# 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

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

- \chi_(x,y) = t = \frac.

- \begin

- t = \frac . \,\!

### 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.

### 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

allotropic, assorted, divers, diverse, diversified, diversiform, heteromorphic, heteromorphous, increased, many, many-sided, metamorphic, metamorphotic, miscellaneous, multifarious, multifold, multiform, multiphase, multiple, multiplex, multiplied, multitudinous, numerous, polymorphic, polymorphous, polynomial, protean, proteiform, sundry, varied, various