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Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the next properties

  • $(A)$ Let $\sigma_n: [0,a_n]\to \Sigma$ be minimizing geodesics such that $p_n:=\sigma_n(0)\to p$ and $q_n:=\sigma_n(a_n)\to q$, $p,q\in \Sigma$. For any such sequence there is a minimizing geodesic $\sigma:[0,a]\to\Sigma$, $\sigma(0)=p$, $\sigma(a)=q$ to which some subsequence $\sigma_{n_k}$ converges.

  • $(A')$ Same as $(A)$ but the final part ``to which some subsequence $\sigma_{n_k}$ converges'' is dropped,

  • $(B)$ for any two points $p,q\in \Sigma$, there is a minimizing geodesic connecting them,

Clearly $(A) \Rightarrow (A')$. Questions: 1) is $(A) \Rightarrow (B)$ true? $\quad$ 2) Is the stronger $(A') \Rightarrow (B)$ true?

$(\Sigma, g)$ is not assumed to be complete otherwise $(B)$ follows from the Hopt-Rinow theorem (cf. Jost 2011 Riemannian geometry and geometric analysis). My impression is that $(A) \Rightarrow (B)$ holds for open subsets of Euclidean space because condition $(A)$ somehow implies local convexity at the boundary (see Tietze-Nakajima's theorem) but I wonder whether there are more general results of this type for Riemannian manifolds. Observe that $(A') \Rightarrow (B)$ has the following nice reformulation: let $C\subset \Sigma \times \Sigma$ be the set of pairs connected by a minimizing geodesic. If $C$ is closed then $C=\Sigma \times \Sigma$.

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We have now clarified this question which was connected to some other questions in Lorentzian geometry. It turns out that $(A) \Rightarrow (B)$ is false (and so $(A) \Rightarrow (B)$ is false too). We provided a counterexample here arXiv:2105.08998. It consists of two cupped cylinders connected by an infinite series of tubes, see Example 4.1. However, on the positive side we proved that $(B) \Rightarrow (A)$ is true (Theorem 2.8).

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