Inmathematics,orientability is a property of sometopological spaces such asreal vector spaces,Euclidean spaces,surfaces, and more generallymanifolds that allows a consistent definition of "clockwise" and "anticlockwise".^[1] A space isorientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is anorientation of the space. Real vector spaces, Euclidean spaces, andspheres are orientable. A space isnon-orientable if "clockwise" is changed into "counterclockwise" after running through someloops in it, and coming back to the starting point. This means that ageometric shape, such as, that moves continuously along such a loop is changed into its ownmirror image. AMöbius 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 ofhomology theory, whereas fordifferentiable manifolds more structure is present, allowing a formulation in terms ofdifferential 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 (afiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

A surfaceS in theEuclidean spaceR³ isorientable if achiral 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 isnon-orientable. An abstract surface (i.e., a two-dimensionalmanifold) 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 ishomeomorphic to theMöbius 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 anorientation, and the surface is calledoriented. For surfaces embedded in Euclidean space, an orientation is specified by the choice of a continuously varyingsurface normaln at every point. If such a normal exists at all, then there are always two ways to select it:n or −n. 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.

Most surfaces encountered in the physical world are orientable.Spheres,planes, andtori are orientable, for example. ButMöbius strips,real projective planes, andKlein bottles are non-orientable. They, as visualized in 3-dimensions, all have just one side. The real projective plane and Klein bottle cannot be embedded inR³, onlyimmersed 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 asR³ above) is orientable. For example, a torus embedded in

${\displaystyle K^2\times S^1}$

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

Any surface has atriangulation: 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 anyn-manifold having a triangulation. However, some 4-manifolds do not have a triangulation, and in general forn > 4 somen-manifolds have triangulations that are inequivalent.

IfH₁(S) denotes the firsthomology group of a closed surfaceS, thenS is orientable if and only ifH₁(S) has a trivialtorsion subgroup. More precisely, ifS is orientable thenH₁(S) is afree abelian group, and if not thenH₁(S) =F +Z/2Z whereF is free abelian, and theZ/2Z factor is generated by the middle curve in aMöbius band embedded inS.

LetM be a connected topologicaln-manifold. There are several possible definitions of what it means forM to be orientable. Some of these definitions require thatM has extra structure, like being differentiable. Occasionally,n = 0 must be made into a special case. When more than one of these definitions applies toM, thenM is orientable under one definition if and only if it is orientable under the others.^[2][3]

The most intuitive definitions require thatM be a differentiable manifold. This means that the transition functions in the atlas ofM areC¹-functions. Such a function admits aJacobian determinant. When the Jacobian determinant is positive, the transition function is said to beorientation preserving. Anoriented atlas onM is an atlas for which all transition functions are orientation preserving.M isorientable if it admits an oriented atlas. Whenn > 0, anorientation ofM is a maximal oriented atlas. (Whenn = 0, an orientation ofM is a functionM → {±1}.)

Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent bundle is avector bundle, so it is afiber bundle withstructure groupGL(n,R). That is, the transition functions of the manifold induce transition functions on the tangent bundle which are fiberwise linear transformations. If the structure group can be reduced to the groupGL⁺(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 manifoldM is orientable. Conversely,M 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 throughvolume forms. A volume form is a nowhere vanishing sectionω of⋀ⁿT^∗M, the top exterior power of the cotangent bundle ofM. For example,Rⁿ has a standard volume form given bydx¹ ∧ ⋯ ∧dxⁿ. Given a volume form onM, the collection of all chartsU →Rⁿ for which the standard volume form pulls back to a positive multiple ofω is an oriented atlas. The existence of a volume form is therefore equivalent to orientability of the manifold.

Volume forms and tangent vectors can be combined to give yet another description of orientability. IfX₁, …,X_n is a basis of tangent vectors at a pointp, then the basis is said to beright-handed ifω(X₁, …,X_n) > 0. A transition function is orientation preserving if and only if it sends right-handed bases to right-handed bases. The existence of a volume form implies a reduction of the structure group of the tangent bundle or the frame bundle toGL⁺(n,R). As before, this implies the orientability ofM. Conversely, ifM 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.

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 pointp 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 nearp but not atp. For the general case, letM be a topologicaln-manifold. Alocal orientation ofM around a pointp is a choice of generator of the group

${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right).}$

To see the geometric significance of this group, choose a chart aroundp. In that chart there is a neighborhood ofp which is an open ballB around the originO. By theexcision theorem,${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$ is isomorphic to${\displaystyle H_n\left(B,B\setminus \{O\};\mathbf{Z}\right)}$. The ballB is contractible, so its homology groups vanish except in degree zero, and the spaceB \O is an(n − 1)-sphere, so its homology groups vanish except in degreesn − 1 and0. A computation with thelong exact sequence inrelative homology shows that the above homology group is isomorphic to${\displaystyle H_{n-1}\left(S^{n-1};\mathbf{Z}\right)\cong \mathbf{Z}}$. A choice of generator therefore corresponds to a decision of whether, in the given chart, a sphere aroundp is positive or negative. A reflection ofRⁿ through the origin acts by negation on${\displaystyle H_{n-1}\left(S^{n-1};\mathbf{Z}\right)}$, so the geometric significance of the choice of generator is that it distinguishes charts from their reflections.

On a topological manifold, a transition function isorientation preserving if, at each pointp in its domain, it fixes the generators of${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$. From here, the relevant definitions are the same as in the differentiable case. Anoriented atlas is one for which all transition functions are orientation preserving,M isorientable if it admits an oriented atlas, and whenn > 0, anorientation ofM is a maximal oriented atlas.

Intuitively, an orientation ofM ought to define a unique local orientation ofM at each point. This is made precise by noting that any chart in the oriented atlas aroundp can be used to determine a sphere aroundp, and this sphere determines a generator of${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$. Moreover, any other chart aroundp 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.

Purely homological definitions are also possible. Assuming thatM is closed and connected,M isorientable if and only if thenth homology group${\displaystyle H_n(M;\mathbf{Z})}$ is isomorphic to the integersZ. Anorientation ofM is a choice of generatorα of this group. This generator determines an oriented atlas by fixing a generator of the infinite cyclic group${\displaystyle H_n(M;\mathbf{Z})}$ and taking the oriented charts to be those for whichα pushes forward to the fixed generator. Conversely, an oriented atlas determines such a generator as compatible local orientations can be glued together to give a generator for the homology group${\displaystyle H_n(M;\mathbf{Z})}$.^[4]

A manifoldM is orientable if and only if the firstStiefel–Whitney class${\displaystyle w_1(M)\in H^1(M;\mathbf{Z}/2)}$ vanishes. In particular, if the first cohomology group withZ/2 coefficients is zero, then the manifold is orientable. Moreover, ifM is orientable andw₁ vanishes, then${\displaystyle H^0(M;\mathbf{Z}/2)}$ parametrizes the choices of orientations.^[5] This characterization of orientability extends toorientability of general vector bundles overM, not just the tangent bundle.

Around each point ofM there are two local orientations. Intuitively, there is a way to move from a local orientation at a pointp to a local orientation at a nearby pointp′: when the two points lie in the same coordinate chartU →Rⁿ, that coordinate chart defines compatible local orientations atp andp′. The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.

More precisely, letO be the set of all local orientations ofM. To topologizeO we will specify a subbase for its topology. LetU be an open subset ofM chosen such that${\displaystyle H_n(M,M\setminus U;\mathbf{Z})}$ is isomorphic toZ. Assume that α is a generator of this group. For eachp inU, there is a pushforward function${\displaystyle H_n(M,M\setminus U;\mathbf{Z})\to H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$. The codomain of this group has two generators, and α maps to one of them. The topology onO is defined so that

${\displaystyle \{\text{Image of }\alpha \text{ in }{H}_n\left(M,M\setminus \{p\};\mathbf{Z}\right):p\in U\}}$

is open.

There is a canonical mapπ :O →M that sends a local orientation atp top. It is clear that every point ofM has precisely two preimages underπ. In fact,π is even a local homeomorphism, because the preimages of the open setsU mentioned above are homeomorphic to the disjoint union of two copies ofU. IfM is orientable, thenM itself is one of these open sets, soO is the disjoint union of two copies ofM. IfM is non-orientable, however, thenO is connected and orientable. The manifoldO is called theorientation double cover.

IfM is a manifold with boundary, then an orientation ofM is defined to be an orientation of its interior. Such an orientation induces an orientation of ∂M. Indeed, suppose that an orientation ofM is fixed. LetU →Rⁿ₊ be a chart at a boundary point ofM which, when restricted to the interior ofM, is in the chosen oriented atlas. The restriction of this chart to ∂M is a chart of ∂M. Such charts form an oriented atlas for ∂M.

WhenM is smooth, at each pointp of ∂M, the restriction of the tangent bundle ofM to ∂M is isomorphic toT_p∂M ⊕R, where the factor ofR is described by the inward pointing normal vector. The orientation ofT_p∂M is defined by the condition that a basis ofT_p∂M is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis ofT_pM.

A closely related notion uses the idea ofcovering space. For a connected manifold${\displaystyle M}$ take${\displaystyle M^{\ast }}$, the set of pairs${\displaystyle (x,o)}$ where${\displaystyle x}$ is a point of${\displaystyle M}$ and${\displaystyle o}$ is an orientation at${\displaystyle x}$; here we assume${\displaystyle M}$ is either smooth so we can choose an orientation on the tangent space at a point or we usesingular homology to define orientation. Then for every open, oriented subset of${\displaystyle M}$ we consider the corresponding set of pairs and define that to be an open set of${\displaystyle M^{\ast }}$. This gives${\displaystyle M^{\ast }}$ a topology and the projection sending${\displaystyle (x,o)}$ to${\displaystyle x}$ is then a 2-to-1 covering map. This covering space is called theorientable double cover, as it is orientable.${\displaystyle M^{\ast }}$ is connected if and only if${\displaystyle M}$ 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 ofindex two. In the latter case (which means there is an orientation-reversing path), the subgroup corresponds to a connected double covering; this cover is orientable by construction. In the former case, one can simply take two copies of${\displaystyle M}$, each of which corresponds to a different orientation.

A realvector bundle, whicha priori has aGL(n)structure group, is calledorientable when thestructure group may bereduced to${\displaystyle G{L}^{+}(n)}$, the group ofmatrices with positivedeterminant. For thetangent 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 asmooth realmanifold: a smooth manifold is defined to be orientable if itstangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle isalways orientable, even over nonorientable manifolds.

InLorentzian geometry, there are two kinds of orientability:space orientability andtime orientability. These play a role in thecausal structure of spacetime.^[6] In the context ofgeneral relativity, aspacetime 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.^[7]

Formally, thepseudo-orthogonal group O(p,q) has a pair ofcharacters: the space orientation character σ₊ and the time orientation character σ₋,

${\displaystyle {\sigma }_{\pm }:\mathrm{O}(p,q)\to \{-1,+1\}.}$

Their product σ = σ₊σ₋ is the determinant, which gives the orientation character. A space-orientation of a pseudo-Riemannian manifold is identified with asection of theassociated bundle

${\displaystyle \mathrm{O}(M){\times }_{{\sigma }_{+}}\{-1,+1\}}$

where O(M) is the bundle of pseudo-orthogonal frames. Similarly, a time orientation is a section of the associated bundle

${\displaystyle \mathrm{O}(M){\times }_{{\sigma }_{-}}\{-1,+1\}.}$

• Curve orientation
• Orientation sheaf

• 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 onOrientation.

• Differential topology
• Surfaces

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