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May 22, 2013 at 20:34 comment added Matthias Ludewig Do you mean that $x_n$ is in the inverse image under $f$ of the closed geodesic ball about $y$ of radius $\mathrm{dist}(y, f(N)) + 2$? Because otherwise, one can construct a counterexample with a circle of radius $>1$ embedded in the plane.
May 22, 2013 at 17:44 history answered Peter Michor CC BY-SA 3.0