In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". It generalizes the concept of curve orientation, which for a plane simple closed curve is defined based on whether the curve interior is to the left or to the right of the curve. A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. 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 . A Möbius strip is an example of a non-orientable space.
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. A generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.
Contents
Orientable surfaces
A surface
S
{\displaystyle S}
in the Euclidean space
R
3
{\displaystyle \mathbb {R} ^{3}}
is orientable if a chiral two-dimensional figure (for example, ) cannot be moved around the surface and back to where it started so that it looks like its own mirror image (). Otherwise the surface is non-orientable. An abstract surface (i.e., a two-dimensional manifold) is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous manner. That is to say that a loop going around one way on the surface can never be continuously deformed (without overlapping itself) to a loop going around the opposite way. This turns out to be equivalent to the question of whether the surface contains no subset that is homeomorphic to the Möbius strip. Thus, for surfaces, the Möbius strip may be considered the source of all non-orientability.
For an orientable surface, a consistent choice of "clockwise" (as opposed to counter-clockwise) is called an orientation, and the surface is called oriented. For surfaces embedded in Euclidean space, an orientation is specified by the choice of a continuously varying surface normal
n
{\displaystyle \mathbf {n} }
at every point. If such a normal exists at all, then there are always two ways to select it:
n
{\displaystyle \mathbf {n} }
or
−
n
{\displaystyle -\mathbf {n} }
Examples
Most surfaces encountered in the physical world are orientable. Spheres, planes, and tori are orientable, for example. But Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in
3
{\displaystyle 3}
-dimensions, all have just one side. The real projective plane and Klein bottle cannot be embedded in
R
3
{\displaystyle \mathbb {R} ^{3}}
, only immersed with nice intersections.
Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or flipping over an edge, but simply by crawling far enough.
In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as
R
3
{\displaystyle R^{3}}
above) is orientable. For example, a torus embedded in
K
2
×
S
1
{\displaystyle K^{2}\times S^{1}}
Orientation by triangulation
Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. Each triangle is oriented by choosing a direction around the perimeter of the triangle, associating a direction to each edge of the triangle. 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. Such a choice is only possible if the surface is orientable, and in this case there are exactly two different orientations.
If the figure can be consistently positioned at all points of the surface without turning into its mirror image, then this will induce an orientation in the above sense on each of the triangles of the triangulation by selecting the direction of each of the triangles based on the order red-green-blue of colors of any of the figures in the interior of the triangle.
This approach generalizes to any
n
{\displaystyle n}
-manifold having a triangulation. However, some 4-manifolds do not have a triangulation, and in general for
n
>
4
{\displaystyle n>4}
some
n
{\displaystyle n}
-manifolds have triangulations that are inequivalent.
Orientability and homology
If
H
1
(
S
)
{\displaystyle H_{1}(S)}
denotes the first homology group of a closed surface
S
{\displaystyle S}
, then
S
{\displaystyle S}
is orientable if and only if
H
1
(
S
)
{\displaystyle H_{1}(S)}
has a trivial torsion subgroup. More precisely, if
S
{\displaystyle S}
is orientable then
H
1
(
S
)
{\displaystyle H_{1}(S)}
is a free abelian group, and if not then
Orientability of manifolds
Let M be a connected topological n-manifold. There are several possible definitions of what it means for M to be orientable. Some of these definitions require that M has extra structure, like being differentiable. Occasionally, n = 0 must be made into a special case. When more than one of these definitions applies to M, then M is orientable under one definition if and only if it is orientable under the others.
Orientability of differentiable manifolds
The most intuitive definitions require that
M
{\displaystyle M}
be a differentiable manifold. This means that the transition functions in the atlas of
M
{\displaystyle M}
are
C
1
{\displaystyle C^{1}}
-functions. Such a function admits a Jacobian determinant. When the Jacobian determinant is positive, the transition function is said to be orientation preserving. An oriented atlas on
M
{\displaystyle M}
is an atlas for which all transition functions are orientation preserving.
M
{\displaystyle M}
is orientable if it admits an oriented atlas. When
n
>
0
{\displaystyle n>0}
, an orientation of
M
{\displaystyle M}
is a maximal oriented atlas. (When
Homology and the orientability of general manifolds
At the heart of all the above definitions of orientability of a differentiable manifold is the notion of an orientation preserving transition function. This raises the question of what exactly such transition functions are preserving. They cannot be preserving an orientation of the manifold because an orientation of the manifold is an atlas, and it makes no sense to say that a transition function preserves or does not preserve an atlas of which it is a member.
This question can be resolved by defining local orientations. On a one-dimensional manifold, a local orientation around a point
p
{\displaystyle p}
corresponds to a choice of left and right near that point. On a two-dimensional manifold, it corresponds to a choice of clockwise and counter-clockwise. These two situations share the common feature that they are described in terms of top-dimensional behavior near
p
{\displaystyle p}
but not at
p
{\displaystyle p}
. For the general case, let
M
{\displaystyle M}
be a topological
n
{\displaystyle n}
-manifold. A local orientation of
M
{\displaystyle M}
Orientation and cohomology
A manifold
M
{\displaystyle M}
is orientable if and only if the first Stiefel–Whitney class
w
1
(
M
)
∈
H
1
(
M
;
Z
/
2
Z
)
{\displaystyle w_{1}(M)\in H^{1}(M;\mathbb {Z} /2\mathbb {Z} )}
vanishes. In particular, if the first cohomology group with
Z
/
2
Z
{\displaystyle \mathbb {Z} /2\mathbb {Z} }
coefficients is zero, then the manifold is orientable. Moreover, if
M
{\displaystyle M}
is orientable and
The orientation double cover
Around each point of
M
{\displaystyle M}
there are two local orientations. Intuitively, there is a way to move from a local orientation at a point
p
{\displaystyle p}
to a local orientation at a nearby point
p
′
{\displaystyle p^{\prime }}
: when the two points lie in the same coordinate chart
U
→
R
n
{\displaystyle U\to \mathbb {R} ^{n}}
, that coordinate chart defines compatible local orientations at
p
{\displaystyle p}
and
p
′
{\displaystyle p^{\prime }}
. The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.
More precisely, let
O
Manifolds with boundary
If
M
{\displaystyle M}
is a manifold with boundary, then an orientation of
M
{\displaystyle M}
is defined to be an orientation of its interior. Such an orientation induces an orientation of
∂
M
{\displaystyle \partial M}
. Indeed, suppose that an orientation of
M
{\displaystyle M}
is fixed. Let
U
→
R
+
n
{\displaystyle U\to \mathbb {R} _{+}^{n}}
be a chart at a boundary point of
M
{\displaystyle M}
which, when restricted to the interior of
M
{\displaystyle M}
, is in the chosen oriented atlas. The restriction of this chart to
Orientable double cover
A closely related notion uses the idea of covering space. For a connected manifold
M
{\displaystyle M}
take
M
∗
{\displaystyle M^{*}}
, the set of pairs
(
x
,
o
)
{\displaystyle (x,o)}
where
x
{\displaystyle x}
is a point of
M
{\displaystyle M}
and
o
{\displaystyle o}
is an orientation at
x
{\displaystyle x}
; here we assume
M
{\displaystyle M}
is either smooth so we can choose an orientation on the tangent space at a point or we use singular homology to define orientation. Then for every open, oriented subset of
Orientation of vector bundles
A real vector bundle, which a priori has a
GL
(
n
)
{\displaystyle \operatorname {GL} (n)}
structure group, is called orientable when the structure group may be reduced to
GL
+
(
n
)
{\displaystyle \operatorname {GL} ^{+}(n)}
, the group of matrices with positive determinant. For the tangent bundle, this reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a smooth real manifold: a smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even over nonorientable manifolds.
Related concepts
Lorentzian geometry
In Lorentzian geometry, there are two kinds of orientability: space orientability and time orientability. These play a role in the causal structure of spacetime. 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.
Formally, the pseudo-orthogonal group
O
(
p
,
q
)
{\displaystyle \operatorname {O} (p,q)}
has a pair of characters: the space orientation character
σ
+
{\displaystyle \sigma _{+}}
and the time orientation character
σ
−
{\displaystyle \sigma _{-}}
,
σ
±
:
O