Curve

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Curve

A parabola, one of the simplest curves, after (straight) lines

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.

Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width."

This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. This definition encompasses most curves that are studied in mathematics; notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations.

Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. This is the case of space-filling curves and fractal curves. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve.

A plane algebraic curve is the zero set of a polynomial in two indeterminates. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic curves defined over other fields have been widely studied. In particular, algebraic curves over a finite field are widely used in modern cryptography.

History

Megalithic art from Newgrange showing an early interest in curves

Interest in curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, for example with a stick on the sand on a beach.

Historically, the term line was used in place of the more modern term curve. Hence the terms straight line and right line were used to distinguish what are today called lines from curved lines. For example, in Book I of Euclid's Elements, a line is defined as a "breadthless length" (Def. 2), while a straight line is defined as "a line that lies evenly with the points on itself" (Def. 4). Euclid's idea of a line is perhaps clarified by the statement "The extremities of a line are points," (Def. 3). Later commentators further classified lines according to various schemes. For example:

• Composite lines (lines forming an angle)
• Incomposite lines
  • Determinate (lines that do not extend indefinitely, such as the circle)
  • Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)

The curves created by slicing a cone (conic sections) were among the curves studied in ancient Greek mathematics.

The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction. These curves include:

• The conic sections, studied in depth by Apollonius of Perga
• The cissoid of Diocles, studied by Diocles and used as a method to double the cube.
• The conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle.
• The Archimedean spiral, studied by Archimedes as a method to trisect an angle and square the circle.
• The spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius.

Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction.

A fundamental advance in the theory of curves was the introduction of analytic geometry by René Descartes in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between algebraic curves that can be defined using polynomial equations, and transcendental curves that cannot. Previously, curves had been described as "geometrical" or "mechanical" according to how they were, or supposedly could be, generated.

Conic sections were applied in astronomy by Kepler. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

Since the nineteenth century, curve theory is viewed as the special case of dimension one of the theory of manifolds and algebraic varieties. Nevertheless, many questions remain specific to curves, such as space-filling curves, Jordan curve theorem and Hilbert's sixteenth problem.

Topological curve

A topological curve can be specified by a continuous function ${\displaystyle \gamma :I\to X}$ from an interval I of the real numbers into a topological space X. Properly speaking, the curve is the image of ${\displaystyle \gamma .}$ However, in some contexts, ${\displaystyle \gamma }$ itself is called a curve, especially when the image does not look like what is generally called a curve and does not characterize sufficiently ${\displaystyle \gamma .}$

For example, the image of the Peano curve or, more generally, a space-filling curve completely fills a square, and therefore does not give any information on how ${\displaystyle \gamma }$ is defined.

A curve ${\displaystyle \gamma }$ is closed or is a loop if ${\displaystyle I=[a,b]}$ and ${\displaystyle \gamma (a)=\gamma (b)}$. A closed curve is thus the image of a continuous mapping of a circle. A non-closed curve may also be called an open curve.

If the domain of a topological curve is a closed and bounded interval ${\displaystyle I=[a,b]}$, the curve is called a path, also known as topological arc (or just arc).

A curve is simple if it is the image of an interval or a circle by an injective continuous function. In other words, if a curve is defined by a continuous function ${\displaystyle \gamma }$ with an interval as a domain, the curve is simple if and only if any two different points of the interval have different images, except, possibly, if the points are the endpoints of the interval. Intuitively, a simple curve is a curve that "does not cross itself and has no missing points" (a continuous non-self-intersecting curve).

A plane curve is a curve for which ${\displaystyle X}$ is the Euclidean plane—these are the examples first encountered—or in some cases the projective plane. A space curve is a curve for which ${\displaystyle X}$ is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves, although the above definition of a curve does not apply (a real algebraic curve may be disconnected).

A dragon curve with a positive area

A plane simple closed curve is also called a Jordan curve. It is also defined as a non-self-intersecting continuous loop in the plane. The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both connected).

The definition of a curve includes figures that can hardly be called curves in common usage. For example, the image of a curve can cover a square in the plane (space-filling curve), and a simple curve may have a positive area. Fractal curves can have properties that are strange for the common sense. For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake) and even a positive area. An example is the dragon curve, which has many other unusual properties.

Differentiable curve

Main article: Differentiable curve

Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function ${\displaystyle \gamma :I\to X}$ from an interval I of the real numbers into a differentiable manifold X, often ${\displaystyle {\mathbb{R}}^n.}$

More precisely, a differentiable curve is a subset C of X where every point of C has a neighborhood U such that ${\displaystyle C\cap U}$ is diffeomorphic to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one.

Differentiable arc

"Arc (geometry)" redirects here. For the use in finite projective geometry, see Arc (projective geometry). For other uses, see Arc (disambiguation).

In Euclidean geometry, an arc (symbol: ⌒) is a connected subset of a differentiable curve.

Arcs of lines are called segments, rays, or lines, depending on how they are bounded.

A common curved example is an arc of a circle, called a circular arc.

In a sphere (or a spheroid), an arc of a great circle (or a great ellipse) is called a great arc.

Length of a curve

Main article: Arc length

Further information: Differentiable curve § Length

If ${\displaystyle X={\mathbb{R}}^n}$ is the ${\displaystyle n}$-dimensional Euclidean space, and if ${\displaystyle \gamma :[a,b]\to {\mathbb{R}}^n}$ is an injective and continuously differentiable function, then the length of ${\displaystyle \gamma }$ is defined as the quantity

${\displaystyle \mathrm{Length}(\gamma )\stackrel{\text{def}}{=}{\int }_a^b|\gamma {}^{\prime }(t)|\mathrm{d}t.}$

The length of a curve is independent of the parametrization ${\displaystyle \gamma }$.

In particular, the length ${\displaystyle s}$ of the graph of a continuously differentiable function ${\displaystyle y=f(x)}$ defined on a closed interval ${\displaystyle [a,b]}$ is

${\displaystyle s={\int }_a^b\sqrt{1+[f^{\prime }(x){]}^2}\mathrm{d}x.}$

More generally, if ${\displaystyle X}$ is a metric space with metric ${\displaystyle d}$, then we can define the length of a curve ${\displaystyle \gamma :[a,b]\to X}$ by

${\displaystyle \mathrm{Length}(\gamma )\stackrel{\text{def}}{=}sup\left\{\sum _{i=1}^{n}d(\gamma (t_i),\gamma (t_{i-1}))|n\in \mathbb{N}\text{and}a=t_0<t_1<\dots <t_n=b\right\},}$

where the supremum is taken over all ${\displaystyle n\in \mathbb{N}}$ and all partitions ${\displaystyle t_0<t_1<\dots <t_n}$ of ${\displaystyle [a,b]}$.

A rectifiable curve is a curve with finite length. A curve ${\displaystyle \gamma :[a,b]\to X}$ is called natural (or unit-speed or parametrized by arc length) if for any ${\displaystyle t_1,t_2\in [a,b]}$ such that ${\displaystyle t_1\le t_2}$, we have

${\displaystyle \mathrm{Length}\left(\gamma {|}_{[t_1,t_2]}\right)=t_2-t_1.}$

If ${\displaystyle \gamma :[a,b]\to X}$ is a Lipschitz-continuous function, then it is automatically rectifiable. Moreover, in this case, one can define the speed (or metric derivative) of ${\displaystyle \gamma }$ at ${\displaystyle t\in [a,b]}$ as

${\displaystyle {\mathrm{Speed}}_{\gamma }(t)\stackrel{\text{def}}{=}\underset{s\to t}{lim sup}\frac{d(\gamma (s),\gamma (t))}{|s-t|}}$

and then show that

${\displaystyle \mathrm{Length}(\gamma )={\int }_a^b{\mathrm{Speed}}_{\gamma }(t)\mathrm{d}t.}$

Differential geometry

Main article: Differential geometry of curves

While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime.

If ${\displaystyle X}$ is a differentiable manifold, then we can define the notion of differentiable curve in ${\displaystyle X}$. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take ${\displaystyle X}$ to be Euclidean space. On the other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to ${\displaystyle X}$ by means of this notion of curve.

If ${\displaystyle X}$ is a smooth manifold, a smooth curve in ${\displaystyle X}$ is a smooth map

${\displaystyle \gamma :I\to X}$.

This is a basic notion. There are less and more restricted ideas, too. If ${\displaystyle X}$ is a ${\displaystyle C^k}$ manifold (i.e., a manifold whose charts are ${\displaystyle k}$ times continuously differentiable), then a ${\displaystyle C^k}$ curve in ${\displaystyle X}$ is such a curve which is only assumed to be ${\displaystyle C^k}$ (i.e. ${\displaystyle k}$ times continuously differentiable). If ${\displaystyle X}$ is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and ${\displaystyle \gamma }$ is an analytic map, then ${\displaystyle \gamma }$ is said to be an analytic curve.

A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two ${\displaystyle C^k}$ differentiable curves

${\displaystyle {\gamma }_1:I\to X}$ and

${\displaystyle {\gamma }_2:J\to X}$

are said to be equivalent if there is a bijective ${\displaystyle C^k}$ map

${\displaystyle p:J\to I}$

such that the inverse map

${\displaystyle p^{-1}:I\to J}$

is also ${\displaystyle C^k}$, and

${\displaystyle {\gamma }_2(t)={\gamma }_1(p(t))}$

for all ${\displaystyle t}$. The map ${\displaystyle {\gamma }_2}$ is called a reparametrization of ${\displaystyle {\gamma }_1}$; and this makes an equivalence relation on the set of all ${\displaystyle C^k}$ differentiable curves in ${\displaystyle X}$. A ${\displaystyle C^k}$ arc is an equivalence class of ${\displaystyle C^k}$ curves under the relation of reparametrization.

Algebraic curve

Main article: Algebraic curve

Algebraic curves are the curves considered in algebraic geometry. A plane algebraic curve is the set of the points of coordinates x, y such that f(x, y) = 0, where f is a polynomial in two variables defined over some field F. One says that the curve is defined over F. Algebraic geometry normally considers not only points with coordinates in F but all the points with coordinates in an algebraically closed field K.

If C is a curve defined by a polynomial f with coefficients in F, the curve is said to be defined over F.

In the case of a curve defined over the real numbers, one normally considers points with complex coordinates. In this case, a point with real coordinates is a real point, and the set of all real points is the real part of the curve. It is therefore only the real part of an algebraic curve that can be a topological curve (this is not always the case, as the real part of an algebraic curve may be disconnected and contain isolated points). The whole curve, that is the set of its complex point is, from the topological point of view a surface. In particular, the nonsingular complex projective algebraic curves are called Riemann surfaces.

The points of a curve C with coordinates in a field G are said to be rational over G and can be denoted C(G). When G is the field of the rational numbers, one simply talks of rational points. For example, Fermat's Last Theorem may be restated as: For n > 2, every rational point of the Fermat curve of degree n has a zero coordinate.

Algebraic curves can also be space curves, or curves in a space of higher dimension, say n. They are defined as algebraic varieties of dimension one. They may be obtained as the common solutions of at least n–1 polynomial equations in n variables. If n–1 polynomials are sufficient to define a curve in a space of dimension n, the curve is said to be a complete intersection. By eliminating variables (by any tool of elimination theory), an algebraic curve may be projected onto a plane algebraic curve, which however may introduce new singularities such as cusps or double points.

A plane curve may also be completed to a curve in the projective plane: if a curve is defined by a polynomial f of total degree d, then w^df(u/w, v/w) simplifies to a homogeneous polynomial g(u, v, w) of degree d. The values of u, v, w such that g(u, v, w) = 0 are the homogeneous coordinates of the points of the completion of the curve in the projective plane and the points of the initial curve are those such that w is not zero. An example is the Fermat curve uⁿ + vⁿ = wⁿ, which has an affine form xⁿ + yⁿ = 1. A similar process of homogenization may be defined for curves in higher dimensional spaces.

Except for lines, the simplest examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero. Elliptic curves, which are nonsingular curves of genus one, are studied in number theory, and have important applications to cryptography.

Orientability

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Orientability

A torus is an orientable surface

The Möbius strip is a non-orientable surface. Note how the disk flips with every loop.

The Roman surface is non-orientable.

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". 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.

Orientable surfaces

In this animation, a simple analogy is made using a gear that rotates according to the right-hand rule on a surface's normal vector. The orientation of the curves given by the boundaries is given by the direction in which the dots move as they are pushed by the moving gear. On a non-orientable surface, such as the Möbius strip, the boundary would have to move in both directions at once, which is not possible.

A surface S in the Euclidean space R³ 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 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.

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-dimensions, all have just one side. The real projective plane and Klein bottle cannot be embedded in R³, 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 R³ 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.

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-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 H₁(S) denotes the first homology group of a surface S, then S is orientable if and only if H₁(S) has a trivial torsion subgroup. More precisely, if S is orientable then H₁(S) is a free abelian group, and if not then H₁(S) = F + Z/2Z where F is free abelian, and the Z/2Z factor is generated by the middle curve in a Möbius band embedded in S.

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 be a differentiable manifold. This means that the transition functions in the atlas of M are C¹-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 is an atlas for which all transition functions are orientation preserving. M is orientable if it admits an oriented atlas. When n > 0, an orientation of M is a maximal oriented atlas. (When n = 0, an orientation of M is a function M → {±1}.)

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 GL(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 group GL⁺(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 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 forms. A volume form is a nowhere vanishing section ω of ⋀ⁿ T^∗M, the top exterior power of the cotangent bundle of M. For example, Rⁿ has a standard volume form given by dx¹ ∧ ⋯ ∧ dxⁿ. Given a volume form on M, the collection of all charts U → Rⁿ 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 X₁, …, X_n is a basis of tangent vectors at a point p, then the basis is said to be right-handed if ω(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 to GL⁺(n, R). 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

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

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, ${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$ is isomorphic to ${\displaystyle H_n\left(B,B\setminus \{O\};\mathbf{Z}\right)}$. The ball B is contractible, so its homology groups vanish except in degree zero, and the space B \ O is an (n − 1)-sphere, so its homology groups vanish except in degrees n − 1 and 0. A computation with the long exact sequence in relative 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 around p is positive or negative. A reflection of Rⁿ 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 is orientation preserving if, at each point p 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. An oriented atlas is one for which all transition functions are orientation preserving, M is orientable if it admits an oriented atlas, and when n > 0, 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 ${\displaystyle H_n\left(M,M\setminus \{p\};\mathbf{Z}\right)}$. 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 nth homology group ${\displaystyle H_n(M;\mathbf{Z})}$ is isomorphic to the integers Z. An orientation of M is a choice of generator α 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 which α 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 ${\displaystyle H_n(M;\mathbf{Z})}$.

Orientation and cohomology

A manifold M is orientable if and only if the first Stiefel–Whitney class ${\displaystyle w_1(M)\in H^1(M;\mathbf{Z}/2)}$ 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 w₁ vanishes, then ${\displaystyle H^0(M;\mathbf{Z}/2)}$ 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 p to a local orientation at a nearby point p′: when the two points lie in the same coordinate chart U → Rⁿ, that coordinate chart defines compatible local orientations at p and p′. 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 ${\displaystyle H_n(M,M\setminus U;\mathbf{Z})}$ is isomorphic to Z. Assume that α is a generator of this group. For each p in U, 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 on O 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 map π : O → M that sends a local orientation at p to p. It is clear that every point of M has precisely two preimages under π. In fact, π 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 U → Rⁿ₊ 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 T_p∂M ⊕ R, 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_pM.

Orientable double cover

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-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 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

Main article: Orientation of a vector bundle

A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the structure group may be reduced to ${\displaystyle G{L}^{+}(n)}$, 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.

See also: Euler class

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) has a pair of characters: 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 a section of the associated 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

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