Incompleteness and nonexistence of a joining geodesic
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Problem 3.2.20 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.2 "Differential Geometry of Surfaces"
Give an example (e.g., draw a picture) to show that a connected surface can have two points which are not joined by any geodesic. What is the usual topological hypothesis that prevents this problem?
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Origin: Stated according to the original source
References
[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.