![Fiddler crab mobius strip](https://upload.wikimedia.org/wikipedia/commons/b/b7/Fiddler_crab_mobius_strip.gif)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, orientability is a property of some
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s such as
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
s,
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
s,
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s, and more generally
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s that allows a consistent definition of "clockwise" and "counterclockwise". 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
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
s in it, and coming back to the starting point. This means that a
geometric shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie on ...
, such as
![Small pie](https://upload.wikimedia.org/wikipedia/commons/7/71/Small_pie.svg)
, that moves continuously along such a loop is changed into its own
mirror image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
![pie 2](https://upload.wikimedia.org/wikipedia/commons/1/14/Pie_2.svg)
. A
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
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
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
, whereas for
differentiable manifolds more structure is present, allowing a formulation in terms of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s. 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
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
) 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
![Surface orientation](https://upload.wikimedia.org/wikipedia/commons/1/14/Surface_orientation.gif)
A surface ''S'' in the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
R
3 is orientable if a 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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
) 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
. 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
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
n 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 orient''ed'' surface and an orient''able'' 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.
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s,
planes, and
tori are orientable, for example. But
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...
s,
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
s, and
Klein bottle
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
s are non-orientable. They, as visualized in 3-dimensions, all have just one side. The real projective plane and Klein bottle cannot be embedded in 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 above) is orientable. For example, a torus embedded in
:
can be one-sided, and a Klein bottle in the same space can be two-sided; here
refers to the Klein bottle.
Orientation by triangulation
Any surface has a
triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle me ...
: 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
![Small pie](https://upload.wikimedia.org/wikipedia/commons/7/71/Small_pie.svg)
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''-manifold having a triangulation. However, some 4-manifolds do not have a triangulation, and in general for ''n'' > 4 some ''n''-manifolds have triangulations that are inequivalent.
Orientability and homology
If ''H''
1(''S'') denotes the first
homology
Homology may refer to:
Sciences
Biology
*Homology (biology), any characteristic of biological organisms that is derived from a common ancestor
* Sequence homology, biological homology between DNA, RNA, or protein sequences
*Homologous chrom ...
group of a surface ''S'', then ''S'' is orientable if and only if ''H''
1(''S'') has a trivial
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
. More precisely, if ''S'' is orientable then ''H''
1(''S'') is a
free abelian group
In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
, and if not then ''H''
1(''S'') = ''F'' + Z/2Z where ''F'' is free abelian, and the Z/2Z factor is generated by the middle curve in a
Möbius band
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
embedded in ''S''.
Orientability of manifolds
Let ''M'' be a connected topological ''n''-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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
The most intuitive definitions require that ''M'' be a differentiable manifold. This means that the transition functions in the atlas of ''M'' are ''C''
1-functions. Such a function admits a
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
. When the Jacobian determinant is positive, the transition function is said to be orientation preserving. An oriented atlas on ''M'' is an atlas for which all transition functions are orientation preserving. ''M'' is orientable if it admits an oriented atlas. When , an orientation of ''M'' is a maximal oriented atlas. (When , an orientation of ''M'' is a function .)
Orientability and orientations can also be expressed in terms of the tangent bundle. The tangent bundle is a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
, so it is a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
with
structure group . 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 group 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. Similar observations can be made for the frame bundle.
Another way to define orientations on a differentiable manifold is through
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
s. A volume form is a nowhere vanishing section ''ω'' of , the top exterior power of the cotangent bundle of ''M''. For example, R
''n'' has a standard volume form given by . Given a volume form on ''M'', the collection of all charts for which the standard volume form pulls back to a positive multiple of ''ω'' 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. If is a basis of tangent vectors at a point ''p'', then the basis is said to be right-handed if . 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 to . As before, this implies the orientability of ''M''. Conversely, if ''M'' 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.
This question can be resolved by defining local orientations. On a one-dimensional manifold, a local orientation around a point ''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'' 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''. By the
excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem ...
,
is isomorphic to
. The ball ''B'' is contractible, so its homology groups vanish except in degree zero, and the space is an -sphere, so its homology groups vanish except in degrees and . A computation with the
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
in
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
shows that the above homology group is isomorphic to
. A choice of generator therefore corresponds to a decision of whether, in the given chart, a sphere around ''p'' is positive or negative. A reflection of through the origin acts by negation on
, 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
. From here, the relevant definitions are the same as in the differentiable case. An oriented atlas is one for which all transition functions are orientation preserving, ''M'' is orientable if it admits an oriented atlas, and when , an orientation of ''M'' is a maximal oriented atlas.
Intuitively, an orientation of ''M'' ought to define a unique local orientation of ''M'' at each point. This is made precise by noting that any chart in the oriented atlas around ''p'' can be used to determine a sphere around ''p'', and this sphere determines a generator of
. Moreover, any other chart around ''p'' 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 that ''M'' is closed and connected, ''M'' is orientable if and only if the ''n''th homology group
is isomorphic to the integers Z. An orientation of ''M'' is a choice of generator of this group. This generator determines an oriented atlas by fixing a generator of the infinite cyclic group
and taking the oriented charts to be those for which 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
.
Orientation and cohomology
A manifold ''M'' is orientable if and only if the first
Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
vanishes. In particular, if the first cohomology group with Z/2 coefficients is zero, then the manifold is orientable. Moreover if ''M'' is orientable and ''w''
1 vanishes, then
parametrizes the choices of orientations. This characterization of orientability extends to
orientability of general vector bundles over ''M'', not just the tangent bundle.
The orientation double cover
Around each point of ''M'' there are two local orientations. Intuitively, there is a way to move from a local orientation at a point to a local orientation at a nearby point : when the two points lie in the same coordinate chart , that coordinate chart defines compatible local orientations at and . The set of local orientations can therefore be given a topology, and this topology makes it into a manifold.
More precisely, let ''O'' be the set of all local orientations of ''M''. To topologize ''O'' we will specify a subbase for its topology. Let ''U'' be an open subset of ''M'' chosen such that
is isomorphic to Z. Assume that α is a generator of this group. For each ''p'' in ''U'', there is a pushforward function
. The codomain of this group has two generators, and α maps to one of them. The topology on ''O'' is defined so that
:
is open.
There is a canonical map that sends a local orientation at ''p'' to ''p''. It is clear that every point of ''M'' has precisely two preimages under . In fact, is even a local homeomorphism, because the preimages of the open sets ''U'' mentioned above are homeomorphic to the disjoint union of two copies of ''U''. If ''M'' is orientable, then ''M'' itself is one of these open sets, so ''O'' is the disjoint union of two copies of ''M''. If ''M'' is non-orientable, however, then ''O'' is connected and orientable. The manifold ''O'' is called the orientation double cover.
Manifolds with boundary
If ''M'' is a manifold with boundary, then an orientation of ''M'' is defined to be an orientation of its interior. Such an orientation induces an orientation of ∂''M''. Indeed, suppose that an orientation of ''M'' is fixed. Let be a chart at a boundary point of ''M'' which, when restricted to the interior of ''M'', 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''.
When ''M'' is smooth, at each point ''p'' of ∂''M'', the restriction of the tangent bundle of ''M'' to ∂''M'' is isomorphic to , where the factor of R is described by the inward pointing normal vector. The orientation of ''T''
''p''∂''M'' is defined by the condition that a basis of ''T''
''p''∂''M'' is positively oriented if and only if it, when combined with the inward pointing normal vector, defines a positively oriented basis of ''T''
''p''''M''.
Orientable double cover
A closely related notion uses the idea of
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
. 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
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
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-to-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.
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
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
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 ''M'', each of which corresponds to a different orientation.
Orientation of vector bundles
A real
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
, which ''a priori'' has a
GL(n)
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
structure group, is called ''orientable'' when the
structure group may be
reduced to
, the group of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with positive
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
. For the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
, 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
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
real
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
: a smooth manifold is defined to be orientable if its
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
. These play a role in the
causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Introduction
In modern physics (especially general relativity) spacetime is represented by a Lorentzian ma ...
of spacetime. In the context of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, a
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
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
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''- dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the ...
O(''p'',''q'') has a pair of
characters
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
: the space orientation character σ
+ and the time orientation character σ
−,
:
Their product σ = σ
+σ
− is the determinant, which gives the orientation character. A space-orientation of a pseudo-Riemannian manifold is identified with a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces wit ...
:
where O(''M'') is the bundle of pseudo-orthogonal frames. Similarly, a time orientation is a section of the associated bundle
:
See also
*
Curve orientation
*
Orientation sheaf In the mathematical field of algebraic topology, the orientation sheaf on a manifold ''X'' of dimension ''n'' is a locally constant sheaf ''o'X'' on ''X'' such that the stalk of ''o'X'' at a point ''x'' is
:o_ = \operatorname_n(X, X - \)
(in t ...
References
External links
Orientation of manifoldsat the Manifold Atlas.
Orientation coveringat the Manifold Atlas.
Orientation of manifolds in generalized cohomology theoriesat the Manifold Atlas.
* The Encyclopedia of Mathematics article o
Orientation
{{Manifolds
Differential topology
Surfaces
Articles containing video clips
de:Orientierung (Mathematik)#Orientierung einer Mannigfaltigkeit