Geodesic
A Flat Nonmetrizable Connexion
Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 205-206

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A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R 2\{0} is covered by a smooth atlas with two charts 
Helfenstein, H. G. A Flat Nonmetrizable Connexion. Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 205-206. doi: 10.4153/CMB-1972-037-2
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     title = {A {Flat} {Nonmetrizable} {Connexion}},
     journal = {Canadian mathematical bulletin},
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     year = {1972},
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     doi = {10.4153/CMB-1972-037-2},
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