When I posed this question for my student in midterm examination of undergraduate differential geometry I was not aware of this theorem so it was a spantaneous observation.

This remind me of a question I posed in my course of "differential geometry" in midterm as follows: Let $\gamma$ be closed curve included in unit circle prove that there is a point whose curvature is $\geq 1$

I am sorry if my comment is not relevant. But what do you mean by map? What other extra conditions you expect from this map?a graph preserving map?Otherwise every injective map from graph to an epsilon neighborhood of 0 works

you may be also interested in the remark at the bottom of the page 2 of arxiv.org/pdf/math/0507516.pdf and item 3 in page 3 of the latter