NationMaster - Encyclopedia: Differentiable manifold

Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. It is important to note that differentiable can mean slightly different things in different contexts, such as continuously differentiable, k times differentiable, infinitely differentiable (also known as smooth), or complex differentiable (also known as holomorphic). On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... For other uses, see Calculus (disambiguation). ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Any manifold can be described by a collection (or atlas) of charts. Each chart specifies a coordinate system on a piece of the manifold, which is a function from that piece of the manifold into a Euclidean space. One may then apply ideas from calculus while working within the individual charts, since these lie in Euclidean spaces to which the usual rules of calculus apply. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

Problems can arise, however, when passing from one chart to the other. Consider the image to the right. Here the results of calculus clearly do not carry over from one chart to the other. In the middle chart, for instance, the Tropic of Cancer is a smooth curve, whereas in the first chart it has a sharp corner. The notion of a differentiable manifold refines the notion of a manifold by requiring the transition from one chart to another to be differentiable. Image File history File links No higher resolution available. ...

More formally, a differentiable manifold is a topological manifold with a globally defined differentiable structure. Any topological manifold can be given a differentiable structure locally by using the homeomorphisms in its atlas, combined with the standard differentiable structure on the Euclidean space. In other words, the homeomorphism can be used to give a local coordinate system. To induce a global differentiable structure, the compositions of the homeomorphisms on overlaps between charts in the atlas must be differentiable functions on Euclidean 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 other chart. These maps that relate the coordinates defined by the various charts to each other in areas of intersection are called transition maps. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

This allows one to extend the meaning of differentiability to spaces without global coordinate systems. Specifically, a differentiable structure allows one to define a global differentiable tangent space, and consequently, differentiable functions, and differentiable tensor fields (including vector fields). Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the arena for physical theories such as classical mechanics (Hamiltonian mechanics, Lagrangian mechanics), general relativity and Yang-Mills theory (gauge theory). It is possible to develop calculus on differentiable manifolds, leading to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry. For a non-technical overview of the subject, see Calculus. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... For other uses, see Calculus (disambiguation). ... This article may be too technical for most readers to understand. ... It has been suggested that this article be split into multiple articles. ...

History

The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture^[1] before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments: Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Bernhard Riemann. ...

The works of physicists such as James Clerk Maxwell^{[citation needed]}, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita^[2] lead to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in Einstein's theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces^[3]. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney^[4]. James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ... Tullio Levi-Civita. ... For more technical Wiki articles on tensors, see the section later in this article. ... In category theory, see covariant functor. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... Einstein redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ... Hermann Klaus Hugo Weyl (November 9, 1885 â€“ December 9, 1955) was a German mathematician. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Hassler Whitney (23 March 1907 â€“ 10 May 1989) was an American mathematician who was one of the founders of singularity theory, PhB, Yale University, 1928; MusB, 1929; ScD (Honorary), 1947; PhD, Harvard University, under G.D. Birkhoff, 1932. ...

For more on the history of manifolds see the history section of the primary manifold entry. On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ...

Definition

A topological manifold is a second countable Hausdorff space which is locally homeomorphic to Euclidean space, by a collection (called an atlas) of homeomorphisms called charts. The composition of one chart with the inverse of another chart is a function called a transition map, and defines a homeomorphism of an open subset of Euclidean space onto another open subset of Euclidean space. In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second countable if its topology has a countable base. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following.

These definitions, however, leave out an important feature. They each still involve a preferred choice of atlas. Given a differentiable manifold (in any of the above senses), there is a notion of when two atlases are equivalent. Then, strictly speaking, a differentiable manifold is an equivalence class of such atlases. (See below.) A differentiability class in mathematics is a class of functions which share differentiability features. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

Atlases

An atlas on a topological space X is a collection of pairs {(U_α,φ_α)} called charts, where the U_α are open sets which cover X, and for each index α In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset...

is a homeomorphism of U_α onto an open subset of n-dimensional Euclidean space. The transition maps of the atlas are the functions In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

Every topological manifold has an atlas. A C^k-atlas is an atlas for which all transition maps are C^k. A topological manifold has a C⁰-atlas and generally a C^k-manifold has a C^k-atlas. A continuous atlas is a C⁰ atlas, a smooth atlas is a C^∞ atlas and an analytic atlas is a C^ω atlas. If the atlas is at least C¹, it is also called a differentiable structure. An holomorphic atlas is an atlas whose underlying Euclidean space is defined on the complex field and whose transition maps are biholomorphic. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Compatible atlases

Different atlases can give rise to essentially the same manifold. The circle can be mapped by two coordinate charts, but if the domains of these charts are changed slightly a different atlas for the same manifold is obtained. These different atlases can be combined into a bigger atlas. It can happen that the transition maps of such a combined atlas are not as smooth as those of the constituent atlases. If C^k atlases can be combined to form a C^k atlas, then they are called compatible. Compatibility of atlases is an equivalence relation; by combining all the atlases in an equivalence class, a maximal atlas can be constructed. Each C^k atlas belongs to a unique maximal C^k atlas. In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

Alternative definitions

Pseudogroups

The notion of a pseudogroup^[5] provides a flexible generalization of atlases in order to allow a variety of different structures to be defined on manifolds in a uniform way. A pseudogroup consists of a topological space S and a collection Γ consisting of homeomorphisms from open subsets of S to other open subsets of S such that In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ...

These last three conditions are analogous to the definition of a group. Note that Γ need not be a group, however, since the functions are not globally defined on S. For example, the collection of all local C^k diffeomorphisms on Rⁿ form a pseudogroup. All biholomorphisms between open sets in Cⁿ form a pseudogroup. More examples include: orientation preserving maps of Rⁿ, symplectomorphisms, Moebius transformations, affine transformations, and so on. Thus a wide variety of function classes determine pseudogroups. This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In the mathematical theory of functions of several complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a holomorphic function whose inverse is also holomorphic. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ... Möbius transformations should not be confused with the Möbius transform. ... In geometry, an affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation: In the finite-dimensional case each affine transformation is given by a matrix A and a vector b...

An atlas (U_i, φ_i) of homeomorphisms φ_i from U_i ⊂ M to open subsets of a topological space S is said to be compatible with a pseudogroup Γ provided that the transition functions φ_j o φ_i^-1 : φ_i(U_i ∩ U_j) → φ_j(U_i ∩ U_j) are all in Γ.

A differentiable manifold is then an atlas compatible with the pseudogroup of C^k functions on Rⁿ. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in Cⁿ. And so forth. Thus pseudogroups provide a single framework in which to describe many structures on manifolds of importance to differential geometry and topology.

Structure sheaf

Sometimes it can be useful to use an alternate approach to endow a manifold with a C^k-structure. Here k = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The structure sheaf of M, denoted C^k, is a sort of functor which defines, for each open set U ⊂ M, an algebra C^k(U) of continuous functions U → R. A structure sheaf C^k is said to give M the structure of a C^k manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x¹,...,xⁿ ∈ C^k(U) such that the map f = (x¹, ..., xⁿ) : U → Rⁿ is a homeomorphism onto an open set in Rⁿ, and such that C^k|_U is the pullback of the sheaf of k-times continuously differentiable functions on Rⁿ.^[6] In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

In particular, this latter condition means that any function h in C^k(V), for V, can be written uniquely as h(x) = H(x¹(x),...,xⁿ(x)), where H is a k-times differentiable function on f(V) (an open set in Rⁿ). Thus, intuitively, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on Rⁿ, and a fortiori this is sufficient to characterize the differentiable structure on the manifold.

Sheaves of local rings

A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space. This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions. It is especially popular in the context of complex manifolds. In mathematics, a ringed space is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

We begin by describing the basic structure sheaf on Rⁿ. If U is an open set in Rⁿ, let

consist of all real-valued k-times continuously differentiable functions on U. As U varies, this determines a sheaf of rings on Rⁿ. The stalk O_p for p ∈ Rⁿ consists of germs of functions near p, and is an algebra over R. In particular, this is a local ring whose unique maximal ideal consists of those functions which vanish at p. The pair (Rⁿ, O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings. In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

A differentiable manifold (of class C^k) consists of a pair (M, O_M) where M is a topological space, and O_M is a sheaf of local R-algebras defined on M, such that the locally ringed space (M,O_M) is locally isomorphic to (Rⁿ, O). In this way, differentiable manifolds can be thought of as schemes modelled on Rⁿ. This means that,^[7] for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f,f^#) where In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be Rⁿ. For example (particularly in algebraic geometry), one could take this to be the space of complex numbers Cⁿ equipped with the sheaf of holomorphic functions (thus arriving at the spaces of complex analytic geometry), or the sheaf of polynomials (thus arriving at the spaces of interest in complex algebraic geometry). More generally, this concept can be adapted for any suitable notion of a scheme (see topos theory). Secondly, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair (f, f^#), but these merely quantify the idea of local isomorphism rather than being central to the discussion (as in the case of charts and atlases). Thirdly, the sheaf O_M is not manifestly a sheaf of functions at all. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals). Hence it is a more primitive definition of the structure (see synthetic differential geometry). Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, complex analytic geometry sometimes denotes the application of complex numbers to plane geometry. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... For discussion of topoi in literary theory, see literary topos. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ...

A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.

In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... A variety of dualities in mathematics are listed at duality (mathematics). ...

Differentiable functions

A real valued function f on an m-dimensional differentiable manifold M is called differentiable at a point p ∈ M if it is differentiable in any coordinate chart defined around p. More precisely, if (U, φ) is a chart where U in an open set in M containing p and φ : U → Rⁿ is the map defining the chart, then f is differentiable if

is differentiable at φ(p). Ostensibly, the definition of differentiability depends on the choice of chart at p; in general there will be many available charts. However, it follows from the chain rule applied to the transition functions between one chart and another that if f is differentiable in any particular chart at p, then it is differentiable in all charts at p. Analogous considerations apply to defining C^k functions, smooth functions, and analytic functions. In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

Differentiation of functions

There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. The directional derivative therefore looks at curves in the manifold instead of vectors. For a non-technical overview of the subject, see Calculus. ... In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

Directional differentiation

Given a real valued function f on an m dimensional differentiable manifold M, the directional derivative of f at a point p in M is defined as follows. Suppose that γ(t) is a curve in M with γ(0) = p, which is differentiable in the sense that its composition with any chart is a differentiable curve in R^m. Then the directional derivative of f at p along γ is

If γ₁ and γ₂ are two curves such that γ₁(0) = γ₂(0) = p, and in any coordinate chart φ,

then, by the chain rule, f has the same directional derivative at p along γ₁ as along γ₂. Intuitively, this means that the directional derivative depends only on the tangent vector of the curve at p. Thus the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space. In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

Tangent vectors and the differential

A tangent vector at p ∈ M is an equivalence class of differentiable curves γ with γ(0) = p, modulo the equivalence relation of first-order contact between the curves. Explicitly, In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. ...

in any (and hence all) coordinate charts φ. Intuitively, the equivalence classes are curves through p with a prescribed velocity vector at p. The collection of all tangent vectors at p forms a vector space: the tangent space to M at p, denoted T_pM. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

If X is a tangent vector at p and f a differentiable function defined near p, then differentiating f along any curve in the equivalence class defining X gives a well-defined directional derivative along X:

Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.

is a linear functional on the tangent space. This linear functional is often denoted by df(p) and is called the differential of f at p: In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...

Partitions of unity

One of the topological features of the sheaf of differentiable functions on a differentiable manifold is that it admits partitions of unity. This distinguishes the differentiable structure on a manifold from stronger structures (such as analytic and holomorphic structures) which generally fail to have partitions of unity. In mathematics, a partition of unity of a topological space X is a set of continuous functions {Ïi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all...

Suppose that M is a manifold of class C^k, where 0 ≤ k ≤ ∞. Let {U_α} be an open covering of M. Then a partition of unity subordinate to the cover {U_α }is a collection of real-valued C^k functions φ_i on M satisfying the following conditions

(Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φ_i.) In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... Compact redirects here. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

Every open covering of a C^k manifold M has a C^k partition of unity. This allows for certain constructions from the topology of C^k functions on Rⁿ to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of Rⁿ. Partitions of unity therefore allow for certain other kinds of function spaces to be considered: for instance L^p spaces, Sobolev spaces, and other kinds of spaces that require integration. In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. ...

Differentiability of mappings between manifolds

Suppose M and N are two differentiable manifolds with dimensions m and n respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. But what does "f is C^k(M, N)" mean for k≥1? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map which goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be C^k(R^m, Rⁿ). We define "f is C^k(M, N)" to mean that all such compositions of f with charts are C^k(R^m, Rⁿ). Once again the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already an Euclidean space, then we don't need a chart to map it to one.

Algebra of scalars

For a C^k manifold M, the set of real-valued C^k functions on the manifold forms an algebra under pointwise addition and multiplication, called the algebra of scalar fields or simply the algebra of scalars. This algebra has the constant function 1 as unit. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

It is possible to reconstruct a manifold from its algebra of scalars. In fact, there is a one-to-one correspondence between the points of M and the algebra homomorphisms φ : C^k(M) → R. For suppose that φ is such a homomorphism. Then the kernel of φ is a codimension one ideal in C_k(M), which is necessarily a maximal ideal. Every maximal ideal in this algebra is an ideal of functions vanishing at a single point.

Bundles

Tangent bundle

The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension n as the manifold does. The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. The tangent bundle is where tangent vectors live, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle. One can also define the tangent bundle as the bundle of 1- jets from R (the real line) to M. In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... 2-dimensional renderings (ie. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ... In mathematics, the real line is simply the set of real numbers. ...

One may construct an atlas for the tangent bundle consisting of charts based on U_α × Rⁿ, where U_α denotes one of the charts in the atlas for M. Each of these new charts is the tangent bundle for the charts U_α. The transition maps on this atlas are defined from the transition maps on the original manifold, and retain the original differentiability class.

Cotangent bundle

The dual space of a vector space is the set of real valued linear functions on the vector space. In particular, if the vector space is finite and has an inner product then the linear functionals can be realized by the functions f_v(w) = <v,w>. In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ...

The cotangent bundle is the dual tangent bundle in the sense that at each point, the cotangent space is the dual of the tangent space. The cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle naturally has the structure of a symplectic manifold. Cotangent vectors are sometimes called covectors. One can also define the cotangent bundle as the bundle of 1-jets of functions from M to R. In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... A one-form, also called a covector, is a linear function which maps each vector in a vector space to a real number, such that the mapping is invariant with respect to coordinate transformations of the vector space. ... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ...

Elements of the cotangent space can be thought of as infinitesimal displacements: if f is a differentiable function we can define at each point p a cotangent vector df_p which sends a tangent vector X_p to the derivative of f associated with X_p. However, not every covector field can be expressed this way. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

Tensor bundle

The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field, which can act as a multilinear operator on vector fields, or on other tensor fields. In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

The tensor bundle cannot be a differentiable manifold, since it is infinite dimensional. It is however an algebra over the ring of scalar functions. Each tensor is characterized by its ranks, which indicate how many tangent and cotangent factors it has. Sometimes these ranks are referred to as covariant and contravariant ranks, signifying tangent and cotangent ranks, respectively. In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

Frame bundle

A frame (or more precisely, a tangent frame) is an ordered basis of particular tangent space. Equivalently, a tangent frame is a linear isomorphism of Rⁿ to this tangent space. A moving tangent frame is an ordered list of vector fields that give a basis at every point of their domain. One may also regard a moving frame as a section of the frame bundle F(M), a GL_nR principal bundle made up of the set of all frames over M. The frame bundle is useful because tensor fields on M can be regarded as equivariant vector-valued functions on F(M). In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In mathematics, the general linear group of degree n is the set of nÃ—n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X Ã— G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with An action of G on P, analogous to... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

Jet bundles

On a manifold which is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. Analogously, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Similarly, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential operators on manifolds. In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ... In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ...

The notion of a frame also generalizes to the case of higher-order jets. Define a k-th order frame to be the k-jet of a diffeomorphism from Rⁿ to M.^[8] The collection of all k-th order frames, F^k(M), is a principle G^k bundle over M, where G^k is the group of k-jets; i.e, the group made up of k-jets of diffeomorphisms of Rⁿ that fix the origin. Note that GL_nR is naturally isomorphic to G¹, and a subgroup of every G^k, k≥2. In particular, a section of F²(M) gives the frame components of a connection on M. Thus, the quotient bundle F²(M)/GL_nR is the bundle of linear connections over M. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. ... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ... Look up connection, connected, connectivity in Wiktionary, the free dictionary. ...

Calculus on manifolds

Many of the techniques from multivariate calculus also apply, mutatis mutandis, to differentiable manifolds. One can define the directional derivative of a differentiable function along a tangent vector to the manifold, for instance, and this leads to a means of generalizing the total derivative of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least locally. For example, there are versions of the implicit and inverse function theorems for such functions. Multivariate calculus is a means of analyzing deterministic systems with multiple degrees of freedom. ... In mathematics, a total derivative may be either. ... In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... In mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. ...

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

Ideas from integral calculus also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and differential forms. The fundamental theorems of integral calculus in several variables — namely Green's theorem, the divergence theorem, and Stokes' theorem — generalize to a theorem (also called Stokes' theorem) relating the exterior derivative and integration over submanifolds. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... This article deals with the concept of an integral in calculus. ... This article may be too technical for most readers to understand. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... This is a glossary of terms specific to differential geometry and differential topology. ...

Differential calculus of functions

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If f : M → N is a differentiable function from a differentiable manifold M of dimension m to another differentiable manifold N of dimension n, then the differential of f is a mapping df : TM → TN. At each point of M, this is a linear transformation from one tangent space to another: This is a glossary of terms specific to differential geometry and differential topology. ... Suppose that Ï† : M â†’ N is a smooth map between smooth manifolds; then the differential of Ï† at a point x is, in some sense, the best linear approximation of Ï† near x. ...

The rank of f at p is the rank of this linear transformation. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...

Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions: Sards lemma, also known as Sards theorem or the Morse-Sard theorem, is a result of mathematical analysis characterising the image of the critical points of a smooth function F from one Euclidean space to another as having Lebesgue measure 0 (and so small, in a definite sense). ... In mathematics, an immersion is a differentiable map between differentiable manifolds whose derivative is everywhere injective. ... In mathematics, a submersion is a differentiable map between differentiable manifolds whose derivative is everywhere surjective. ...

In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... This is a glossary of terms specific to differential geometry and differential topology. ... This article may be too technical for most readers to understand. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Lie derivative

A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... Marius Sophus Lie (IPA pronunciation: , pronounced Lee) (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way round, the group of diffeomorphisms of M has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group theory. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... An active transformation is one which actually changes the physical state of a system and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Exterior calculus

The exterior calculus allows for a generalization of the gradient, divergence and curl operators. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... For other uses, see Gradient (disambiguation). ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... For other uses, see Curl (disambiguation). ...

The bundle of differential forms, at each point, consists of all totally antisymmetric multilinear maps on the tangent space at that point. It is naturally divided into n-forms for each n at most equal to the dimension of the manifold; an n-form is an n-variable form, also called a form of degree n. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. More generally, an n-form is a tensor with cotangent rank n and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In set theory, the adjective antisymmetric usually refers to an antisymmetric relation. ... In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ...

Exterior derivative

There is a map from scalars to covectors called the exterior derivative In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

This map is the one which relates covectors to infinitesimal displacements, mentioned above; some covectors are the exterior derivatives of scalar functions. It can be generalized into a map from the n-forms onto the n+1-forms. Applying this derivative twice will produce a zero form. Forms with zero derivative are called closed forms, while forms which are themselves exterior derivatives are known as exact forms.

The space of differential forms at a point is the archetypal example of an exterior algebra; thus it possesses a wedge product, mapping a k-form and l-form to a k+l-form. The exterior derivative extends to this algebra, and satisfies a version of the product rule: In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold. The rank n cohomology group is the quotient group of the closed forms by the exact forms. In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

Topology of differentiable manifolds

Relationship with topological manifolds

Every topological manifold in dimension 1, 2, or 3 has a unique differentiable structure (up to diffeomorphism); thus the concepts of topological and differentiable manifold are distinct only in higher dimensions. It is known that in each higher dimension, there are some topological manifolds with no differentiable structure ^[9], and some with multiple non-diffeomorphic structures. The classic example of manifolds with multiple incompatible structures are the exotic 7-spheres of John Milnor ^[10]. In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ... John Willard Milnor (b. ...

Classification

Every second-countable 1-manifold without boundary is homeomorphic to a disjoint union of countably many copies of R (the real line) and S (the circle); the only connected examples are R and S, and of these only S is compact. In higher dimensions, classification theory normally focuses only on compact connected manifolds. In mathematics, the real line is simply the set of real numbers. ... Circle illustration This article is about the shape and mathematical concept of circle. ...

For a classification of 2-manifolds, see surface: in particular compact connected oriented 2-manifolds are classified by their genus, which is a nonnegative integer. An open surface with X-, Y-, and Z-contours shown. ...

A classification of 3-manifolds follows in principle from the geometrization of 3-manifolds and various recognition results for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.^[11] In mathematics, a 3-manifold is a 3-dimensional manifold. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In mathematics, Mostows rigidity theorem, sometimes called the strong rigidity theorem, essentially states that the geometry of a finite volume hyperbolic manifold (for dimension greater than two) is determined by the fundamental group and hence unique. ...

The classification of n-manifolds for n greater than three is known to be impossible, even up to homotopy equivalence. Given any finitely presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decide the isomorphism problem for finitely presented groups, there is no algorithm to decide if two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds which are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable. Additionally, since even recognizing the trivial group is undecidable, it is not even possible in general to decide if a manifold has trivial fundamental group, i.e. is simply-connected. An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, one method of defining a group is by a presentation. ... In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

Simply-connected 4-manifolds have been classified up to homeomorphism by Freedman using the intersection form and Kirby-Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as the exotic smooth structures on $mathbb R^4$ demonstrate. In mathematics, 4-manifold is a 4-dimensional topological manifold. ... Michael Hartley Freedman (born 21 April 1951 in Los Angeles, California, USA) is a mathematician at Microsoft Research. ... In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. ... In mathematics, the Kirby-Siebenmann class is an element of the fourth cohomology group which must vanish if a topological manifold M is to have a piecewise linear structure. ...

Somewhat surprisingly, the situation becomes more tractible for simply connected smooth manifolds of dimension ≥ 5, where the h-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied^[12]. This has been carried out to provide an explicit classification of simply connected 5-manifolds by Dennis Barden. A cobordism W between M and N is an h-cobordism if the inclusion maps M â†’ W and N â†’ W are homotopy equivalences. ... In mathematics, specifically in topology, surgery theory is the name given to a collection of techniques used to produce one manifold from another in a controlled way. ... There are very few or no other articles that link to this one. ...

Structures on manifolds

(Pseudo-)Riemannian manifolds

A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. This metric can be used to interconvert vectors and covectors, and to define a rank 4 Riemann curvature tensor. On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one). Pseudo-Riemannian manifolds of signature (3, 1) are important in general relativity. Not every differentiable manifold can be given a pseudo-Riemannian structure; there are topological restrictions on doing so. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

A Finsler manifold is a generalization of a Riemannian manifold, in which the inner product is replaced with a vector norm; this allows the definition of length, but not angle. In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...

Symplectic manifolds

A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional. Cotangent bundles, which arise as phase spaces in Hamiltonian mechanics, are the motivating example, but many compact manifolds also have symplectic structure. All orientable surfaces embedded in Euclidean space have a symplectic structure, the signed area form on each tangent space induced by the ambient Euclidean inner product. (This form is clearly nondegenerate, and it must be closed because it is top-dimensional with respect to the surface.) Every Riemann surface is an example of such a surface, and hence a symplectic manifold, when considered as a real manifold. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... This article or section should be merged with Orientable manifold. ... In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... An open surface with X-, Y-, and Z-contours shown. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

Lie groups

A Lie group is C^∞ manifold which also carries a group structure whose product and inversion operations are smooth as maps of manifolds. These objects arise naturally in describing symmetries. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

Generalizations

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. The diffeological spaces, (differential spaces) use a different notion of chart known as "plot". Frölicher spaces and orbifolds are other attempts. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a diffeology is a generalization of smooth manifolds to a category that is more stable. ... In mathematics, FrÃ¶licher spaces extend the notions of calculus and smooth manifolds. ... In topology and group theory, an orbifold (for orbit-manifold) is a generalization of a manifold. ...

A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds. In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

Notes

References

Categories: Mathematics articles needing expert attention | Articles needing expert attention | All articles with unsourced statements | Articles with unsourced statements since February 2007 | Articles with sections needing expansion | Differential geometry | Differential structures Simon Kirwan Donaldson, born in Cambridge in 1957, is an English mathematician famous for his work on the topology of smooth (differentiable) four-dimensional manifolds. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ... The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ...

Contents

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