Orientable vector bundle

Definition

A vector bundle over a topological space is termed orientable if it satisfies the following equivalent conditions:

• The associated orientation sheaf is trivial
• For every fiber, we can choose an orientation, in such a manner that in any open neighbourhood on which the bundle is trivial, the choice of orientation is continuous (equivalently, smooth)

Relation with other properties

Stronger properties

• Trivial vector bundle
• Stably trivial vector bundle