   #copyright

Manifold

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   On a sphere, the sum of the angles of a triangle is not equal to 180°.
   A sphere is not a Euclidean space, but locally the laws of the
   Euclidean geometry are good approximations. In a small triangle on the
   face of the earth, the sum of the angles is very nearly 180°. A sphere
   can be represented by a collection of two dimensional maps, therefore a
   sphere is a manifold.
   Enlarge
   On a sphere, the sum of the angles of a triangle is not equal to 180°.
   A sphere is not a Euclidean space, but locally the laws of the
   Euclidean geometry are good approximations. In a small triangle on the
   face of the earth, the sum of the angles is very nearly 180°. A sphere
   can be represented by a collection of two dimensional maps, therefore a
   sphere is a manifold.

   A manifold is an abstract mathematical space in which every point has a
   neighbourhood 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
   neighbourhood 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 neighbourhood 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 technical 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

   Figure 1: The four charts each map part of the circle to an open
   interval, and together cover the whole circle. The origin is understood
   to be at the center of the circle.
   Enlarge
   Figure 1: The four charts each map part of the circle to an open
   interval, and together cover the whole circle. The origin is understood
   to be at the centre of the 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, x^2 + y^2 = 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, by
   projecting onto the first coordinate, one obtains a continuous mapping
   between the semicircle and the open interval (−1, 1):

          \chi_{\mathrm{top}}(x,y) = x. \,

   Such a function is called a chart. 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 χ[top] and χ[right] each map this part
   bijectively to the interval (0, 1). Thus a function T from (0, 1) to
   itself can be constructed, which first inverts the yellow chart to
   reach the circle and then follows the green chart back to the interval.
   Let a be any number in (0, 1), then:

          T(a) =
          \chi_{\mathrm{right}}\left(\chi_{\mathrm{top}}^{-1}(a)\right) =
          \chi_{\mathrm{right}}\left(a, \sqrt{1-a^2}\right) =
          \sqrt{1-a^2}.

   Such a function is called a transition map.
   Figure 2: A circle manifold chart based on slope, covering all but one
   point of the circle.
   Enlarge
   Figure 2: A circle manifold chart based on slope, covering all but one
   point of the circle.

   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_{\mathrm{minus}}(x,y) = s = {y\over{1+x}}

   and

          \chi_{\mathrm{plus}}(x,y) = t = {y\over{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 is the mirror image, with pivot
   point (+1,0). The inverse mapping from s to (x,y) is given by

          x = {{1-s^2}\over{1+s^2}},\qquad y = {{2s}\over{1+s^2}};

   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 = {1\over 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 is
   not possible to cover the full circle with a single chart, since the
   circle is doubly connected and the line is only simply connected. Note
   that it is possible to construct a circle by "gluing" together a single
   piece of the line; this does not produce a chart, since a portion of
   the circle will be mapped to both "glued" regions at once.

Other curves

   Four manifolds from algebraic curves: ■ circles, ■ parabola,
   ■ hyperbola, ■ cubic.
   Enlarge
   Four manifolds from algebraic curves: ■ circles, ■ parabola,
   ■ hyperbola, ■ cubic.

   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 ends 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 y^2 = x^3−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. Certain special classes of manifolds also
   have additional algebraic structure; they may behave like groups, for
   instance.

Prehistory

   Before the modern concept of a manifold there were several important
   results.

   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 the Euler characteristic. For a non-intersecting graph in
   Euclidean 2-dimensional space, with V vertices (or corners), E edges
   and F faces (counting the exterior) Euler showed that V-E+F= 2. Thus 2
   is called the Euler characteristic of Euclidean 2-dimensional space. By
   contrast, the Euler characteristic of the torus is 0, since the
   complete graph on seven points can be embedded into the torus. The
   Euler characteristic of other 2-dimensional spaces is a useful
   topological invariant, which can be extended to higher dimensions using
   Betti numbers.

   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 manifolds with constant negative and
   positive curvature, respectively.

Synthesis

   Bernhard Riemann was the first 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.

   In the study of complex variables, the process of analytic continuation
   leads to the construction of manifolds.

   Abelian varieties were already implicitly known in Riemann's time as
   complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when
   considered geometrically, are also naturally manifold theories. All
   these use the notion of several characteristic axes or dimensions
   (known as generalized coordinates in the latter two cases), but these
   dimensions do not lie along the physical dimensions of width, height,
   and breadth.

   Henri Poincaré studied three-dimensional manifolds and raised a
   question, today known as the Poincaré conjecture. As of 2006, a
   consensus among experts is that recent work by Grigori Perelman may
   have answered this question, after nearly a century of effort by many
   mathematicians.

   Hermann Weyl gave an intrinsic definition for differentiable manifolds
   in 1912. 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.

Mathematical definition

   In topology, an n-manifold is a second countable Hausdorff space in
   which every point has a neighbourhood homeomorphic to an open Euclidean
   n-ball,

          \mathbf{B}^n = \{ (x_1, x_2, ..., x_n) | x_1^2 + x_2^2 + ... +
          x_n^2 < 1 \}.

   Unless otherwise stated, a manifold is an n-manifold for some positive
   integer n, perhaps with additional structure. However, some authors
   admit manifolds which are not n-manifolds, in the sense that they allow
   different connected components to have different topological dimension.

   The second countable condition excludes spaces such as the long line.
   The Hausdorff condition avoids spaces such as the one formed by
   identifying two real lines at every point except the origin:

          R × {a} and R × {b}

   with the equivalence relation

          (x,a) \sim (x,b)\; if \,x \neq 0\;

   which has a single point for each nonzero real number r plus two points
   0[a] and 0[b]. In this space all neighbourhoods of 0[a] intersect all
   neighbourhoods of 0[b].

   There are many different kinds of manifold. All manifolds are
   topological manifolds, which locally have the topology of some
   Euclidean space. If there is additional structure, the structure on
   each map must be consistent with the overlapping maps. 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 neighbourhood, and so
   differentiable on the manifold as a whole.

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 properly represent the entire Earth. 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 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 χ[top] mentioned in
   the section above, a chart for the circle.

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.

   The atlas containing all possible charts consistent with a given atlas
   is called the maximal atlas. Unlike an ordinary atlas, the maximal
   atlas of a given atlas is unique. Though it is useful for definitions,
   it is a very 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.

   In general the structure on the manifold depends on the atlas, but
   sometimes different atlases give rise to the same structure. Such
   atlases are called compatible.

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.

Charts

   The chart maps the part of the sphere with positive z coordinate to a
   disc.
   Enlarge
   The chart maps the part of the sphere with positive z coordinate to a
   disc.

   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:

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 = \{ (x,y,z) \in \mathbf{R}^3 | x^2 + y^2 + z^2 = 1 \}.

   The sphere is two-dimensional, so each chart will map part of the
   sphere to an open subset of R^2. Consider the northern hemisphere,
   which is the part with positive z coordinate (coloured red in the
   picture on the right). The function χ defined by

          χ(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 ^1⁄[s] on the first copy (the point t = 0 is not
   identified with any point on the first copy). 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.

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.

   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).

Cartesian products

   The Cartesian product of manifolds is also a manifold. Not every
   manifold can be written as a product.

   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.
   A finite cylinder is a manifold with boundary.
   Enlarge
   A finite cylinder is a manifold with boundary.

Manifold with boundary

   A manifold with boundary is a manifold with an edge. For example a
   sheet of paper with rounded corners is a 2-manifold with a
   1-dimensional boundary. The edge of an n-manifold is an (n-1)-manifold.
   A disk (circle plus interior) is a 2-manifold with boundary. Its
   boundary is a circle, a 1-manifold. 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
   neighbourhood homeomorphic to the open n-ball {(x[1], x[2], …, x[n]) |
   Σ x[i]^2 < 1}. Every boundary point has a neighbourhood homeomorphic to
   the "half" n-ball {(x[1], x[2], …, x[n]) | Σ x[i]^2 < 1 and x[1] ≥ 0}.
   The homeomorphism must send the boundary point to a point with x[1] =
   0.

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 R × [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.

Classes of manifolds

Topological manifolds

   The simplest kind of manifold to define is the topological manifold,
   which looks locally like some "ordinary" Euclidean space R^n. 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.

   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 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 the same Euclidean space. 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, 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 (a technical definition which loosely means that Taylor's
   theorem holds). The sphere can be given analytic structure, as can most
   familiar curves and surfaces.

   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 an analytic 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 u and v
   the inner product 〈u,v〉 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, gradients of functions and divergence of vector fields.

   Most familiar curves and surfaces, including n-spheres and Euclidean
   space, can be given the structure of a Riemannian manifold.

Finsler manifolds

   A Finsler manifold allows the definition of distance, but not 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 are a particularly important class of manifolds. They were
   named after Sophus Lie (last name pronounced Lee). As well as having an
   inner product they also have the structure of a topological group,
   allowing a notion of multiplication of points on the manifold. Any
   compact Lie group can be given a Riemannian manifold structure. The
   circle can be given the structure of a Lie group — the circle group.
   The group structure is then the multiplicative group of all complex
   numbers with modulus 1.

   A Euclidean vector space with the group operation of vector addition is
   an example of a non-compact Lie group. 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 modeled on C^n with holomorphic
       transition functions on chart overlaps. These manifolds are the
       basic objects of study in complex geometry. A
       one-complex-dimensional manifold is called a Riemann surface. (Note
       that an n-dimensional complex manifold has dimension 2n as a
       differentiable manifold.)
     * 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.

Invariant properties

   Unlike curves and surfaces, higher dimensional manifolds cannot be
   understood by means of visual intuition. Indeed, it is difficult or
   even impossible to decide whether two different descriptions of a
   higher-dimensional manifold refer to the same object. For this reason
   it is useful to develop concepts and criteria that describe intrinsic
   geometric and topological aspects of these mathematical objects. Such
   criteria are commonly referred to as being invariant, because they are
   the same relative to all possible descriptions of a particular
   manifold. Thus, it is possible to distinguish two manifolds if they
   disagree with respect to some invariant property. 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 single
   decision procedure can be used to decide whether two manifolds have the
   same topological configuration.

   Some invariant properties are local, and serve to characterize
   manifolds at the smallest of scales. Other invariant properties are
   global, and take account of a manifold's overall spatial structure.
   Many invariant properties relevant to manifold theory come from point
   set topology. Separability of points, or the Hausdorff property is one
   such invariant, dimension (see below) is another. Compactness,
   connectedness, and paracompactness are important global properties.
   However, many mathematicians consider separability of points and
   paracompactness to be so essential that they include them in the very
   definition of manifold.

   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.

   It is often said that, aside from dimension, a differential manifold
   has no local invariants. However, if a manifold is endowed with some
   geometric information, such Riemannian structure, then invariant local
   properties may arise. Examples include the notions of flatness and
   constant curvature for Riemannian manifolds, and the absence of torsion
   for manifolds equipped with an affine connection.

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 3-space, and (3) the real projective
   plane, which arises naturally in geometry.
   Möbius strip
   Enlarge
   Möbius strip

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.

Klein bottle

   The Klein bottle immersed in three-dimensional space.
   Enlarge
   The Klein bottle immersed in three-dimensional space.

   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. Note that 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 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.

Dimension

   Dimensionality is built right into the definition of an n-manifold.
   Dimension is a local invariant, but it does not change as one moves
   inside the manifold. However in some settings it is convenient to allow
   a single manifold to consist of several disconnected pieces, each of
   its own dimension.

Generalizations of manifolds

     * 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: 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 using sheaves instead of atlases. Because of singular
       points one cannot assume a variety is a manifold (even though
       linguistically the French variété, German Mannigfaltigkeit and
       English manifold are much the same thing).
     * CW-complexes: A CW complex is a topological space formed by gluing
       objects of different dimensionality together; for this reason they
       generally are not manifolds. However, they are of central interest
       in algebraic topology, especially in homotopy theory, where such
       dimensional defects are acceptable.

   Retrieved from " http://en.wikipedia.org/wiki/Manifold"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
