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 neighbourhood 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 they have crossing points that are not locally 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 not: manifolds in general are 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.

The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
Polar stereographic projections
The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.

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 (i.e 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 with overlapping manifold charts
Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole 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, x2 + y2 = 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):

Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle:

Together, these parts cover the whole circle and the four charts form an atlas for the circle.

The top and right charts, and respectively, overlap in their domain: their intersection lies in the quarter of the circle where both the - and the -coordinates are positive. Each map this part into the interval , though differently. Thus a function can be constructed, which takes values from the co-domain of back to the circle using the inverse, followed by the back to the interval. Let a be any number in , then:

Such a function is called a transition map.

Circle manifold chart from slope
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 choice. Consider the charts


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 (xy) is given by

It can easily be confirmed that x2 + y2 = 1 for all values of the slope s. These two charts provide a second atlas for the circle, with

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.


The sphere is an example of a surface. The unit sphere of implicit equation

x2 + y2 + z2 – 1 = 0

may be covered by an atlas of six charts: the plane z = 0 divides the sphere into two half spheres (z > 0 and z < 0), which may both be mapped on the disc x2 + y2 < 1 by the projection on the xy 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

Conics and cubic
Four manifolds from algebraic curves:  circles,  parabola,  hyperbola,  cubic.

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 y2 = x3x (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,

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 -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.[1] 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.


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.[2] For a topological manifold, the simple space is a subset of some Euclidean space Rn 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 χtop mentioned in the section above, a chart for the circle.


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 Rn to the manifold and then back to another (or perhaps the same) open ball in Rn. 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 Rn (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 (n − 1)-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 {(x1, x2, …, xn) | Σxi2 < 1} . Every boundary point has a neighborhood homeomorphic to the "half" n-ball {(x1, x2, …, xn) | Σxi2 < 1 and x1 ≥ 0} . The homeomorphism must send each boundary point to a point with x1 = 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 Rn. 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 (xn = 0) of Rn+ 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 n − 1.


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 chart
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:

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 χ defined by

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.


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 = 1s 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 = 1t 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 Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sn can be constructed by gluing together two copies of Rn. The transition map between them is defined as

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.

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.

Red cylinder
A finite cylinder is a manifold with boundary.


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,

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.[3] 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.


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,[4] Henri Poincaré gave a definition of a (differentiable) manifold (variété) which served as a precursor to the modern concept of a manifold.[5]

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 and . If these manifolds overlap (a une partie commune), then he requires that the coordinates depend continuously differentiably on the coordinates and vice versa ('...les sont fonctions analytiques des et inversement'). In this way he introduces a precursor to the notion of a chart and of a transition map. Note that 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 or else the function 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, and . 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 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[6] 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.

Additional structure

Topological manifolds

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. 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 Rn. These homeomorphisms are the charts of the manifold.

It is to be noted that 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.[7]

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 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 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 n-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 n2-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

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.



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 Rn. Given an ordered basis for Rn, 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

Moebius 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

Klein bottle
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 (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

Spherical harmonics
3D color plot of the spherical harmonics of degree

A basic example of maps between manifolds are scalar-valued functions on a manifold,


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.
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.
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.[8]
Differential spaces
Let be a nonempty set. Suppose that some family of real functions on was chosen. Denote it by . It is an algebra with respect to the pointwise addition and multiplication. Let be equipped with the topology induced by . Suppose also that the following conditions hold. First: for every , where , and arbitrary , the composition . Second: every function, which in every point of locally coincides with some function from , also belongs to . A pair for which the above conditions hold, is called a Sikorski differential space.[9][10]

See also

By dimension


  1. ^ E.g. see Riaza, Ricardo (2008), Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scientific, p. 110, ISBN 9789812791818; Gunning, R. C. (1990), Introduction to Holomorphic Functions of Several Variables, Volume 2, CRC Press, p. 73, ISBN 9780534133092.
  2. ^ Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0-8218-1045-6.
  3. ^ The notion of a map can formalized as a cell decomposition.
  4. ^ Poincaré, H. (1895). "Analysis Situs". Journal de l'Ecole Polytechnique. Serié 11 (in French). Gauthier-Villars.
  5. ^ Arnolʹd, V. I. (1998). "О преподавании математики" [On Teaching Mathematics]. Uspekhi Mat. Nauk (in Russian). 53 (319): 229–234. doi:10.4213/rm5.; translation in Russian Math. Surveys 53 (1998), no. 1, 229–236
  6. ^ Whitney, H. (1936). "Differentiable Manifolds". Ann. of Math. 2. 37 (3): 645–680. doi:10.2307/1968482. JSTOR 1968482.
  7. ^ Kervaire, M. (1961). "A Manifold which does not admit any differentiable structure". Comment. Math. Helv. 35 (1): 1–14. doi:10.1007/BF02565940.
  8. ^ Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S. (1996). "Topology of homology manifolds". Annals of Mathematics. 143 (3): 435–467. JSTOR 2118532.
  9. ^ Sikorski, R. (1967). "Abstract covariant derivative". Coll. Math. 18: 251–272. doi:10.4064/cm-18-1-251-272.
  10. ^ Drachal, K. (2013). "Introduction to d–spaces theory" (PDF). Math. Aeterna. 3: 753–770.


  • Freedman, Michael H., and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3.
  • Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. Advanced undergraduate / first-year graduate text inspired by Milnor.
  • Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1.
  • Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. 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. ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
  • Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2. Detailed and comprehensive first-year graduate text.
  • Lee, John M. (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3. Detailed and comprehensive first-year graduate text; sequel to Introduction to Topological Manifolds.
  • Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6.
  • Milnor, John (1997) Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 0-691-04833-9. Classic brief introduction to differential topology.
  • Munkres, James R. (1991) Analysis on Manifolds. Addison-Wesley (reprinted by Westview Press) ISBN 0-201-51035-9. Undergraduate text treating manifolds in Rn.
  • Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2.
  • Neuwirth, L. P., ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox. Princeton University Press. ISBN 978-0-691-08170-0.
  • Riemann, Bernhard, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN 3-253-03059-8.
  • Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press). ISBN 0-8053-9021-9. 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.
  • Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7399-3.. Concise first-year graduate text.

External links

Calabi–Yau manifold

In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi (1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.

Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used.

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold.

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called transition maps.

Differentiability means different things in different contexts including: continuously differentiable, k times differentiable, smooth, and holomorphic. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields. Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.

Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.

Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and similar spaces of higher dimension. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions were also used to define rational numbers as ratios of commensurable lengths. When algebra and mathematical analysis became developed enough, this relation reversed and now it is more common to define Euclidean spaces from vector spaces, which allows using Cartesian coordinates and the power of algebra and calculus. This means that points are specified with tuples of real numbers, called coordinate vectors, and geometric shapes are defined by equations and inequalities relating these coordinates. This approach has also the advantage of allowing easily the generalization of geometry to Euclidean spaces of more than three dimensions.

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. While Euclidean space is defined by a set of axioms, these axioms do not specify how the points are to be represented. Euclidean space can, as one possible choice of representation, be modeled using Cartesian coordinates. In this case, the Euclidean space is then modeled by the real coordinate space (Rn) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.

Exhaust manifold

In automotive engineering, an exhaust manifold collects the exhaust gases from multiple cylinders into one pipe. The word manifold comes from the Old English word manigfeald (from the Anglo-Saxon manig [many] and feald [fold]) and refers to the folding together of multiple inputs and outputs (in contrast, an inlet or intake manifold supplies air to the cylinders).

Exhaust manifolds are generally simple cast iron or stainless steel units which collect engine exhaust gas from multiple cylinders and deliver it to the exhaust pipe. For many engines, there are aftermarket tubular exhaust manifolds known as headers in American English, as extractor manifolds in British and Australian English, and simply as "tubular manifolds" in British English. These consist of individual exhaust headpipes for each cylinder, which then usually converge into one tube called a collector. Headers that do not have collectors are called zoomie headers.

The most common types of aftermarket headers are made of mild steel or stainless steel tubing for the primary tubes along with flat flanges and possibly a larger diameter collector made of a similar material as the primaries. They may be coated with a ceramic-type finish (sometimes both inside and outside), or painted with a heat-resistant finish, or bare. Chrome plated headers are available but these tend to blue after use. Polished stainless steel will also color (usually a yellow tint), but less than chrome in most cases.

Another form of modification used is to insulate a standard or aftermarket manifold. This decreases the amount of heat given off into the engine bay, therefore reducing the intake manifold temperature. There are a few types of thermal insulation but three are particularly common:

Ceramic paint is sprayed or brushed onto the manifold and then cured in an oven. These are usually thin, so have little insulatory properties; however, they reduce engine bay heating by lessening the heat output via radiation.

A ceramic mixture is bonded to the manifold via thermal spraying to give a tough ceramic coating with very good thermal insulation. This is often used on performance production cars and track-only racers.

Exhaust wrap is wrapped completely around the manifold. Although this is cheap and fairly simple, it can lead to premature degradation of the manifold.The goal of performance exhaust headers is mainly to decrease flow resistance (back pressure), and to increase the volumetric efficiency of an engine, resulting in a gain in power output. The processes occurring can be explained by the gas laws, specifically the ideal gas law and the combined gas law.

Inlet manifold

In automotive engineering, an inlet manifold or intake manifold (in American English) is the part of an engine that supplies the fuel/air mixture to the cylinders. The word manifold comes from the Old English word manigfeald' (from the Anglo-Saxon manig [many] and feald [repeatedly]) and refers to the multiplying of one (pipe) into many.

In contrast, an exhaust manifold collects the exhaust gases from multiple cylinders into a smaller number of pipes – often down to one pipe.

The primary function of the intake manifold is to evenly distribute the combustion mixture (or just air in a direct injection engine) to each intake port in the cylinder head(s). Even distribution is important to optimize the efficiency and performance of the engine. It may also serve as a mount for the carburetor, throttle body, fuel injectors and other components of the engine.

Due to the downward movement of the pistons and the restriction caused by the throttle valve, in a reciprocating spark ignition piston engine, a partial vacuum (lower than atmospheric pressure) exists in the intake manifold. This manifold vacuum can be substantial, and can be used as a source of automobile ancillary power to drive auxiliary systems: power assisted brakes, emission control devices, cruise control, ignition advance, windshield wipers, power windows, ventilation system valves, etc.

This vacuum can also be used to draw any piston blow-by gases from the engine's crankcase. This is known as a positive crankcase ventilation system, in which the gases are burned with the fuel/air mixture.

The intake manifold has historically been manufactured from aluminium or cast iron, but use of composite plastic materials is gaining popularity (e.g. most Chrysler 4-cylinders, Ford Zetec 2.0, Duratec 2.0 and 2.3, and GM's Ecotec series).

Kähler manifold

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.

MAP sensor

The manifold absolute pressure sensor (MAP sensor) is one of the sensors used in an internal combustion engine's electronic control system.

Engines that use a MAP sensor are typically fuel injected. The manifold absolute pressure sensor provides instantaneous manifold pressure information to the engine's electronic control unit (ECU). The data is used to calculate air density and determine the engine's air mass flow rate, which in turn determines the required fuel metering for optimum combustion (see stoichiometry) and influence the advance or retard of ignition timing. A fuel-injected engine may alternatively use a mass airflow sensor (MAF sensor) to detect the intake airflow. A typical naturally aspirated engine configuration employs one or the other, whereas forced induction engines typically use both; a MAF sensor on the intake tract pre-turbo and a MAP sensor on the charge pipe leading to the throttle body.

MAP sensor data can be converted to air mass data using the speed-density method. Engine speed (RPM) and air temperature are also necessary to complete the speed-density calculation. The MAP sensor can also be used in OBD II (on-board diagnostics) applications to test the EGR (exhaust gas recirculation) valve for functionality, an application typical in OBD II equipped General Motors engines.

Manifold vacuum

Manifold vacuum, or engine vacuum in an internal combustion engine is the difference in air pressure between the engine's intake manifold and Earth's atmosphere.

Manifold vacuum is an effect of a piston's movement on the induction stroke and the choked flow through a throttle in the intake manifold of an engine. It is a measure of the amount of restriction of airflow through the engine, and hence of the unused power capacity in the engine. In some engines, the manifold vacuum is also used as an auxiliary power source to drive engine accessories and for the crankcase ventilation system.

Manifold vacuum should not be confused with venturi vacuum, which is an effect exploited in carburetors to establish a pressure difference roughly proportional to mass airflow and to maintain a somewhat constant air/fuel ratio. It is also used in light airplanes to provide airflow for pneumatic gyroscopic instruments.


In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a "clockwise" orientation for all loops in the manifold. Equivalently, a surface is orientable if a two-dimensional figure such as in the space cannot be moved (continuously) around the space and back to where it started so that it looks like its own mirror image .

The notion of orientability can be generalised to higher-dimensional manifolds as well. A manifold is orientable if it has a consistent choice of orientation, and a connected orientable manifold has exactly two different possible orientations. In this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds often employ methods of homology theory, whereas for differentiable manifolds more structure is present, allowing a formulation in terms of differential forms. An important generalization of the notion of orientability of a space is that of orientability of a family of spaces parameterized by some other space (a fiber bundle) for which an orientation must be selected in each of the spaces which varies continuously with respect to changes in the parameter values.

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.

Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean space described by a quadratic form, which may be isotropic.

A special case of great importance to general relativity is a Lorentzian manifold, in which one dimension has a sign opposite to that of the rest. This allows tangent vectors to be classified into timelike, null, and spacelike. Spacetime can be modeled as a four-dimensional Lorentzian manifold.

Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold (Besse 1987, p. 43).In relativity theory, the Ricci tensor is the part of the curvature of spacetime that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. Besse 1987). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric; the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.

Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M, g) is a real, smooth manifold M equipped with an inner product gp on the tangent space TpM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ gp(X|p, Y|p) is a smooth function. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an insection, length of a curve, area of a surface and higher-dimensional analogues (volume, etc.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself.

River Manifold

The River Manifold is a river in Staffordshire, England. It is a tributary of the River Dove (which also flows through the Peak District, forming the boundary between Derbyshire and Staffordshire).

The Manifold rises at Flash Head just south of Buxton near Axe Edge, at the northern edge of the White Peak, known for its limestone beds. It continues for 12 miles (19 km) before it joins the Dove. For part of its course, it runs underground (except when in spate), from Wetton Mill to Ilam. During this section it is joined by its major tributary, the River Hamps.

Villages on the river include Longnor, Hulme End and Ilam.

Its name may come from Anglo-Saxon manig-fald = "many folds", referring to its meanders.

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

Any real-valued differentiable function, H, on a symplectic manifold can serve as an energy function or Hamiltonian. Associated to any Hamiltonian is a Hamiltonian vector field; the integral curves of the Hamiltonian vector field are solutions to Hamilton's equations. The Hamiltonian vector field defines a flow on the symplectic manifold, called a Hamiltonian flow or symplectomorphism. By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in M. As a set, it is given by the disjoint union of the tangent spaces of M. That is,

where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at . There is a natural projection

defined by . This projection maps each tangent space to the single point .

The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of . By definition, a manifold is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum TME is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).

Variable-length intake manifold

In internal combustion engines, a variable-length intake manifold (VLIM),variable intake manifold (VIM), or variable intake system (VIS) is an automobile internal combustion engine manifold technology. As the name implies, VLIM/VIM/VIS can vary the length of the intake tract - in order to optimise power and torque across the range of engine speed operation, as well as help provide better fuel efficiency. This effect is often achieved by having two separate intake ports, each controlled by a valve, that open two different manifolds - one with a short path that operates at full engine load, and another with a significantly longer path that operates at lower load. The first patent issued for a variable length intake manifold was published in 1958, US Patent US2835235 by Daimler Benz AG.There are two main effects of variable intake geometry:


Variable geometry can create a beneficial air swirl pattern, or turbulence in the combustion chamber. The swirling helps distribute the fuel and form a homogeneous air-fuel mixture - this aids the initiation of the combustion process, helps minimise engine knocking, and helps facilitate complete combustion. At low revolutions per minute (rpm), the speed of the airflow is increased by directing the air through a longer path with limited capacity (i.e., cross-sectional area) - and this assists in improving low engine speed torque. At high rpms, the shorter and larger path opens when the load increases, so that a greater amount of air with least resistance can enter the chamber - this helps maximise 'top-end' power. In double overhead camshaft (DOHC) designs, the air paths may sometimes be connected to separate intake valves so the shorter path can be excluded by de-activating the intake valve itself.


A tuned intake path can have a light pressurising effect similar to a low-pressure supercharger - due to Helmholtz resonance. However, this effect occurs only over a narrow engine speed band. A variable intake can create two or more pressurized "hot spots", increasing engine output. When the intake air speed is higher, the dynamic pressure pushing the air (and/or mixture) inside the engine is increased. The dynamic pressure is proportional to the square of the inlet air speed, so by making the passage narrower or longer the speed/dynamic pressure is increased.

Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane (for instance), can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).

In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).

More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.

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