In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfacess, 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 loopss 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.

Orientable surfaces

in the Euclidean space 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. More generally, an orientable surface admits exactly two orientations, and the distinction between an oriented surface and an orientable surface is subtle and frequently blurred. An orientable surface is an abstract surface that admits an orientation, while an oriented surface is a surface that is abstractly orientable, and has the additional datum of a choice of one of the two possible orientations.

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, 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.

above) is orientable. For example, a torus embedded in

refers to the Klein bottle.

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.

-manifolds have triangulations that are inequivalent.

Orientability and homology

denotes the first homology has a trivial torsion subgroup is a free abelian group factor is generated by the middle curve in a Möbius band.

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, 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

-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 is orientable, an orientation.)

Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle, so it is a fiber bundle with structure group is orientable if and only if the structure group of the tangent bundle can be reduced in this way. Similar observations can be made for the frame bundle.

Another way to define orientations on a differentiable manifold is through volume forms is an oriented atlas. The existence of a volume form is therefore equivalent to orientability of the manifold.

, then the basis is said to be right-handed is orientable, then local volume forms can be patched together to create a global volume form, orientability being necessary to ensure that the global form is nowhere vanishing.

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.

-manifold. A local orientation is a choice of generator of the group

. By the excision theorem. A computation with the long exact sequence in relative homology, 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. From here, the relevant definitions are the same as in the differentiable case. An oriented atlas is orientable, an orientation is a maximal oriented atlas.

is related to the first chart by an orientation preserving transition function, and this implies that the two charts yield the same generator, whence the generator is unique.

is orientable. An orientation.

Orientation and cohomology

is orientable if and only if the first Stiefel–Whitney class parametrizes the choices of orientations. This characterization of orientability extends to orientability of general vector bundles, not just the tangent bundle.

The orientation double cover

. The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.

is defined so that

is open.

is called the orientation double cover.

Manifolds with boundary

.

.

Orientable double cover

A closely related notion uses the idea of covering space is either smooth so we can choose an orientation on the tangent space at a point or we use singular homology is then a 2-to-1 covering map. This covering space is called the orientable double cover is not orientable.

Another way to construct this cover is to divide the loops based at a basepoint into either orientation-preserving or orientation-reversing loops. The orientation preserving loops generate a subgroup of the fundamental group which is either the whole group or of index, each of which corresponds to a different orientation.

Orientation of vector bundles

A real vector bundle, which a priori has a \operatorname{GL}(n) structure group, is called orientable when the structure group may be reduced, 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 has a pair of characters,

is the determinant, which gives the orientation character. A space-orientation of a pseudo-Riemannian manifold is identified with a section of the associated bundle

is the bundle of pseudo-orthogonal frames. Similarly, a time orientation is a section of the associated bundle

See also

• Curve orientation

• Orientation sheaf

References

External links

• Orientation of manifolds at the Manifold Atlas.

• Orientation covering at the Manifold Atlas.

• Orientation of manifolds in generalized cohomology theories at the Manifold Atlas.

• The Encyclopedia of Mathematics article on Orientation.

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