Orientation

In General
$Def: An equivalence class of continuous, nowhere-vanishing n-forms on an n-manifold M, where two such forms ω1 ~ ω2 iff there is a (strictly) positive function f on M such that ω1 = f ω2. * Applications: A specific choice of n-form is necessary for defining integrals on the manifold. * And other structure: A vector space does not come with a natural orientation. Orientability$ Def: A differentiable manifold M is orientable if, equivalently,
- There exists a continuous n-form ω ≠ 0 on M;
- There exists an atlas such that, for any two charts (Ui, φi) and (Uj, φj), the Jacobian of φj $$\circ$$ (φi)–1 is positive; or
- The frame bundle F(M) is reducible to a principal fiber bundle with group the connected component of the identity of GL(n, $$\mathbb R$$).
* Sufficient condition: The manifold M is simply connected.
* Necessary and sufficient condition: The Stiefel-Whitney class W1(TM) = 0.
* Properties: The product of orientable manifolds is also orientable.