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For orientation of vector spaces, see orientation (mathematics).
For other uses, see Orientation (disambiguation).
A torus is an orientable surface
The Möbius strip is a non-orientable surface
The Roman surface is non-orientable

In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as Small pie.svg in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image Pie 2.svg.

The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, 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. An important 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[edit]








K^2 \times S^1

can be one-sided, and a Klein bottle in the same space can be two-sided; here K^2 refers to the Klein bottle.

Orientation by triangulation




Orientability of manifolds[edit]

Topological definitions[edit]

An n-dimensional manifold (either embedded in a finite-dimensional vector space, or an abstract manifold) is called non-orientable if it is possible to take the homeomorphic image of an n-dimensional ball in the manifold and move it through the manifold and back to itself, so that at the end of the path, the ball has been reflected, using the same definition as for surfaces above. Equivalently, a n-dimensional manifold is non-orientable if it contains a homeomorphic image of the space formed by taking the direct product of a (n-1)-dimensional ball B and the unit interval [0,1] and gluing the ball B×{0} at one end to the ball B×{1} at other end with a single reflection. For surfaces, this space is a Möbius strip; for 3-manifolds, this is a solid Klein bottle.

As another alternative definition, in the language of structure groups, an orientable manifold is one whose structure group (a priori GL(n)) can be reduced to the subgroup GL+(n) of orientation-preserving transforms. Concretely, an orientable manifold is one that has a cover of open n-dimensional balls with consistent orientations (i.e. all transition maps are orientation preserving). Here one needs to define what a local orientation means, which can be done using orientations of vector bundles (a local orientation is an orientation of the tangent spaces at a point) or using singular homology (an orientation is a choice of generator of the n-th relative homology group

H_n(M, M\setminus\{p\}; \mathbb{Z})

at a point p). A manifold is then said to be orientable if one can choose local orientations consistently throughout the manifold.

Using homology allows one to define orientability for compact n-manifolds without considering local orientations. A compact n-manifold M with boundary is orientable if and only if the top homology group,

H_n(M, \partial M; \mathbb{Z}), is isomorphic to \mathbb{Z}.

Considering simplicial homology, which applies to any triangulable manifold, allows one to consider this a concrete statement about coherently orienting top-dimensional simplices in a triangulation, as done in the surface case above.

If the manifold has a differentiable structure, one can use the language of differential forms (see below).

Orientation of differential manifolds[edit]

Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in the manifold. A differentiable manifold is said to be orientable if it is possible to select coordinate transitions so that there is a consistent choice of "right-hand" in each coordinate patch. More precisely, the manifold has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A maximal such atlas gives an orientation on the manifold, and the manifold so equipped is then called oriented.[1]

Equivalently, a n-dimensional differentiable manifold is orientable if there is a consistent choice of oriented basis of tangent vectors at every point of the manifold. This can be formalized in a variety of ways, one of which is the condition that M should possess a volume form: a differential form ω of degree n which is nonzero at every point on the manifold. Given such an n-form, the atlas consisting of local diffeomorphisms sending ω to a positive multiple of the Euclidean volume form on Rn is oriented.

Orientable double cover[edit]

Animation of the Orientable double cover of the Möbius strip.

A closely related notion uses the idea of covering space. For a connected manifold M take M*, the set of pairs (x, o) where x is a point of M and o is an orientation at x; here we assume 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 M we consider the corresponding set of pairs and define that to be an open set of M*. This gives M* a topology and the projection sending (x, o) to x is then a 2-1 covering map. This covering space is called the orientable double cover, as it is orientable. M* is connected if and only if M is not orientable.


Orientation of vector bundles[edit]


See also: Euler class

Related concepts[edit]

Linear algebra[edit]

The notion of orientability is essentially derived from the topology of the real general linear group

\operatorname{GL}(n,\mathbf{R}), specifically that the lowest homotopy group is \pi_0(\operatorname{GL}(n,\mathbf{R}))=\mathbf{Z}/2

an invertible transform of a real vector space is either orientation-preserving or orientation-reversing.

This holds not only for differentiable manifolds but for topological manifolds, as the space of self-homotopy equivalences of a sphere also has two connected components, which can be denoted the "orientation-preserving" and "orientation-reversing" maps.

The analogous notion for the symmetric group is the alternating group of even permutations.

Lorentzian geometry[edit]

In Lorentzian geometry, there are two kinds of orientability: space orientability and time orientability. These play a role in the causal structure of spacetime.[2] In the context of general relativity, a space-time manifold is space orientable if, whenever two right-handed observers head off in rocket ships starting at the same space-time point, and then meet again at another point, they remain right-handed with respect to one another. If a space-time is time-orientable then the two observers will always agree on the direction of time at both points of their meeting. In fact, a space-time is time-orientable if and only if any two observers can agree which of the two meetings preceded the other.[3]


\sigma_{\pm} : \operatorname{O}(p,q)\to \{-1,+1\}.


\operatorname{O}(M) \times_{\sigma_+} \{-1,+1\}


\operatorname{O}(M) \times_{\sigma_-} \{-1,+1\}.

See also[edit]


  2. ^ S.W. Hawking, G.F.R. Ellis, (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press. ISBN 0-521-20016-4. 
  3. ^ Mark J. Hadley.The Orientability of Spacetime. Class Quantum Grav.19(2002)4565-4571 arXiv:gr-qc/0202031v4

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