In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).