Orientability

A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point.

This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image .

Various equivalent formulations of orientability can be given, depending on the desired application and level of generality.

Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms.

This turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip.

But Möbius strips, real projective planes, and Klein bottles are non-orientable.

The real projective plane and Klein bottle cannot be embedded in R3, only immersed with nice intersections.

For example, a torus embedded in can be one-sided, and a Klein bottle in the same space can be two-sided; here

If this is done in such a way that, when glued together, neighboring edges are pointing in the opposite direction, then this determines an orientation of the surface.

If H1(S) denotes the first homology group of a closed surface S, then S is orientable if and only if H1(S) has a trivial torsion subgroup.

More precisely, if S is orientable then H1(S) is a free abelian group, and if not then H1(S) = F + Z/2Z where F is free abelian, and the Z/2Z factor is generated by the middle curve in a Möbius band embedded in S. Let M be a connected topological n-manifold.

When the Jacobian determinant is positive, the transition function is said to be orientation preserving.

If the structure group can be reduced to the group GL+(n, R) of positive determinant matrices, or equivalently if there exists an atlas whose transition functions determine an orientation preserving linear transformation on each tangent space, then the manifold M is orientable.

Conversely, M is orientable if and only if the structure group of the tangent bundle can be reduced in this way.

Another way to define orientations on a differentiable manifold is through volume forms.

A volume form is a nowhere vanishing section ω of ⋀n T∗M, the top exterior power of the cotangent bundle of M. For example, Rn has a standard volume form given by dx1 ∧ ⋯ ∧ dxn.

Given a volume form on M, the collection of all charts U → Rn for which the standard volume form pulls back to a positive multiple of ω is an oriented atlas.

Volume forms and tangent vectors can be combined to give yet another description of orientability.

These two situations share the common feature that they are described in terms of top-dimensional behavior near p but not at p. For the general case, let M be a topological n-manifold.

A local orientation of M around a point p is a choice of generator of the group To see the geometric significance of this group, choose a chart around p. In that chart there is a neighborhood of p which is an open ball B around the origin O.

A choice of generator therefore corresponds to a decision of whether, in the given chart, a sphere around p is positive or negative.

, so the geometric significance of the choice of generator is that it distinguishes charts from their reflections.

On a topological manifold, a transition function is orientation preserving if, at each point p in its domain, it fixes the generators of

Assuming that M is closed and connected, M is orientable if and only if the nth homology group

and taking the oriented charts to be those for which α pushes forward to the fixed generator.

Intuitively, there is a way to move from a local orientation at a point p to a local orientation at a nearby point p′: when the two points lie in the same coordinate chart U → Rn, that coordinate chart defines compatible local orientations at p and p′.

Let U → Rn+ be a chart at a boundary point of M which, when restricted to the interior of M, is in the chosen oriented atlas.

[6] In the context of general relativity, a spacetime manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at the same spacetime point, and then meet again at another point, they remain right-handed with respect to one another.

If a spacetime is time-orientable then the two observers will always agree on the direction of time at both points of their meeting.

In fact, a spacetime is time-orientable if and only if any two observers can agree which of the two meetings preceded the other.