Orientation

In General
$ Def: An equivalence class of continuous, non-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 (U_i, φ_i) and (U_j, φ_j), the Jacobian of φ_j \(\circ\) (φ_i)⁻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.
> Online resources: see Wikipedia page (2D surfaces in 3D).

Time Orientability > s.a. diffeomorphisms [changing time orientation].
* Idea: A manifold M is time orientable if we can choose a Lorentzian metric on it and, in a continuous way throughout M, one of the two half-light cones at each point as the future one.
* Conditions: A sufficient condition is that the manifold be simply connected.
@ Non-time-orientable: Hadley CQG(02)gq [phenomenology].

Related Topics > s.a. Orientifold.
* Synge's theorem: If M is an even-dimensional, orientable manifold with a Riemannian metric that has positive sectional curvatures, then any closed geodesic of M is unstable (it can be shortened by a variation); Corollary: A compact, orientable, even-dimensional manifold with positive sectional curvatures is simply connected.
@ In physics: Marmo et al RPMP(05)-a0708 [electrodynamics]; Nawarajan & Visser IJMPD(16)-a1601 [physical motivation]; Lemos & Rebouças a2009 [testing space orientability from electromagnetic quantum fluctuations].

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