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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
======================================================================
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
========
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
interval (�1, 1):
: \chi_{\mathrm{top}}(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:
: \begin{align}
\chi_\mathrm{bottom}(x, y) &= x \\
\chi_\mathrm{left}(x, y) &= y \\
\chi_\mathrm{right}(x, y) &= y.
\end{align}
Together, these parts cover the whole circle and the four charts form
an atlas for the circle.
The top and right charts, \chi_\mathrm{top} and \chi_\mathrm{right}
respectively, overlap in their domain: their intersection lies in the
quarter of the circle where both the x- and the y-coordinates are
positive. Each map this part into the interval (0,1), though
differently. Thus a function
T:(0,1)\rightarrow(0,1)=\chi_\mathrm{right} \circ
\chi^{-1}_\mathrm{top} can be constructed, which takes values from the
co-domain of \chi_\mathrm{top} back to the circle using the inverse,
followed by the \chi_\mathrm{right} back to the interval. Let 'a' be
any number in (0,1), then:
: \begin{align}
T(a) &=
\chi_\mathrm{right}\left(\chi_\mathrm{top}^{-1}\left[a\right]\right)
\\
&= \chi_\mathrm{right}\left(a, \sqrt{1 - a^2}\right) \\
&= \sqrt{1 - a^2}
\end{align}
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
choice. Consider the charts
: \chi_\mathrm{minus}(x, y) = s = \frac{y}{1 + x}
and
: \chi_\mathrm{plus}(x, y) = t = \frac{y}{1 - x}
Here 's' is the slope of the line through the point at coordinates
('x','y') and the fixed pivot point (�1, 0); 't' follows similarly,
but with pivot point (+1, 0). The inverse mapping from 's' to ('x',
'y') is given by
: \begin{align}
x &= \frac{1 - s^2}{1 + s^2} \\[5pt]
y &= \frac{2s}{1 + s^2}
\end{align}
It can easily be confirmed that 'x'2 + 'y'2 = 1 for all values of the
slope 's'. These two charts provide a second atlas for the circle,
with
: t = \frac{1}{s}
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.
It can be proved 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 "gluing" the
ends, this does not produce a chart; a portion of the circle will be
mapped to both ends at once, losing invertibility.
Sphere
========
The sphere is an example of a surface. The unit sphere of implicit
equation
:
may be covered by an atlas of six charts: the plane divides the
sphere into two half spheres ( and ), which may both be mapped on the
disc by the projection on the plane of coordinates. This provides
two charts; the four other charts are provided by a similar
construction with the two other coordinate planes.
As for the circle, one may define one chart that covers the whole
sphere excluding one point. Thus two charts are sufficient, but the
sphere cannot be covered by a single chart.
This example is historically significant, as it has motivated the
terminology; it became apparent that the whole surface of the Earth
cannot have a plane representation consisting of a single map (also
called "chart", see nautical chart), and therefore one needs atlases
for covering the whole Earth surface.
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'.
Other curves
==============
Manifolds need not be connected (all in "one piece"); an example is a
pair of separate circles.
Manifolds need not be closed; thus a line segment without its end
points is a manifold. And they are never countable, unless the
dimension of the manifold is 0. Putting these freedoms together, other
examples of manifolds are a parabola, a hyperbola (two open, infinite
pieces), and the locus of points on a cubic curve (a closed loop
piece and an open, infinite piece).
However, excluded are examples like two touching circles that share a
point to form a figure-8; at the shared point a satisfactory chart
cannot be created. Even with the bending allowed by topology, the
vicinity of the shared point looks like a "+", not a line. A "+" is
not homeomorphic to a closed interval (line segment), since deleting
the center point from the "+" gives a space with four components (i.e.
pieces), whereas deleting a point from a closed interval gives a space
with at most two pieces; topological operations always preserve the
number of pieces.
Mathematical definition
======================================================================
Informally, a manifold is a space that is "modeled on" Euclidean
space.
There are many different kinds of manifolds, depending on the context.
In geometry and topology, all manifolds are topological manifolds,
possibly with additional structure, such as a differentiable
structure. A manifold can be constructed by giving a collection of
coordinate charts, that is a covering by open sets with homeomorphisms
to a Euclidean space, and patching functions: homeomorphisms from one
region of Euclidean space to another region if they correspond to the
same part of the manifold in two different coordinate charts. A
manifold can be given additional structure if the patching functions
satisfy axioms beyond 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 which are in some sense 'too large' such as
the long line, while 'Hausdorff' excludes spaces such as "the line
with two origins" (these generalizations of 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{B}^n = \left\{ (x_1, x_2, \dots, x_n)\in\mathbb{R}^n \mid
x_1^2 + x_2^2 + \cdots + x_n^2 < 1 \right\}.
More precisely, locally homeomorphic here means that each point 'm' in
the manifold 'M' has an open neighborhood homeomorphic to an open
'neighborhood' in Euclidean space, not to the unit ball specifically.
However, given such a homeomorphism, the pre-image of an \epsilon-ball
gives a homeomorphism between the unit ball and a smaller neighborhood
of 'm', so this is no loss of generality. For topological or
differentiable manifolds, one can also ask that every point have a
neighborhood homeomorphic to all of Euclidean space (as this is
diffeomorphic to the unit ball), but this cannot be done for complex
manifolds, as the complex unit ball is not holomorphic to complex
space.
Generally manifolds are taken to have a fixed dimension (the space
must be locally homeomorphic to a fixed 'n'-ball), and such a space is
called an 'n'-manifold; however, some authors admit manifolds where
different points can have different dimensions. If a manifold has a
fixed dimension, it is called a pure manifold. For example, the sphere
has a constant dimension of 2 and is therefore a pure manifold whereas
the disjoint union of a sphere and a line in three-dimensional space
is 'not' a pure manifold. Since dimension is a local invariant (i.e.
the map sending each point to the dimension of its neighbourhood over
which a chart is defined, is locally constant), each connected
component has a fixed dimension.
Scheme-theoretically, a manifold is a locally ringed space, whose
structure sheaf is locally isomorphic to the sheaf of continuous (or
differentiable, or complex-analytic, etc.) functions on Euclidean
space. This definition is mostly used when discussing analytic
manifolds in algebraic geometry.
Charts, atlases, and transition maps
======================================================================
The spherical Earth is navigated using flat maps or charts, collected
in an atlas. Similarly, a differentiable manifold can be described
using mathematical maps, called 'coordinate charts', collected in a
mathematical 'atlas'. It is not generally possible to describe a
manifold with just one chart, because the global structure of the
manifold is different from the simple structure of the charts. For
example, no single flat map can represent the entire Earth without
separation of adjacent features across the map's boundaries or
duplication of coverage. When a manifold is constructed from multiple
overlapping charts, the regions where they overlap carry information
essential to understanding the global structure.
Charts
========
A coordinate map, a coordinate chart, or simply a chart, of a manifold
is an invertible map between a subset of the manifold and a simple
space such that both the map and its inverse preserve the desired
structure. For a topological manifold, the simple space is a subset of
some Euclidean space R'n' and interest focuses on the topological
structure. This structure is preserved by homeomorphisms, invertible
maps that are continuous in both directions.
In the case of a differentiable manifold, a set of charts called an
atlas allows us to do calculus on manifolds. Polar coordinates, for
example, form a chart for the plane R2 minus the positive 'x'-axis and
the origin. Another example of a chart is the map �top mentioned in
the section above, a chart for the circle.
Perhaps the simplest way to construct a manifold is the one used in
the example above of the circle. First, a subset of R2 is identified,
and then an atlas covering this subset is constructed. The concept of
'manifold' grew historically from constructions like this. Here is
another example, applying this method to the construction of a sphere:
Atlases
=========
The description of most manifolds requires more than one chart (a
single chart is adequate for only the simplest manifolds). A specific
collection of charts which covers a manifold is called an atlas. An
atlas is not unique as all manifolds can be covered multiple ways
using different combinations of charts. Two atlases are said to be
equivalent if their union is also an atlas.
The atlas containing all possible charts consistent with a given atlas
is called the maximal atlas (i.e. an equivalence class containing that
given atlas (under the already defined equivalence relation given in
the previous paragraph)). Unlike an ordinary atlas, the maximal atlas
of a given manifold is unique. Though it is useful for definitions, it
is an abstract object and not used directly (e.g. in calculations).
Transition maps
=================
Charts in an atlas may overlap and a single point of a manifold may be
represented in several charts. If two charts overlap, parts of them
represent the same region of the manifold, just as a map of Europe and
a map of Asia may both contain Moscow. Given two overlapping charts,
a transition function can be defined which goes from an open ball in
R'n' to the manifold and then back to another (or perhaps the same)
open ball in R'n'. The resultant map, like the map 'T' in the circle
example above, is called a change of coordinates, a coordinate
transformation, a transition function, or a transition map.
Additional structure
======================
An atlas can also be used to define additional structure on the
manifold. The structure is first defined on each chart separately. If
all the transition maps are compatible with this structure, the
structure transfers to the manifold.
This is the standard way differentiable manifolds are defined. If the
transition functions of an atlas for a topological manifold preserve
the natural differential structure of R'n' (that is, if they are
diffeomorphisms), the differential structure transfers to the manifold
and turns it into a differentiable manifold. Complex manifolds are
introduced in an analogous way by requiring that the transition
functions of an atlas are holomorphic functions. For symplectic
manifolds, the transition functions must be symplectomorphisms.
The structure on the manifold depends on the atlas, but sometimes
different atlases can be said to give rise to the same structure. Such
atlases are called compatible.
These notions are made precise in general through the use of
pseudogroups.
Manifold with boundary
======================================================================
A manifold with boundary is a manifold with an edge. For example, a
sheet of paper is a 2-manifold with a 1-dimensional boundary. The
boundary of an 'n'-manifold with boundary is an -manifold. A disk
(circle plus interior) is a 2-manifold with boundary. Its boundary is
a circle, a 1-manifold. A square with interior is also a 2-manifold
with boundary. A ball (sphere plus interior) is a 3-manifold with
boundary. Its boundary is a sphere, a 2-manifold. (See also Boundary
(topology)).
In technical language, a manifold with boundary is a space containing
both interior points and boundary points. Every interior point has a
neighborhood homeomorphic to the open 'n'-ball {{nowrap|{('x'1, 'x'2,
�, 'x''n') Σ'x''i'2 < 1}.}} Every boundary point has a
neighborhood homeomorphic to the "half" 'n'-ball {{nowrap| {('x'1,
'x'2, �, 'x''n') Σ'x''i'2 < 1 and 'x'1 � 0} }}. The homeomorphism
must send each boundary point to a point with 'x'1 = 0.
Boundary and interior
=======================
Let 'M' be a manifold with boundary. The interior of 'M', denoted Int
'M', is the set of points in 'M' which have neighborhoods homeomorphic
to an open subset of R'n'. The boundary of 'M', denoted �'M', is the
complement of Int'M' in 'M'. The boundary points can be characterized
as those points which land on the boundary hyperplane of R'n'+ under
some coordinate chart.
If 'M' is a manifold with boundary of dimension 'n', then Int'M' is a
manifold (without boundary) of dimension 'n' and �'M' is a manifold
(without boundary) of dimension .
Construction
======================================================================
A single manifold can be constructed in different ways, each stressing
a different aspect of the manifold, thereby leading to a slightly
different viewpoint.
Sphere with charts
====================
A sphere can be treated in almost the same way as the circle. In
mathematics a sphere is just the surface (not the solid interior),
which can be defined as a subset of R3:
: S = \left\{ (x,y,z) \in \mathbf{R}^3 \mid x^2 + y^2 + z^2 = 1
\right\}.
The sphere is two-dimensional, so each chart will map part of the
sphere to an open subset of R2. Consider the northern hemisphere,
which is the part with positive 'z' coordinate (coloured red in the
picture on the right). The function � defined by
: \chi(x, y, z) = (x, y),\
maps the northern hemisphere to the open unit disc by projecting it on
the ('x', 'y') plane. A similar chart exists for the southern
hemisphere. Together with two charts projecting on the ('x', 'z')
plane and two charts projecting on the ('y', 'z') plane, an atlas of
six charts is obtained which covers the entire sphere.
This can be easily generalized to higher-dimensional spheres.
Patchwork
===========
A manifold can be constructed by gluing together pieces in a
consistent manner, making them into overlapping charts. This
construction is possible for any manifold and hence it is often used
as a characterisation, especially for differentiable and Riemannian
manifolds. It focuses on an atlas, as the patches naturally provide
charts, and since there is no exterior space involved it leads to an
intrinsic view of the manifold.
The manifold is constructed by specifying an atlas, which is itself
defined by transition maps. A point of the manifold is therefore an
equivalence class of points which are mapped to each other by
transition maps. Charts map equivalence classes to points of a single
patch. There are usually strong demands on the consistency of the
transition maps. For topological manifolds they are required to be
homeomorphisms; if they are also diffeomorphisms, the resulting
manifold is a differentiable manifold.
This can be illustrated with the transition map 't' = 1�'s' from the
second half of the circle example. Start with two copies of the line.
Use the coordinate 's' for the first copy, and 't' for the second
copy. Now, glue both copies together by identifying the point 't' on
the second copy with the point 's' = 1�'t' on the first copy (the
points 't' = 0 and 's' = 0 are not identified with any point on the
first and second copy, respectively). This gives a circle.
Intrinsic and extrinsic view
==============================
The first construction and this construction are very similar, but
they represent rather different points of view. In the first
construction, the manifold is seen as embedded in some Euclidean
space. This is the 'extrinsic view'. When a manifold is viewed in this
way, it is easy to use intuition from Euclidean spaces to define
additional structure. For example, in a Euclidean space it is always
clear whether a vector at some point is tangential or normal to some
surface through that point.
The patchwork construction does not use any embedding, but simply
views the manifold as a topological space by itself. This abstract
point of view is called the 'intrinsic view'. It can make it harder to
imagine what a tangent vector might be, and there is no intrinsic
notion of a normal bundle, but instead there is an intrinsic stable
normal bundle.
''n''-Sphere as a patchwork
=============================
The 'n'-sphere S'n' is a generalisation of the idea of a circle
(1-sphere) and sphere (2-sphere) to higher dimensions. An 'n'-sphere
S'n' can be constructed by gluing together two copies of R'n'. The
transition map between them is defined as
:\mathbf{R}^n \setminus \{0\} \to \mathbf{R}^n \setminus \{0\}: x
\mapsto x/\|x\|^2.
This function is its own inverse and thus can be used in both
directions. As the transition map is a smooth function, this atlas
defines a smooth manifold.
In the case 'n' = 1, the example simplifies to the circle example
given earlier.
Identifying points of a manifold
==================================
It is possible to define different points of a manifold to be same.
This can be visualized as gluing these points together in a single
point, forming a quotient space. There is, however, no reason to
expect such quotient spaces to be manifolds. Among the possible
quotient spaces that are not necessarily manifolds, orbifolds and CW
complexes are considered to be relatively well-behaved. An example of
a quotient space of a manifold that is also a manifold is the real
projective space identified as a quotient space of the corresponding
sphere.
One method of identifying points (gluing them together) is through a
right (or left) action of a group, which acts on the manifold. Two
points are identified if one is moved onto the other by some group
element. If 'M' is the manifold and 'G' is the group, the resulting
quotient space is denoted by 'M' / 'G' (or 'G' \ 'M').
Manifolds which can be constructed by identifying points include tori
and real projective spaces (starting with a plane and a sphere,
respectively).
Gluing along boundaries
=========================
Two manifolds with boundaries can be glued together along a boundary.
If this is done the right way, the result is also a manifold.
Similarly, two boundaries of a single manifold can be glued together.
Formally, the gluing is defined by a bijection between the two
boundaries. Two points are identified when they are mapped onto each
other. For a topological manifold this bijection should be a
homeomorphism, otherwise the result will not be a topological
manifold. Similarly for a differentiable manifold it has to be a
diffeomorphism. For other manifolds other structures should be
preserved.
A finite cylinder may be constructed as a manifold by starting with a
strip [0, 1] � [0, 1] and gluing a pair of opposite edges on the
boundary by a suitable diffeomorphism. A projective plane may be
obtained by gluing a sphere with a hole in it to a Möbius strip along
their respective circular boundaries.
{{Anchor|Cartesian products}} Cartesian products
==================================================
The Cartesian product of manifolds is also a manifold.
The dimension of the product manifold is the sum of the dimensions of
its factors. Its topology is the product topology, and a Cartesian
product of charts is a chart for the product manifold. Thus, an atlas
for the product manifold can be constructed using atlases for its
factors. If these atlases define a differential structure on the
factors, the corresponding atlas defines a differential structure on
the product manifold. The same is true for any other structure defined
on the factors. If one of the factors has a boundary, the product
manifold also has a boundary. Cartesian products may be used to
construct tori and finite cylinders, for example, as S1 � S1 and S1 �
[0, 1], respectively.
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.
Early development
===================
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 such geometries in 1733 but
sought only to disprove them. Gauss, Bolyai and Lobachevsky
independently discovered 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 topological 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.
Another important source of manifolds in 19th century mathematics was
analytical mechanics, as developed by Siméon 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-Louis 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
Riemann.
Poincaré's definition
=======================
In his very influential paper, Analysis Situs, Henri Poincaré gave a
definition of a (differentiable) manifold ('variété') which served as
a precursor to the modern concept of a manifold.
In the first section of Analysis Situs, Poincaré defines a manifold as
the level set of a continuously differentiable function between
Euclidean spaces that satisfies the nondegeneracy hypothesis of the
implicit function theorem. In the third section, he begins by
remarking that the graph of a continuously differentiable function is
a manifold in the latter sense. He then proposes a new, more general,
definition of manifold based on a 'chain of manifolds' ('une chaîne
des variétés').
Poincaré's notion of a 'chain of manifolds' is a precursor to the
modern notion of atlas. In particular, he considers two manifolds
defined respectively as graphs of functions \theta(y) and
\theta'\left(y'\right) . If these manifolds overlap ('a une partie
commune'), then he requires that the coordinates y depend
continuously differentiably on the coordinates y' and vice versa
(''...les y sont fonctions analytiques des y' et inversement''). In
this way he introduces a precursor to the notion of a chart and of a
transition map. It is implicit in Analysis Situs that a manifold
obtained as a 'chain' is a subset of Euclidean space.
For example, the unit circle in the plane can be thought of as the
graph of the function y = \sqrt{1 - x^2} or else the function y =
-\sqrt{1 - x^2} in a neighborhood of every point except the points (1,
0) and (�1, 0); and in a neighborhood of those points, it can be
thought of as the graph of, respectively, x = \sqrt{1 - y^2} and x =
-\sqrt{1 - y^2}. The reason the circle can be represented by a graph
in the neighborhood of every point is because the left hand side of
its defining equation x^2 + y^2 - 1 = 0 has nonzero gradient at every
point of the circle. By the implicit function theorem, every
submanifold of Euclidean space is locally the graph of a function.
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. Notably, the
Whitney embedding theorem showed that the intrinsic definition in
terms of charts was equivalent to Poincaré's definition in terms of
subsets of Euclidean space.
Topology of manifolds: highlights
===================================
Two-dimensional manifolds, also known as a 2D 'surfaces' embedded in
our common 3D space, 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, Grigori Perelman proved the Poincaré conjecture (see
the Solution of the Poincaré conjecture). William 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' spacetime. Andrey Markov Jr. showed in 1960 that no algorithm
exists for classifying four-dimensional manifolds. 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.
Topological manifolds
=======================
The simplest kind of manifold to define is the topological manifold,
which looks locally like some "ordinary" Euclidean space R'n'. By
definition, all manifolds are topological manifolds, so the phrase
"topological manifold" is usually used to emphasize that a manifold
lacks additional structure, or that only its topological properties
are being considered. Formally, a topological manifold is a
topological space locally homeomorphic to a Euclidean space. This
means that every point has a neighbourhood for which there exists a
homeomorphism (a bijective continuous function whose inverse is also
continuous) mapping that neighbourhood to R'n'. These homeomorphisms
are the charts of the manifold.
A 'topological' manifold looks locally like a Euclidean space in a
rather weak manner: while for each individual chart it is possible to
distinguish differentiable functions or measure distances and angles,
merely by virtue of being a topological manifold a space does not have
any 'particular' and 'consistent' choice of such concepts. In order to
discuss such properties for a manifold, one needs to specify further
structure and consider differentiable manifolds and Riemannian
manifolds discussed below. In particular, the same underlying
topological manifold can have several mutually incompatible classes of
differentiable functions and an infinite number of ways to specify
distances and angles.
Usually additional technical assumptions on the topological space are
made to exclude pathological cases. It is customary to require that
the space be Hausdorff and second countable.
The 'dimension' of the manifold at a certain point is the dimension of
the Euclidean space that the charts at that point map to (number 'n'
in the definition). All points in a connected manifold have the same
dimension. Some authors require that all charts of a topological
manifold map to Euclidean spaces of same dimension. In that case every
topological manifold has a topological invariant, its dimension. Other
authors allow disjoint unions of topological manifolds with differing
dimensions to be called manifolds.
Differentiable manifolds
==========================
For most applications a special kind of topological manifold, namely a
differentiable manifold, is used. If the local charts on a manifold
are compatible in a certain sense, one can define directions, tangent
spaces, and differentiable functions on that manifold. In particular
it is possible to use calculus on a differentiable manifold. Each
point of an 'n'-dimensional differentiable manifold has a tangent
space. This is an 'n'-dimensional Euclidean space consisting of the
tangent vectors of the curves through the point.
Two important classes of differentiable manifolds are smooth and
analytic manifolds. For smooth manifolds the transition maps are
smooth, that is infinitely differentiable. Analytic manifolds are
smooth manifolds with the additional condition that the transition
maps are analytic (they can be expressed as power series). The sphere
can be given analytic structure, as can most familiar curves and
surfaces.
There are also topological manifolds, i.e., locally Euclidean spaces,
which possess no differentiable structures at all.
A rectifiable set generalizes the idea of a piecewise smooth or
rectifiable curve to higher dimensions; however, rectifiable sets are
not in general manifolds.
Riemannian manifolds
======================
To measure distances and angles on manifolds, the manifold must be
Riemannian. A 'Riemannian manifold' is a differentiable manifold in
which each tangent space is equipped with an inner product in a
manner which varies smoothly from point to point. Given two tangent
vectors and , the inner product gives a real number. The dot (or
scalar) product is a typical example of an inner product. This allows
one to define various notions such as length, angles, areas (or
volumes), curvature and divergence of vector fields.
All differentiable manifolds (of constant dimension) can be given the
structure of a Riemannian manifold. The Euclidean space itself carries
a natural structure of Riemannian manifold (the tangent spaces are
naturally identified with the Euclidean space itself and carry the
standard scalar product of the space). Many familiar curves and
surfaces, including for example all -spheres, are specified as
subspaces of a Euclidean space and inherit a metric from their
embedding in it.
Finsler manifolds
===================
A Finsler manifold allows the definition of distance but does not
require the concept of angle; it is an analytic manifold in which each
tangent space is equipped with a norm, ||·||, in a manner which varies
smoothly from point to point. This norm can be extended to a metric,
defining the length of a curve; but it cannot in general be used to
define an inner product.
Any Riemannian manifold is a Finsler manifold.
Lie groups
============
Lie groups, named after Sophus Lie, are differentiable manifolds that
carry also the structure of a group which is such that the group
operations are defined by smooth maps.
A Euclidean vector space with the group operation of vector addition
is an example of a non-compact Lie group. A simple example of a
compact Lie group is the circle: the group operation is simply
rotation. This group, known as U(1), can be also characterised as the
group of complex numbers of modulus 1 with multiplication as the group
operation.
Other examples of Lie groups include special groups of matrices, which
are all subgroups of the general linear group, the group of 'n' by 'n'
matrices with non-zero determinant. If the matrix entries are real
numbers, this will be an 'n'2-dimensional disconnected manifold. The
orthogonal groups, the symmetry groups of the sphere and hyperspheres,
are 'n'('n'�1)/2 dimensional manifolds, where 'n'�1 is the dimension
of the sphere. Further examples can be found in the table of Lie
groups.
Other types of manifolds
==========================
* A 'complex manifold' is a manifold whose charts take values in
\Complex^n and whose transition functions are holomorphic on the
overlaps. These manifolds are the basic objects of study in complex
geometry. A one-complex-dimensional manifold is called a Riemann
surface. An 'n'-dimensional complex manifold has dimension 2'n' as a
real differentiable manifold.
* A 'CR manifold' is a manifold modeled on boundaries of domains in
\Complex^n.
* 'Infinite dimensional manifolds': to allow for infinite dimensions,
one may consider Banach manifolds which are locally homeomorphic to
Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic
to Fréchet spaces.
* A 'symplectic manifold' is a kind of manifold which is used to
represent the phase spaces in classical mechanics. They are endowed
with a 2-form that defines the Poisson bracket. A closely related type
of manifold is a contact manifold.
* A 'combinatorial manifold' is a kind of manifold which is
discretization of a manifold. It usually means a piecewise linear
manifold made by simplicial complexes.
* A 'digital manifold' is a special kind of combinatorial manifold
which is defined in digital space. See digital topology
Classification and invariants
======================================================================
Different notions of manifolds have different notions of
classification and invariant; in this section we focus on smooth
closed manifolds.
The classification of smooth closed manifolds is well understood 'in
principle', except in dimension 4: in low dimensions (2 and 3) it is
geometric, via the uniformization theorem and the solution of the
Poincaré conjecture, and in high dimension (5 and above) it is
algebraic, via surgery theory. This is a classification in principle:
the general question of whether two smooth manifolds are diffeomorphic
is not computable in general. Further, specific computations remain
difficult, and there are many open questions.
Orientable surfaces can be visualized, and their diffeomorphism
classes enumerated, by genus. Given two orientable surfaces, one can
determine if they are diffeomorphic by computing their respective
genera and comparing: they are diffeomorphic if and only if the genera
are equal, so the genus forms a complete set of invariants.
This is much harder in higher dimensions: higher-dimensional manifolds
cannot be directly visualized (though visual intuition is useful in
understanding them), nor can their diffeomorphism classes be
enumerated, nor can one in general determine if two different
descriptions of a higher-dimensional manifold refer to the same
object.
However, one can determine if two manifolds are 'different' if there
is some intrinsic characteristic that differentiates them. Such
criteria are commonly referred to as invariants, because, while they
may be defined in terms of some presentation (such as the genus in
terms of a triangulation), they are the same relative to all possible
descriptions of a particular manifold: they are 'invariant' under
different descriptions.
Naively, one could hope to develop an arsenal of invariant criteria
that would definitively classify all manifolds up to isomorphism.
Unfortunately, it is known that for manifolds of dimension 4 and
higher, no program exists that can decide whether two manifolds are
diffeomorphic.
Smooth manifolds have a rich set of invariants, coming from point-set
topology,
classic algebraic topology, and geometric topology. The most familiar
invariants, which are visible for surfaces, are orientability (a
normal invariant, also detected by homology) and genus (a homological
invariant).
Smooth closed manifolds have no local invariants (other than
dimension), though geometric manifolds have local invariants, notably
the curvature of a Riemannian manifold and the torsion of a manifold
equipped with an affine connection.
This distinction between local invariants and no local invariants is a
common way to distinguish between geometry and topology. All
invariants of a smooth closed manifold are thus global.
Algebraic topology is a source of a number of important global
invariant properties. Some key criteria include the 'simply
connected' property and orientability (see below). Indeed, several
branches of mathematics, such as homology and homotopy theory, and the
theory of characteristic classes were founded in order to study
invariant properties of manifolds.
Orientability
===============
In dimensions two and higher, a simple but important invariant
criterion is the question of whether a manifold admits a meaningful
orientation. Consider a topological manifold with charts mapping to
R'n'. Given an ordered basis for R'n', a chart causes its piece of the
manifold to itself acquire a sense of ordering, which in 3-dimensions
can be viewed as either right-handed or left-handed. Overlapping
charts are not required to agree in their sense of ordering, which
gives manifolds an important freedom. For some manifolds, like the
sphere, charts can be chosen so that overlapping regions agree on
their "handedness"; these are 'orientable' manifolds. For others, this
is impossible. The latter possibility is easy to overlook, because any
closed surface embedded (without self-intersection) in
three-dimensional space is orientable.
Some illustrative examples of non-orientable manifolds include: (1)
the Möbius strip, which is a manifold with boundary, (2) the Klein
bottle, which must intersect itself in its 3-space representation, and
(3) the real projective plane, which arises naturally in geometry.
Möbius strip
==============
Begin with an infinite circular cylinder standing vertically, a
manifold without boundary. Slice across it high and low to produce two
circular boundaries, and the cylindrical strip between them. This is
an orientable manifold with boundary, upon which "surgery" will be
performed. Slice the strip open, so that it could unroll to become a
rectangle, but keep a grasp on the cut ends. Twist one end 180°,
making the inner surface face out, and glue the ends back together
seamlessly. This results in a strip with a permanent half-twist: the
Möbius strip. Its boundary is no longer a pair of circles, but
(topologically) a single circle; and what was once its "inside" has
merged with its "outside", so that it now has only a 'single' side.
Similarly to the Klein Bottle below, this two dimensional surface
would need to intersect itself in two dimensions, but can easily be
constructed in three or more dimensions.
Klein bottle
==============
Take two Möbius strips; each has a single loop as a boundary.
Straighten out those loops into circles, and let the strips distort
into cross-caps. Gluing the circles together will produce a new,
closed manifold without boundary, the Klein bottle. Closing the
surface does nothing to improve the lack of orientability, it merely
removes the boundary. Thus, the Klein bottle is a closed surface with
no distinction between inside and outside. In three-dimensional space,
a Klein bottle's surface must pass through itself. Building a Klein
bottle which is not self-intersecting requires four or more dimensions
of space.
Real projective plane
=======================
Begin with a sphere centered on the origin. Every line through the
origin pierces the sphere in two opposite points called 'antipodes'.
Although there is no way to do so physically, it is possible (by
considering a quotient space) to mathematically merge each antipode
pair into a single point. The closed surface so produced is the real
projective plane, yet another non-orientable surface. It has a number
of equivalent descriptions and constructions, but this route explains
its name: all the points on any given line through the origin project
to the same "point" on this "plane".
Genus and the Euler characteristic
====================================
For two dimensional manifolds a key invariant property is the genus,
or the "number of handles" present in a surface. A torus is a sphere
with one handle, a double torus is a sphere with two handles, and so
on. Indeed, it is possible to fully characterize compact,
two-dimensional manifolds on the basis of genus and orientability. In
higher-dimensional manifolds genus is replaced by the notion of Euler
characteristic, and more generally Betti numbers and homology and
cohomology.
Maps of manifolds
======================================================================
Just as there are various types of manifolds, there are various types
of maps of manifolds. In addition to continuous functions and smooth
functions generally, there are maps with special properties. In
geometric topology a basic type are embeddings, of which knot theory
is a central example, and generalizations such as immersions,
submersions, covering spaces, and ramified covering spaces.
Basic results include the Whitney embedding theorem and Whitney
immersion theorem.
In Riemannian geometry, one may ask for maps to preserve the
Riemannian metric, leading to notions of isometric embeddings,
isometric immersions, and Riemannian submersions; a basic result is
the Nash embedding theorem.
Scalar-valued functions
=========================
A basic example of maps between manifolds are scalar-valued functions
on a manifold,
:f\colon M \to \mathbf{R} or f\colon M \to \mathbf{C},
sometimes called regular functions or functionals, by analogy with
algebraic geometry or linear algebra. These are of interest both in
their own right, and to study the underlying manifold.
In geometric topology, most commonly studied are Morse functions,
which yield handlebody decompositions, while in mathematical analysis,
one often studies solution to partial differential equations, an
important example of which is harmonic analysis, where one studies
harmonic functions: the kernel of the Laplace operator. This leads to
such functions as the spherical harmonics, and to heat kernel methods
of studying manifolds, such as hearing the shape of a drum and some
proofs of the Atiyah-Singer index theorem.
Generalizations of manifolds
======================================================================
;Infinite dimensional manifolds: The definition of a manifold can be
generalized by dropping the requirement of finite dimensionality.
Thus an infinite dimensional manifold is a topological space locally
homeomorphic to a topological vector space over the reals. This omits
the point-set axioms, allowing higher cardinalities and non-Hausdorff
manifolds; and it omits finite dimension, allowing structures such as
Hilbert manifolds to be modeled on Hilbert spaces, Banach manifolds to
be modeled on Banach spaces, and Fréchet manifolds to be modeled on
Fréchet spaces. Usually one relaxes one or the other condition:
manifolds with the point-set axioms are studied in general topology,
while infinite-dimensional manifolds are studied in functional
analysis.
;Orbifolds: An orbifold is a generalization of manifold allowing for
certain kinds of "singularities" in the topology. Roughly speaking, it
is a space which locally looks like the quotients of some simple space
('e.g.' Euclidean space) by the actions of various finite groups. The
singularities correspond to fixed points of the group actions, and the
actions must be compatible in a certain sense.
;Algebraic varieties and schemes: Non-singular algebraic varieties
over the real or complex numbers are manifolds. One generalizes this
first by allowing singularities, secondly by allowing different
fields, and thirdly by emulating the patching construction of
manifolds: just as a manifold is glued together from open subsets of
Euclidean space, an algebraic variety is glued together from affine
algebraic varieties, which are zero sets of polynomials over
algebraically closed fields. Schemes are likewise glued together from
affine schemes, which are a generalization of algebraic varieties.
Both are related to manifolds, but are constructed algebraically using
sheaves instead of atlases.
:Because of singular points, a variety is in general not a manifold,
though linguistically the French 'variété', German 'Mannigfaltigkeit'
and English 'manifold' are largely synonymous. In French an algebraic
variety is called 'une variété algébrique' (an 'algebraic variety'),
while a smooth manifold is called 'une variété différentielle' (a
'differential variety').
;Stratified space: A "stratified space" is a space that can be divided
into pieces ("strata"), with each stratum a manifold, with the strata
fitting together in prescribed ways (formally, a filtration by closed
subsets). There are various technical definitions, notably a Whitney
stratified space (see Whitney conditions) for smooth manifolds and a
topologically stratified space for topological manifolds. Basic
examples include manifold with boundary (top dimensional manifold and
codimension 1 boundary) and manifold with corners (top dimensional
manifold, codimension 1 boundary, codimension 2 corners). Whitney
stratified spaces are a broad class of spaces, including algebraic
varieties, analytic varieties, semialgebraic sets, and subanalytic
sets.
;CW-complexes: A CW complex is a topological space formed by gluing
disks of different dimensionality together. In general the resulting
space is singular, and hence not a manifold. However, they are of
central interest in algebraic topology, especially in homotopy theory,
as they are easy to compute with and singularities are not a concern.
;Homology manifolds: A homology manifold is a space that behaves like
a manifold from the point of view of homology theory. These are not
all manifolds, but (in high dimension) can be analyzed by surgery
theory similarly to manifolds, and failure to be a manifold is a local
obstruction, as in surgery theory.
;Differential spaces: Let M be a nonempty set. Suppose that some
family of real functions on M was chosen. Denote it by C \subseteq
\R^M. It is an algebra with respect to the pointwise addition and
multiplication. Let M be equipped with the topology induced by C.
Suppose also that the following conditions hold. First: for every H
\in C^\infty\left(\R^i\right), where i \in \N, and arbitrary f_1,
\dots , f_n \in C, the composition H \circ \left(f_1, \dots,
f_n\right) \in C. Second: every function, which in every point of M
locally coincides with some function from C, also belongs to C. A pair
(M, C) for which the above conditions hold, is called a Sikorski
differential space.
See also
======================================================================
* Affine geodesic: paths on manifolds
* Directional statistics: statistics on manifolds
* List of manifolds
* Timeline of manifolds
* Mathematics of general relativity
By dimension
==============
* 3-manifold
* 4-manifold
* 5-manifold
* Manifolds of mappings
References
======================================================================
* Freedman, Michael H., and Quinn, Frank (1990) 'Topology of
4-Manifolds'. Princeton University Press. .
* Guillemin, Victor and Pollack, Alan (1974) 'Differential Topology'.
Prentice-Hall. . Advanced undergraduate / first-year graduate text
inspired by Milnor.
* Hempel, John (1976) '3-Manifolds'. Princeton University Press. .
* Hirsch, Morris, (1997) 'Differential Topology'. Springer Verlag. .
The most complete account, with historical insights and excellent, but
difficult, problems. The standard reference for those wishing to have
a deep understanding of the subject.
* Kirby, Robion C. and Siebenmann, Laurence C. (1977) 'Foundational
Essays on Topological Manifolds. Smoothings, and Triangulations'.
Princeton University Press. . A detailed study of the category of
topological manifolds.
* Lee, John M. (2000) 'Introduction to Topological Manifolds'.
Springer-Verlag. . Detailed and comprehensive first-year graduate
text.
* Lee, John M. (2003)
'[
https://archive.org/details/GraduateTextsInMathematics218LeeJ.M.IntroductionTo
SmoothManifoldsSpringer2012
Introduction to Smooth Manifolds]'. Springer-Verlag. . Detailed and
comprehensive first-year graduate text; sequel to 'Introduction to
Topological Manifolds'.
* Massey, William S. (1977) 'Algebraic Topology: An Introduction'.
Springer-Verlag. .
* Milnor, John (1997) 'Topology from the Differentiable Viewpoint'.
Princeton University Press. . Classic brief introduction to
differential topology.
* Munkres, James R. (1991)
'[
https://archive.org/details/MunkresJ.R.AnalysisOnManifolds Analysis
on Manifolds]'. Addison-Wesley (reprinted by Westview Press) .
Undergraduate text treating manifolds in R'n.'
* Munkres, James R. (2000) 'Topology'. Prentice Hall. .
* Neuwirth, L. P., ed. (1975) 'Knots, Groups, and 3-Manifolds. Papers
Dedicated to the Memory of R. H. Fox'. Princeton University Press. .
* Riemann, Bernhard, 'Gesammelte mathematische Werke und
wissenschaftlicher Nachlass', Sändig Reprint. .
**'[
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/
Grundlagen für eine allgemeine Theorie der Functionen einer
veränderlichen complexen Grösse.]' The 1851 doctoral thesis in which
"manifold" ('Mannigfaltigkeit') first appears.
**'[
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ Ueber
die Hypothesen, welche der Geometrie zu Grunde liegen.]' The 1854
Göttingen inaugural lecture ('Habilitationsschrift').
* Spivak, Michael (1965)
'[
https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint
Calculus on Manifolds: A Modern Approach to Classical Theorems of
Advanced Calculus]'. W.A. Benjamin Inc. (reprinted by Addison-Wesley
and Westview Press). . Famously terse advanced undergraduate /
first-year graduate text.
* Spivak, Michael (1999) 'A Comprehensive Introduction to Differential
Geometry' (3rd edition) Publish or Perish Inc. Encyclopedic
five-volume series presenting a systematic treatment of the theory of
manifolds, Riemannian geometry, classical differential geometry, and
numerous other topics at the first- and second-year graduate levels.
* . Concise first-year graduate text.
External links
======================================================================
*
* [
http://www.dimensions-math.org/Dim_E.htm Dimensions-math.org] (A
film explaining and visualizing manifolds up to fourth dimension.)
* The [
http://www.map.mpim-bonn.mpg.de manifold atlas] project of the
[
http://www.mpim-bonn.mpg.de Max Planck Institute for Mathematics in
Bonn]
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=========
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Original Article:
http://en.wikipedia.org/wiki/Manifold
