Sufficient condition for geodesic convexity/connectedness

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 $$C=\Sigma \times \Sigma$$.

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).