TheInfoList

OR: In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
to allow one to apply
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arit ...
. Any manifold can be described by a collection of charts ( atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector 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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in i ...
), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
with a globally defined
differential structure In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for di ...
. Any topological manifold can be given a differential structure locally by using the
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
s in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their compositions on chart intersections in the atlas must be differentiable functions on the corresponding vector 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.'' The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally differential structure allows one to define the globally differentiable
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, differentiable functions, and differentiable
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...
and vector fields. Differentiable manifolds are very important in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...
. Special kinds of differentiable manifolds form the basis for physical theories such as
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mech ...
,
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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. "Differentiability" of a manifold has been given several meanings, including:
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in i ...
, ''k''-times differentiable, smooth (which itself has many meanings), and analytic.

# 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 Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including ...
lecture before the faculty at
Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony, central Germany, the capital of the eponymous district. The River Leine runs through it. At the end of 2019, the population was 118,911. General information The ori ...
. 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: : ''Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ...'' – B. Riemann The works of physicists such as
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
, and mathematicians Gregorio Ricci-Curbastro and
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
led to the development of tensor analysis and the notion of
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the les ...
, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and its underlying
equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (s ...
. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.

# Definition

## Atlases

Let be a topological space. A chart on consists of an open subset of , and a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
from to an open subset of some
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. Somewhat informally, one may refer to a chart , meaning that the image of is an open subset of , and that is a homeomorphism onto its image; in the usage of some authors, this may instead mean that is itself a homeomorphism. The presence of a chart suggests the possibility of doing
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
on ; for instance, if given a function and a chart on , one could consider the composition , which is a real-valued function whose domain is an open subset of a Euclidean space; as such, if it happens to be differentiable, one could consider its
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s. This situation is not fully satisfactory for the following reason. Consider a second chart on , and suppose that and contain some points in common. The two corresponding functions and are linked in the sense that they can be reparametrized into one another: $u\circ\varphi^=\big(u\circ\psi^\big)\circ\big(\psi\circ\varphi^\big),$ the natural domain of the right-hand side being . Since and are homeomorphisms, it follows that is a homeomorphism from to . Consequently, even if both functions and are differentiable, their differential properties will not necessarily be strongly linked to one another, as is not necessarily sufficiently differentiable for the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
to be applicable. The same problem is found if one considers instead functions ; one is led to the reparametrization formula $\varphi\circ c=\big(\varphi\circ\psi^\big)\circ\big(\psi\circ c\big),$ at which point one can make the same observation as before. This is resolved by the introduction of a "differentiable atlas" of charts, which specifies a collection of charts on for which the transition maps are all differentiable. This makes the situation quite clean: if is differentiable, then due to the reparametrization formula, the map is also differentiable on the region . Moreover, the derivatives of these two maps are linked to one another by the chain rule. Relative to the given atlas, this facilitates a notion of differentiable mappings whose domain or range is , as well as a notion of the derivative of such maps. Formally, the word "differentiable" is somewhat ambiguous, as it is taken to mean different things by different authors; sometimes it means the existence of first derivatives, sometimes the existence of continuous first derivatives, and sometimes the existence of infinitely many derivatives. The following gives a formal definition of various (nonambiguous) meanings of "differentiable atlas". Generally, "differentiable" will be used as a catch-all term including all of these possibilities, provided . Since every real-analytic map is smooth, and every smooth map is for any , one can see that any analytic atlas can also be viewed as a smooth atlas, and every smooth atlas can be viewed as a atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets of can be viewed as a real-analytic map between open subsets of . Given a differentiable atlas on a topological space, one says that a chart is differentiably compatible with the atlas, or differentiable relative to the given atlas, if the inclusion of the chart into the collection of charts comprising the given differentiable atlas results in a differentiable atlas. A differentiable atlas determines a maximal differentiable atlas, consisting of all charts which are differentiably compatible with the given atlas. A maximal atlas is always very large. For instance, given any chart in a maximal atlas, its restriction to an arbitrary open subset of its domain will also be contained in the maximal atlas. A maximal smooth atlas is also known as a smooth structure; a maximal holomorphic atlas is also known as a complex structure. An alternative but equivalent definition, avoiding the direct use of maximal atlases, is to consider equivalence classes of differentiable atlases, in which two differentiable atlases are considered equivalent if every chart of one atlas is differentiably compatible with the other atlas. Informally, what this means is that in dealing with a smooth manifold, one can work with a single differentiable atlas, consisting of only a few charts, with the implicit understanding that many other charts and differentiable atlases are equally legitimate. According to the invariance of domain, each connected component of a topological space which has a differentiable atlas has a well-defined dimension . This causes a small ambiguity in the case of a holomorphic atlas, since the corresponding dimension will be one-half of the value of its dimension when considered as an analytic, smooth, or atlas. For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas.

## Manifolds

A differentiable manifold is a Hausdorff and second countable topological space , together with a maximal differentiable atlas on . Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of
bump function In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all ...
s and
partitions of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0 ...
, both of which are used ubiquitously. The notion of a manifold is identical to that of a
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
. However, there is a notable distinction to be made. Given a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, it is not meaningful to ask whether or not a given topological space is (for instance) a smooth manifold, since the notion of a smooth manifold requires the specification of a smooth atlas, which is an additional structure. It could, however, be meaningful to say that a certain topological space cannot be given the structure of a smooth manifold. It is possible to reformulate the definitions so that this sort of imbalance is not present; one can start with a set (rather than a topological space ), using the natural analogue of a smooth atlas in this setting to define the structure of a topological space on .

## Patching together Euclidean pieces to form a manifold

One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings. Given an indexing set $A,$ let $V_\alpha$ be a collection of open subsets of $\mathbb^n$ and for each $\alpha,\beta \in A$ let $V_$ be an open (possibly empty) subset of $V_\beta$ and let $\phi_:V_ \to V_$ be a smooth map. Suppose that $\phi_$ is the identity map, that $\phi_ \circ \phi_$ is the identity map, and that $\phi_ \circ \phi_ \circ \phi_$ is the identity map. Then define an equivalence relation on the disjoint union $\bigsqcup_ V_\alpha$ by declaring $p \in V_$ to be equivalent to $\phi_\left(p\right) \in V_.$ With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas.

# Differentiable functions

A real valued function ''f'' on an ''n''-dimensional differentiable manifold ''M'' is called differentiable at a point if it is differentiable in any coordinate chart defined around ''p''. In more precise terms, if $\left(U,\phi\right)$ is a differentiable chart where $U$ is an open set in $M$ containing ''p'' and $\phi : U\to ^n$ is the map defining the chart, then ''f'' is differentiable at ''p''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
$f\circ \phi^ \colon \phi(U)\subset ^n \to$ is differentiable at $\phi\left(p\right)$, that is $f\circ \phi^$ is a differentiable function from the open set $\phi\left(U\right)$, considered as a subset of $^n$, to $\mathbf R$. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at ''p''. It follows from the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
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 ''Ck'' functions, smooth functions, and analytic functions.

## Differentiation of functions

There are various ways to define the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
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. Therefore, the directional derivative looks at curves in the manifold instead of vectors.

### Directional differentiation

Given a real valued function ''f'' on an ''n'' 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 , which is ''differentiable'' in the sense that its composition with any chart is a
differentiable curve Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the s ...
in R''n''. Then the directional derivative of ''f'' at ''p'' along γ is $\left.\fracf(\gamma(t))\_.$ If ''γ''1 and ''γ''2 are two curves such that , and in any coordinate chart $\phi$, $\left.\frac\phi\circ\gamma_1(t)\_=\left.\frac\phi\circ\gamma_2(t)\_$ then, by the chain rule, ''f'' has the same directional derivative at ''p'' along ''γ''1 as along ''γ''2. This means that the directional derivative depends only on the
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ele ...
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.

### Tangent vector and the differential

A tangent vector at is an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of differentiable curves ''γ'' with , modulo the equivalence relation of first-order contact between the curves. Therefore, $\gamma_1\equiv \gamma_2 \iff \left.\frac\phi\circ\gamma_1(t)\_ = \left.\frac\phi\circ\gamma_2(t)\_$ in every coordinate chart $\phi$. Therefore, the equivalence classes are curves through ''p'' with a prescribed
velocity vector Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
at ''p''. The collection of all tangent vectors at ''p'' forms a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
: the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to ''M'' at ''p'', denoted ''T''''p''''M''. 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'': $Xf(p) := \left.\fracf(\gamma(t))\_.$ 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. If the function ''f'' is fixed, then the mapping $X\mapsto Xf(p)$ is a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on the tangent space. This linear functional is often denoted by ''df''(''p'') and is called the differential of ''f'' at ''p'': $df(p) \colon T_pM \to .$

### Definition of tangent space and differentiation in local coordinates

Let $M$ be a topological $n$-manifold with a smooth atlas $\_.$ Given $p\in M$ let $A_p$ denote $\.$ A "tangent vector at $p\in M$" is a mapping $v:A_p\to\mathbb^n,$ here denoted $\alpha\mapsto v_\alpha,$ such that $v_\alpha=D\Big, _(\phi_\alpha\circ\phi_\beta^)(v_\beta)$ for all $\alpha,\beta\in A_p.$ Let the collection of tangent vectors at $p$ be denoted by $T_pM.$ Given a smooth function $f:M\to\mathbb$, define $df_p:T_pM\to\mathbb$ by sending a tangent vector $v:A_p\to\mathbb^n$ to the number given by $D\Big, _(f\circ\phi_\alpha^)(v_\alpha),$ which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of $\alpha\in A_p.$ One can check that $T_pM$ naturally has the structure of a $n$-dimensional real vector space, and that with this structure, $df_p$ is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of $v_\beta$ for a single element $\beta$ of $A_p$ automatically determines $v_\alpha$ for all $\alpha\in A.$ The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically : $v^i=\widetilde^j\frac$ and $df_p\left(v\right)=\fracv^i.$ With the idea of the formal definitions understood, this shorthand notation is, for most purposes, much easier to work with.

## 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 In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are 0 ...
. This distinguishes the differential structure on a manifold from stronger structures (such as analytic and holomorphic structures) that in general fail to have partitions of unity. Suppose that ''M'' is a manifold of class ''Ck'', where . Let be an open covering of ''M''. Then a partition of unity subordinate to the cover is a collection of real-valued ''Ck'' functions ''φ''''i'' on ''M'' satisfying the following conditions: * The supports of the ''φ''''i'' are
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
and locally finite; * The support of ''φ''''i'' is completely contained in ''U''''α'' for some ''α''; * The ''φ''''i'' sum to one at each point of ''M'': $\sum_i \phi_i(x) = 1.$ (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''.) Every open covering of a ''Ck'' manifold ''M'' has a ''Ck'' partition of unity. This allows for certain constructions from the topology of ''Ck'' functions on R''n'' 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 R''n''. Partitions of unity therefore allow for certain other kinds of
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s to be considered: for instance L''p'' spaces, Sobolev spaces, and other kinds of spaces that require integration.

## 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 " mean for ? 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 that goes from Euclidean space to ''M'' to ''N'' to Euclidean space we know what it means for that map to be . We define "''f'' is " to mean that all such compositions of ''f'' with charts are . 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 a Euclidean space, then we don't need a chart to map it to one.

# Bundles

## Tangent bundle

The
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and '' tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of a point consists of the possible directional derivatives at that point, and has the same
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
''n'' as does the manifold. For a set of (non-singular) coordinates ''xk'' local to the point, the coordinate derivatives $\partial_k=\frac$ define a holonomic basis of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and o ...
, whose dimension is 2''n''. The tangent bundle is where
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ele ...
s lie, and is itself a differentiable manifold. The
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem wit ...
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''. One may construct an atlas for the tangent bundle consisting of charts based on , 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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ma ...
is the collection of all cotangent vectors, along with the natural differentiable manifold structure. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle has the structure of a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
. Cotangent vectors are sometimes called ''
covector In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
s''. One can also define the cotangent bundle as the bundle of 1- jets of functions from ''M'' to R. Elements of the cotangent space can be thought of as
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
displacements: if ''f'' is a differentiable function we can define at each point ''p'' a cotangent vector ''dfp'', which sends a tangent vector ''Xp'' to the derivative of ''f'' associated with ''Xp''. However, not every covector field can be expressed this way. Those that can are referred to as
exact differential In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&nb ...
s. For a given set of local coordinates ''xk,'' the differentials ''dx'' form a basis of the cotangent space at ''p''.

## Tensor bundle

The tensor bundle is the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
of all
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
s 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. The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
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.

## Frame bundle

A frame (or, in more precise terms, a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of R''n'' 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
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
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'').

## Jet bundles

On a manifold that 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. By analogy, the ''k''-th order tangent bundle is the collection of curves modulo the relation of ''k''-th order contact. Likewise, 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 operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s on manifolds. 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 In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
from R''n'' to ''M''. The collection of all ''k''-th order frames, ''Fk''(''M''), is a principal ''Gk'' bundle over ''M'', where ''Gk'' is the group of ''k''-jets; i.e., the group made up of ''k''-jets of diffeomorphisms of R''n'' that fix the origin. Note that is naturally isomorphic to ''G''1, and a subgroup of every ''Gk'', . In particular, a section of ''F''2(''M'') gives the frame components of a connection on ''M''. Thus, the quotient bundle is the bundle of ''symmetric'' linear connections over ''M''.

# 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 In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
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 In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...
. For example, there are versions of the implicit and inverse function theorems for such functions. 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: * The Lie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation. * An
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
, which is not uniquely defined, but generalizes in a more complete manner the features of ordinary directional differentiation. Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold. Ideas from
integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
also carry over to differential manifolds. These are naturally expressed in the language of
exterior calculus In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
and
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s. The fundamental theorems of integral calculus in several variables—namely
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
, the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
, and
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
—generalize to a theorem (also called Stokes' theorem) relating the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
and integration over submanifolds.

## Differential calculus of functions

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If 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 . It is also denoted by ''Tf'' and called the tangent map. At each point of ''M'', this is a linear transformation from one tangent space to another: $df(p)\colon T_p M \to T_ N.$ The rank of ''f'' at ''p'' is the rank of this linear transformation. 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: * If , and has rank ''m'' at , then ''f'' is called an immersion at ''p''. If ''f'' is an immersion at all points of ''M'' and is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
onto its image, then ''f'' is an
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is giv ...
. Embeddings formalize the notion of ''M'' being a submanifold of ''N''. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities. * If , and has rank ''n'' at , then ''f'' is called a submersion at ''p''. The implicit function theorem states that if ''f'' is a submersion at ''p'', then ''M'' is locally a product of ''N'' and R''m''−''n'' near ''p''. In formal terms, there exist coordinates in a neighborhood of ''f''(''p'') in ''N'', and functions ''x''1, ..., ''x''''m''−''n'' defined in a neighborhood of ''p'' in ''M'' such that $(y_1\circ f,\dotsc,y_n\circ f, x_1, \dotsc, x_)$ is a system of local coordinates of ''M'' in a neighborhood of ''p''. Submersions form the foundation of the theory of
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
s and
fibre bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s.

## Lie derivative

A Lie derivative, named after
Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sop ...
, is a
derivation Derivation may refer to: Language * Morphological derivation, a word-formation process * Parse tree or concrete syntax tree, representing a string's syntax in formal grammars Law * Derivative work, in copyright law * Derivation proceeding, a pro ...
on the
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
of tensor fields over a manifold ''M''. The
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
of all Lie derivatives on ''M'' forms an infinite dimensional
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as infinitesimal generators of flows ( active
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s) on ''M''. Looking at it the other way around, the group of diffeomorphisms of ''M'' has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
theory.

## Exterior calculus

The exterior calculus allows for a generalization of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the grad ...
, divergence and
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
operators. The bundle of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, 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. In general, 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.

### Exterior derivative

The ''exterior derivative'' is a linear operator on the
graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be the ...
of all smooth differential forms on a smooth manifold $M$. It is usually denoted by $d$. More precisely, if $n=\dim\left(M\right)$, for $0 \le k \le n$ the operator $d$ maps the space $\Omega^k\left(M\right)$ of $k$-forms on $M$ into the space $\Omega^\left(M\right)$ of $\left(k+1\right)$-forms (if $k > n$ there are no non-zero $k$-forms on $M$ so the map $d$ is identically zero on $n$-forms). For example, the exterior differential of a smooth function $f$ is given in local coordinates $x_1, \ldots, x_n$, with associated local co-frame $dx_1, \ldots, dx_n$ by the formula : $df = \sum_^n \frac dx_i.$ The exterior differential satisfies the following identity, similar to a
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
with respect to the wedge product of forms: $d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^\omega \wedge d\eta.$ The exterior derivative also satisfies the identity $d \circ d = 0$. That is, if $\omega$ is a $k$-form then the $\left(k+2\right)$-form $d\left(df\right)$ is identically vanishing. A form $\omega$ such that $d\omega = 0$ is called ''closed'', while a form $\omega$ such that $\omega = d\eta$ for some other form $\eta$ is called ''exact''. Another formulation of the identity $d \circ d = 0$ is that an exact form is closed. This allows one to define
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adapt ...
of the manifold $M$, where the $k$th cohomology group is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
of the closed forms on $M$ by the exact forms on $M$.

# Topology of differentiable manifolds

## Relationship with topological manifolds

Suppose that $M$ is a topological $n$-manifold. If given any smooth atlas $\_$, it is easy to find a smooth atlas which defines a different smooth manifold structure on $M;$ consider a homeomorphism $\Phi:M\to M$ which is not smooth relative to the given atlas; for instance, one can modify the identity map localized non-smooth bump. Then consider the new atlas $\_,$ which is easily verified as a smooth atlas. However, the charts in the new atlas are not smoothly compatible with the charts in the old atlas, since this would require that $\phi_\alpha\circ\Phi\circ\phi_\beta^$ and $\phi_\alpha\circ\Phi^\circ\phi_\beta^$ are smooth for any $\alpha$ and $\beta,$ with these conditions being exactly the definition that both $\Phi$ and $\Phi^$ are smooth, in contradiction to how $\Phi$ was selected. With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on $M$ by declaring that smooth atlases $\_$ and $\_$ are equivalent if there is a homeomorphism $\Phi:M\to M$ such that $\_$ is smoothly compatible with $\_,$ and such that $\_$ is smoothly compatible with $\_.$ More briefly, one could say that two smooth atlases are equivalent if there exists a diffeomorphism $M\to M,$ in which one smooth atlas is taken for the domain and the other smooth atlas is taken for the range. Note that this equivalence relation is a refinement of the equivalence relation which defines a smooth manifold structure, as any two smoothly compatible atlases are also compatible in the present sense; one can take $\Phi$ to be the identity map. If the dimension of $M$ is 1, 2, or 3, then there exists a smooth structure on $M$, and all distinct smooth structures are equivalent in the above sense. The situation is more complicated in higher dimensions, although it isn't fully understood. * Some topological manifolds admit no smooth structures, as was originally shown with a ten-dimensional example by . A major application of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
in differential geometry due to
Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
, in combination with results of Michael Freedman, shows that many simply-connected compact topological 4-manifolds do not admit smooth structures. A well-known particular example is the E8 manifold. * Some topological manifolds admit many smooth structures which are not equivalent in the sense given above. This was originally discovered by John Milnor in the form of the exotic 7-spheres.

## Classification

Every one-dimensional connected smooth manifold is diffeomorphic to either $\mathbb$ or $S^1,$ each with their standard smooth structures. For a classification of smooth 2-manifolds, see
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following: $S^2,$ or $\left(S^1\times S^1\right)\sharp\cdots\sharp\left(S^1\times S^1\right),$ or $\mathbb^2\sharp\cdots\sharp\mathbb^2.$ The situation is more nontrivial if one considers complex-differentiable structure instead of smooth structure. The situation in three dimensions is quite a bit more complicated, and known results are more indirect. A remarkable result, proved in 2002 by methods of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
, is the
geometrization conjecture In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...
, stating loosely that any compact smooth 3-manifold can be split up into different parts, each of which admits Riemannian metrics which possess many symmetries. There are also various "recognition results" for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups. 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 whether two 4-manifolds have the same fundamental group. Since the previously described construction results in a class of 4-manifolds that are homeomorphic if and only if their groups are isomorphic, the homeomorphism problem for 4-manifolds is undecidable. In addition, since even recognizing the trivial group is undecidable, it is not even possible in general to decide whether a manifold has trivial fundamental group, i.e. is simply connected. Simply connected
4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. ...
s have been classified up to homeomorphism by
Freedman A freedman or freedwoman is a formerly enslaved person who has been released from slavery, usually by legal means. Historically, enslaved people were freed by manumission (granted freedom by their captor-owners), emancipation (granted freedom a ...
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 R4 demonstrate. However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the
h-cobordism theorem In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M ...
can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied. This has been carried out to provide an explicit classification of simply connected
5-manifold In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups ...
s by Dennis Barden.

# Structures on smooth manifolds

## (Pseudo-)Riemannian manifolds

A
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
consists of a smooth manifold together with a positive-definite
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on each of the individual tangent spaces. This collection of inner products is called the
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
, and is naturally a symmetric 2-tensor field. This "metric" identifies a natural vector space isomorphism $T_pM\to T_p^\ast M$ for each $p\in M.$ On a Riemannian manifold one can define notions of length, volume, and angle. Any smooth manifold can be given many different Riemannian metrics. A
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 ...
is a generalization of the notion of
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
where the inner products are allowed to have an indefinite signature, as opposed to being positive-definite; they are still required to be non-degenerate. Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. I ...
. Pseudo-Riemannian manifolds of signature are fundamental in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. Not every smooth manifold can be given a (non-Riemannian) pseudo-Riemannian structure; there are topological restrictions on doing so. A Finsler manifold is a different generalization of a Riemannian manifold, in which the inner product is replaced with a
vector norm In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ...
; as such, this allows the definition of length, but not angle.

## Symplectic manifolds

A
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
is a manifold equipped with a closed, nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric $\left(2n+1\right)\times\left(2n+1\right)$ matrices all have zero determinant. There are two basic examples: * Cotangent bundles, which arise as phase spaces in
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''moment ...
, are a motivating example, since they admit a natural symplectic form. * All oriented two-dimensional Riemannian manifolds $\left(M,g\right)$ are, in a natural way, symplectic, by defining the form $\omega\left(u,v\right)=g\left(u,J\left(v\right)\right)$ where, for any $v \in T_pM,$ $J\left(v\right)$ denotes the vector such that $v,J\left(v\right)$ is an oriented $g_p$-orthonormal basis of $T_pM.$

## Lie groups

A
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
consists of a ''C'' manifold $G$ together with a group structure on $G$ such that the product and inversion maps $m:G\times G \to G$ and $i:G\to G$ are smooth as maps of manifolds. These objects often arise naturally in describing (continuous) symmetries, and they form an important source of examples of smooth manifolds. Many otherwise familiar examples of smooth manifolds, however, cannot be given a Lie group structure, since given a Lie group $G$ and any $g\in G$, one could consider the map $m\left(g,\cdot\right):G\to G$ which sends the identity element $e$ to $g$ and hence, by considering the differential $T_eG\to T_gG,$ gives a natural identification between any two tangent spaces of a Lie group. In particular, by considering an arbitrary nonzero vector in $T_eG,$ one can use these identifications to give a smooth non-vanishing vector field on $G.$ This shows, for instance, that no even-dimensional sphere can support a Lie group structure. The same argument shows, more generally, that every Lie group must be
parallelizable In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a basis of the tangent space at p. Equiv ...
.

# Alternative definitions

## Pseudogroups

The notion of a pseudogroup 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 # If , and ''U'' is an open subset of the domain of ''f'', then the restriction ''f'', ''U'' is also in Γ. # If ''f'' is a homeomorphism from a union of open subsets of ''S'', $\cup_i \, U_i$, to an open subset of ''S'', then provided $f, _ \in \Gamma$ for every ''i''. # For every open , the identity transformation of ''U'' is in Γ. # If , then . # The composition of two elements of Γ is in Γ. 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 ''Ck''
diffeomorphisms In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
on R''n'' form a pseudogroup. All biholomorphisms between open sets in C''n'' form a pseudogroup. More examples include: orientation preserving maps of R''n'',
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s,
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s,
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s, and so on. Thus, a wide variety of function classes determine pseudogroups. An atlas (''Ui'', ''φ''''i'') of homeomorphisms ''φ''''i'' from to open subsets of a topological space ''S'' is said to be ''compatible'' with a pseudogroup Γ provided that the transition functions are all in Γ. A differentiable manifold is then an atlas compatible with the pseudogroup of ''C''''k'' functions on R''n''. A complex manifold is an atlas compatible with the biholomorphic functions on open sets in C''n''. 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 alternative approach to endow a manifold with a ''Ck''-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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
that defines, for each open set , an algebra C''k''(''U'') of continuous functions . A structure sheaf C''k'' is said to give ''M'' the structure of a ''C''''k'' manifold of dimension ''n'' provided that, for any , there exists a neighborhood ''U'' of ''p'' and ''n'' functions such that the map is a homeomorphism onto an open set in R''n'', and such that C''k'', ''U'' is the pullback of the sheaf of ''k''-times continuously differentiable functions on R''n''. In particular, this latter condition means that any function ''h'' in C''k''(''V''), for ''V'', can be written uniquely as , where ''H'' is a ''k''-times differentiable function on ''f''(''V'') (an open set in R''n''). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on R''n'', and ''a fortiori'' this is sufficient to characterize the differential 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, but uses
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
s of the germs of differentiable functions. It is especially popular in the context of ''complex'' manifolds. We begin by describing the basic structure sheaf on R''n''. If ''U'' is an open set in R''n'', let :O(''U'') = ''C''''k''(''U'', R) 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 consists of germs of functions near ''p'', and is an algebra over R. In particular, this is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic ...
whose unique maximal ideal consists of those functions that vanish at ''p''. The pair is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings. A differentiable manifold (of class ''Ck'') consists of a pair where ''M'' is a second countable
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
, and O''M'' is a sheaf of local R-algebras defined on ''M'', such that the locally ringed space is locally isomorphic to . In this way, differentiable manifolds can be thought of as schemes modeled on R''n''. This means that Hartshorne (1997) for each point , there is a neighborhood ''U'' of ''p'', and a pair of functions , where # ''f'' : ''U'' → ''f''(''U'') ⊂ R''n'' is a homeomorphism onto an open set in R''n''. # ''f''#: O, ''f''(''U'') → ''f'' (O''M'', ''U'') is an isomorphism of sheaves. # The localization of ''f''# is an isomorphism of local rings :: ''f''#''f''(''p'') : O''f''(''p'') → O''M'',''p''. 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, (in particular in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
), one could take this to be the space of complex numbers C''n'' equipped with the sheaf of
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deri ...
s (thus arriving at the spaces of complex analytic geometry), or the sheaf of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
s (thus arriving at the spaces of interest in complex ''algebraic'' geometry). In broader terms, this concept can be adapted for any suitable notion of a scheme (see
topos theory In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
). Second, coordinates are no longer explicitly necessary to the construction. The analog of a coordinate system is the pair , but these merely quantify the idea of ''local isomorphism'' rather than being central to the discussion (as in the case of charts and atlases). Third, 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 In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic da ...
). 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. * The cotangent space at a point is ''Ip''/''Ip''2, where ''Ip'' is the maximal ideal of the stalk O''M'',''p''. * In general, the entire
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ma ...
can be obtained by a related technique (see
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ma ...
for details). *
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
(and jets) can be approached in a coordinate-independent manner using the ''I''''p''-adic filtration on O''M'',''p''. * The
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and o ...
(or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of O''M'' into the ring of dual numbers.

# Generalizations

The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
s are other attempts. 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. Banach manifolds and Fréchet manifolds, in particular manifolds of mappings are infinite dimensional differentiable manifolds.

## Non-commutative geometry

For a ''Ck'' manifold ''M'', the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of real-valued ''Ck'' functions on the manifold forms an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
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 the multiplicative identity, and is a differentiable analog of the ring of
regular function In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regula ...
s in algebraic geometry. It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the Banach–Stone theorem, and is more formally known as the spectrum of a C*-algebra. First, there is a one-to-one correspondence between the points of ''M'' and the algebra homomorphisms , as such a homomorphism ''φ'' corresponds to a codimension one ideal in ''Ck''(''M'') (namely the kernel of ''φ''), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of ''Ck''(''M'') recovers ''M'' as a point set, though in fact it recovers ''M'' as a topological space. One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and operator theory (interpreting Banach spaces geometrically). For example, the tangent bundle to ''M'' can be defined as the derivations of the algebra of smooth functions on ''M''. This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
– a commutative C*-algebra being precisely the ring of scalars of a manifold, by Banach–Stone, and allows one to consider ''non''commutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of noncommutative geometry.

*
Affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
* Atlas (topology) *
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distan ...
* Introduction to the mathematics of general relativity * List of formulas in Riemannian geometry *
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
*
Space (mathematics) In mathematics, a space is a set (sometimes called a universe) with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, ...

# Bibliography

* * * * . * * . * * * * * * * * * * * * {{DEFAULTSORT:Differentiable Manifold