Oriented manifold

It is a manifold $(\mathcal{M},\mathcal{O},\mathcal{A})$ whose atlas can be reduced to an oriented atlas:

Definition: An oriented atlas on a manifold $(M,O,A)$ is a subatlas $A^{+}\subseteq A$ such that for all overlapping charts $(U,x),(V,y)\in A^{+}$,