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 (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.
> 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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