Orientation |

**In General**

$ __Def__: An equivalence
class of continuous, nowhere-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 a1601 [physical motivation].

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jan 2016