Let $M$ be a simply connected complete Riemannian manifold, and let $x\in M$. Does there exist a nondecreasing function $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all paths $\alpha,\beta$ in the ball $B(x,r)$ from $x$ to any $y\in B(x,r)$, there is a homotopy relative to the end points between $\alpha$ and $\beta$ in the ball $B(x,R(r))$?
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$\begingroup$ The following is a simpler variation of the above question: Does there exist a non-decreasing $R:\mathbb R_+\to\mathbb R_+$ such that, for every $r>0$ and all piecewise smooth paths $\alpha,\beta$ from $x$ to any $y\in B(x,r)$ with length $<r$, there is a homotopy relative to the end points between $\alpha$ and $\beta$ consisting of piecewise smooth paths of length $<R$? It seems that an affirmative answer to this question could be given by using convex balls $B(x_i,s_i)$ so that the centers $x_i$ form a discrete subset of $M$ and the balls $B(x_i,s_i/2)$ cover $M$. Is this known? $\endgroup$– Jesús ÁlvarezCommented Mar 6, 2015 at 10:54
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It will follow if we can find a function $R$ so that the map induced by inclusion $\pi_1 (B(x,r)) \to \pi_1 (B(x,R(r)))$ is trivial.
By the density of Morse functions, for any small $\epsilon$ there is a compact smooth manifold with boundary $N$ so that $B(x,r) \subset N \subset B(x, r+ \epsilon)$. The fundamental group $\pi_1 (N)$ is then finitely presented. Now choose $R$ large enough so that each generator of $\pi_1 (N)$ is nullhomotopic in $B(x,R(r))$. This finite value $R$ is then sufficient for the given $r$.
Now for every $r$ choose $R$ minimal. (It might be that actually choosing the infinum causes some problems, but we could simply add a fixed value $\delta > 0$ to cushion it.) This will guarantee the function is non-decreasing.
