Let $M^n$ be a manifold diffeomorphic to $\mathbb R^n$, equipped with a complete Riemannian metric $g$. Suppose for any two points $p, q \in M$, there exists one and only one geodesic connecting $p$ to $q$. Does this imply that $g$ has not conjugate points? ps. the reverse is true.
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2See my answer http://math.stackexchange.com/questions/1865210/uniqueness-of-minimizing-geodesic-rightarrow-uniqueness-of-connecting-geodesi?rq=1 and the reference there. – Moishe Kohan Nov 11 '16 at 19:59
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@Seub , I assume the manifold is diffeomorphic to $R^n$. So "no conjugate points" implies uniqueness of shortest geodesic connecting two points. – user60933 Nov 11 '16 at 23:56
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@Seub, thanks, I change the way of written. – user60933 Nov 12 '16 at 16:40