Geodesic
From Path Lifting and Unique Arc Lifting to Unique Path Lifting
Canadian journal of mathematics, Tome 25 (1973) no. 1, pp. 204-212

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In [4] it was conjectured that a light map p : E → B for which paths can be lifted and lifting of arcs is unique is a Serre fibration. As is well-known this implies that paths have unique liftings. In this paper we shall prove several special cases of this conjecture.The two main theorems are: (3.5) Let p be a light compact map of a metric space E onto a connected semi-locally contractible along arcs metric space B. If arcs can be lifted uniquely then p is locally trivial. 
Bell, Harold; Ungar, Gerald S. From Path Lifting and Unique Arc Lifting to Unique Path Lifting. Canadian journal of mathematics, Tome 25 (1973) no. 1, pp. 204-212. doi: 10.4153/CJM-1973-018-0
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[1] 1. Bing, R. H., A convex metric with unique segments, Proc. Amer. Math. Soc. 4 (1953), 167–174. Google Scholar

[2] 2. Floyd, E., Some characterizations of interior maps, Ann. of Math. 51 (1950), 571–575. Google Scholar

[3] 3. Rolfsen, D., Geometric methods in topological spaces, Topology Conference, Arizona State University, 1967 (Edited by Grace, E. E.). Google Scholar

[4] 4. Ungar, G. S., Light fiber maps, Fund. Math. 62 (1968), 31–45. Google Scholar

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