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Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such that $d(f(x), g(x)) < \epsilon$ for all $x \in N$. Is it true that if $\epsilon$ is sufficiently small then $f, g$ are homotopic to each other?

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    Interesting question! If $N$ is compact, then we can choose finitely many points in $N$ such that, for some positive $\epsilon$, the $\epsilon$-neighbourhoods cover $N$ and the $2 \epsilon$-neighbourhoods are convex. Then there is an obvious patching argument one could try to construct a continuous homotopy... but I haven't checked that it works.2012-08-22

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