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=                              Manifold                              =
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                             Introduction
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In mathematics, a manifold is a topological space that locally
resembles Euclidean space near each point.  More precisely, each point
of an 'n'-dimensional manifold has a neighborhood that is homeomorphic
to the Euclidean space of dimension 'n'. In this more precise
terminology, a manifold is referred to as an 'n'-manifold.

One-dimensional manifolds include lines and circles, but not figure
eights (because no neighborhood of their crossing point is
homeomorphic to Euclidean 1-space).  Two-dimensional manifolds are
also called surfaces.  Examples include the plane, the sphere, and the
torus, which can all be embedded (formed without self-intersections)
in three dimensional real space, but also the Klein bottle and real
projective plane, which will always self-intersect when immersed in
three-dimensional real space.

Although a manifold locally resembles Euclidean space, meaning that
every point has a neighbourhood homeomorphic to an open subset of
Euclidean space, globally it may be not homeomorphic to Euclidean
space.  For example, the surface of the sphere is not homeomorphic to
the Euclidean plane, because (among other properties) it has the
global topological property of compactness that Euclidean space lacks,
but in a region it can be charted by means of map projections of the
region into the Euclidean plane (in the context of manifolds they are
called 'charts').  When a region appears in two neighbouring charts,
the two representations do not coincide exactly and a transformation
is needed to pass from one to the other, called a 'transition map'.

The concept of a manifold is central to many parts of geometry and
modern mathematical physics because it allows complicated structures
to be described and understood in terms of the simpler local
topological properties of Euclidean space.  Manifolds naturally arise
as solution sets of systems of equations and as graphs of functions.

Manifolds can be equipped with additional structure.  One important
class of manifolds is the class of differentiable manifolds; this
differentiable structure allows calculus to be done on manifolds.  A
Riemannian metric on a manifold allows distances and angles to be
measured.  Symplectic manifolds serve as the phase spaces in the
Hamiltonian formalism of classical mechanics, while four-dimensional
Lorentzian manifolds model spacetime in general relativity.


                         Motivating examples
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A surface is a two dimensional manifold, meaning that it locally
resembles the Euclidean plane near each point.  For example, the
surface of a globe can be described by a collection of maps (called
charts), which together form an atlas of the globe.  Although no
individual map is sufficient to cover the entire surface of the globe,
any place in the globe will be in at least one of the charts.

Many places will appear in more than one chart.  For example, a map of
North America will likely include parts of South America and the
Arctic circle.  These regions of the globe will be described in full
in separate charts, which in turn will contain parts of North America.
There is a relation between adjacent charts, called a 'transition map'
that allows them to be consistently patched together to
cover the whole of the globe.

Describing the coordinate charts on surfaces explicitly requires
knowledge of functions of two variables, because these patching
functions must map a region in the plane to another region of the
plane.  However, one-dimensional examples of manifolds (or curves) can
be described with functions of a single variable only.

Manifolds have applications in computer-graphics and augmented-reality
given the need to associate pictures (texture) to coordinates (e.g. CT
scans).
In an augmented reality setting, a picture (tangent plane) can be seen
as something associated with a coordinate  and by using sensors for
detecting movements and rotation one can have knowledge of how the
picture is oriented and placed in space.


 Circle
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After a line, the circle is the simplest example of a topological
manifold. Topology ignores bending, so a small piece of a circle is
treated exactly the same as a small piece of a line. Consider, for
instance, the top part of the unit circle, 'x'2 + 'y'2 = 1, where the
'y'-coordinate is positive (indicated by the yellow circular arc in
'Figure 1'). Any point of this arc can be uniquely described by its
'x'-coordinate. So, projection onto the first coordinate is a
continuous, and invertible, mapping from the upper arc to the open