Orientation |

**In General**

$ __Def__: An equivalence class of
continuous, non-vanishing *n*-forms on an *n*-manifold *M*,
where two such forms *ω*_{1}
~ *ω*_{2} iff there is a (strictly) positive
function *f* on *M* such that *ω*_{1}
= *f* *ω*_{2}.

* __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 *W*_{1}(T*M*) = 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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