AskDefine | Define manifold

Dictionary Definition

manifold adj : many and varied; having many features or forms; "manifold reasons"; "our manifold failings"; "manifold intelligence"; "the multiplex opportunities in high technology" [syn: multiplex]

Noun

1 a pipe that has several lateral outlets to or from other pipes
2 a lightweight paper used with carbon paper to make multiple copies; "an original and two manifolds" [syn: manifold paper]
3 a set of points such as those of a closed surface or and analogue in three or more dimensions

Verb

1 make multiple copies of; "multiply a letter"
2 combine or increase by multiplication; "He managed to multiply his profits" [syn: multiply]

User Contributed Dictionary

English

Etymology

; from manifeald, manigfeald

Pronunciation

  • a GenAm /ˈmænɪˌfold/, /"m

Extensive Definition

A manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important. For example, lines are one-dimensional, and planes two-dimensional.
In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.
Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.
Additional structures are often defined on manifolds. Examples of manifolds with additional structure include differentiable manifolds on which one can do calculus, Riemannian manifolds on which distances and angles can be defined, symplectic manifolds which serve as the phase space in classical mechanics, and the four-dimensional pseudo-Riemannian manifolds which model space-time in general relativity.
A precise mathematical definition of a manifold is given below. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding sets and functions, and helpful to have a working knowledge of calculus and topology.

Motivational examples

Circle

The circle is the simplest example of a topological manifold after a line. Topology ignores bending, so a small piece of a circle is exactly the same as a small piece of a line. Consider, for instance, the top half of the unit circle, x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this semicircle can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the upper semicircle to the open interval (−1,1):
\chi_(x,y) = x . \,\!
Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an atlas for the circle.
The top and right charts overlap: their intersection lies in the quarter of the circle where both the x- and the y-coordinates are positive. The two charts χtop and χright each map this part into the interval (0,1). Thus a function T from (0,1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and then follows the right chart back to the interval. Let a be any number in (0,1), then:
\begin
T(a) &= \chi_\left(\chi_^(a)\right) \\ &= \chi_\left(a, \sqrt\right) \\ &= \sqrt . \end Such a function is called a transition map.
The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts
\chi_(x,y) = s = \frac
and
\chi_(x,y) = t = \frac.
Here s is the slope of the line through the point at coordinates (x,y) and the fixed pivot point (−1,0); t is the mirror image, with pivot point (+1,0). The inverse mapping from s to (x,y) is given by
\begin
x &= \frac \\ y &= \frac . \end It can easily be confirmed that x2+y2 = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with
t = \frac . \,\!
Each chart omits a single point, either (−1,0) for s or (+1,0) for t, so neither chart alone is sufficient to cover the whole circle. Topology can prove that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "glueing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility.

Other curves

Manifolds need not be connected (all in "one piece"); thus a pair of separate circles is also a manifold. They need not be closed; thus a line segment without its end points is a manifold. And they need not be finite; thus a parabola is a manifold. Putting these freedoms together, two other example manifolds are a hyperbola (two open, infinite pieces) and the locus of points on the cubic curve y2 = x3−x (a closed loop piece and an open, infinite piece).
However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line.

Enriched circle

Viewed using calculus, the circle transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable. The transition map T, and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a differentiable manifold. It is also smooth and analytic because the transition functions have these properties as well.
Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a Riemannian manifold.

History

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology.

Prehistory

Before the modern concept of a manifold there were several important results.
Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai, and Riemann developed them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively.
Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.
Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces,
V-E+F= 2.
The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a 'map' with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope. Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V=1 vertex, E=2 edges, and F=1 face. Thus the Euler characteristic of the torus is 1-2+1=0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.

Synthesis

Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables, although these ideas were way ahead of their time.
Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Simeon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology.
Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann.
Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory.

Topology of manifolds: highlights

Two-dimensional manifolds, also known as surfaces, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). Bill Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang-Mills theory), where they serve as a substitute for ordinary 'flat' space-time. Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. One of the most pervasive and flexible techniques underlying much work on the topology of manifolds is Morse theory.

Mathematical definition

details Categories of manifolds Informally, a manifold is a space that is "modeled on" Euclidean space.
There are many different kinds of manifolds and generalizations. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, most often a differentiable structure. In terms of constructing manifolds via patching, a manifold has an additional structure if the transition maps between different patches satisfy axioms beyond just continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole.
Formally, a topological manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space.
Second countable and Hausdorff are point-set conditions; second countable excludes spaces of higher cardinality such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalized manifolds are discussed in non-Hausdorff manifolds).
Locally homeomorphic to Euclidean space means that every point has a neighborhood homeomorphic to an open Euclidean n-ball,
\mathbf^n = \

Synonyms, Antonyms and Related Words

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