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This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points.

DOUBT: Suppose $M\subset \mathbb{R^m}$ is a riemannian manifold with induced riemannian metric from $\mathbb{R^m}$. If given any two point $x,y\in M$ there exists a unique geodesic joining those two points. Then is it true that $M$ is diffeomorphic with $\mathbb{R^n}$ for some $n\leq m$??

Also it will be very helpful if someone can give me some reference from where I can read the relation between number of geodesics and number of genus.

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    $\begingroup$ How about a manifold of constant negative curvature? E.g. any hyperbolic plane. It cannot be imbedded as a whole but some bounded (closed or open) subset can be. $\endgroup$
    – Anton Fetisov
    Commented Nov 9, 2015 at 23:00

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The magic words are "exponential map".

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  • $\begingroup$ By the Nash embedding theorem, this question is actually equivalent to [mathoverflow.net/q/223323/70808]. Please see the comments to Raziel's answer - I don't think it is as trivial as your answer suggests. On the other hand, if you can prove that the exponential map is necessarily a local diffeo in this case, please explain a bit more. $\endgroup$
    – Sebastian Goette
    Commented Nov 12, 2015 at 19:36

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