Geodesic
Two Orientations
Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 95-106

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All trivializations of an Euclidean line bundle $\pi\colon\mathscr R\to B$ over a connected base $B$ split in two classes which can be naturally named orientations of $\pi$. In the case of an orienting sheaf of a manifold or a vector bundle, they admit a natural interpretation as orientations of these objects. This approach establishes an extension of standard classical constructions to all manifolds and vector bundles independently of orientability restrictions in the usual sense.
Mots-clés : orientation, structure group
Keywords: orientability, vector bundle, line bundle, orienting sheaf, twofold covering, cohomology class. 
@article{MZM_2008_83_1_a10,
     author = {E. G. Sklyarenko},
     title = {Two {Orientations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--106},
     publisher = {mathdoc},
     volume = {83},
     number = {1},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a10/}
}
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E. G. Sklyarenko. Two Orientations. Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 95-106. http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a10/