Some Remarks on the Geodesic Completeness of Compact Nonpositively Curved Spaces

Let X be a geodesic space. We say that X is geodesically complete if every geodesic segment β : [0, a] → X from β(0) to β(a) can be extended to a geodesic ray α : [0,∞) → X, (i.e. β(t) = α(t), for 0 ≤ t ≤ a). If X is a compact npc space (“npc” means: “non-positively curved”) then it is almost geodesically complete, see [10]. (X, with metric d, is almost geodesically complete if its universal cover X̃ satisfies the following property: there is a constant C such that for every p, q ∈ X̃ there is a geodesic ray α : [0,∞) → X̃, α(0) = p, and d(q, α) ≤ C.) Then it is natural to ask if in fact every compact npc space has some kind of geodesically complete npc core. In view of this, we ask the following question: (this question was already stated in [1], p. 4)