Orientation of a real vector bundle E can be described in terms of the O(n)- associated principal bundle. Namely, orientability is equivalent to the property that the structure group of the bundle can be reduced to SO(n) ⊂ O(n).
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© 2008 Springer-Verlag Berlin Heidelberg
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Husemöller, D., Joachim, M., Jurčo, B., Schottenloher, M. (2008). Spin Structures. In: Basic Bundle Theory and K-Cohomology Invariants. Lecture Notes in Physics, vol 726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74956-1_13
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DOI: https://doi.org/10.1007/978-3-540-74956-1_13
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