TheInfoList In mathematics, a differentiable manifold (also differential manifold) is a type of
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ... that is locally similar enough to a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to allow one to do
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ... . Any manifold can be described by a collection of charts, also known as an
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. 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 calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... ), 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 which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological sp ...
with a globally defined
differential structureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. Any topological manifold can be given a differential structure locally by using the
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
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 , the tangent space of a 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 manifold at a point can ...
, differentiable functions, and differentiable
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
and
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ... fields. Differentiable manifolds are very important in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ... . Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics,
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, and
Yang–Mills theory In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any Compact Lie algebra, compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, ''k''-times differentiable,
smooth Smooth may refer to: Mathematics * Smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuo ... (which itself has many meanings), and analytic.

# History

The emergence of differential geometry as a distinct discipline is generally credited to
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ... and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
. Riemann first described manifolds in his famous
habilitation Habilitation is the procedure to achieve the highest university degree in many European countries in which the candidate fulfills certain criteria set by the university which require excellence in research, teaching, and further education. Its qua ...
lecture before the faculty at
Göttingen Göttingen (, , ; nds, Chöttingen) is a university city in Lower Saxony Lower Saxony (german: Niedersachsen ; nds, Neddersassen; stq, Läichsaksen) is a German state The Federal Republic of Germany, as a federal state, consist ...
. 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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as num ... , and mathematicians
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ( ...
and
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italians, 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 s ...
led to the development of
tensor analysis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 less ...
, which identifies an intrinsic geometric property as one that is invariant with respect to
coordinate transformation In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as Euclidean space. ...
s. These ideas found a key application in
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ... 's theory of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
and its underlying
equivalence principle In the theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (fr ...
. A modern definition of a 2-dimensional manifold was given by
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of ... in his 1913 book on
Riemann surface In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. The widely accepted general definition of a manifold in terms of an
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is due to
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes, and ...
.

# Definition

## Atlases

Let be a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. A chart on consists of an open subset of , and a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
from to an open subset of some
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... 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: :$u\circ\varphi^=\big\left(u\circ\psi^\big\right)\circ\big\left(\psi\circ\varphi^\big\right),$ 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 to be applicable. The same problem is found if one considers instead functions ; one is led to the reparametrization formula :$\varphi\circ c=\big\left(\varphi\circ\psi^\big\right)\circ\big\left(\psi\circ c\big\right),$ 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 map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s 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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
; 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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... s and partitions of unity, 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 which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological sp ...
. 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 $\varphi_:V_ \to V_$ be a smooth map. Suppose that $\varphi_$ is the identity map, that $\varphi_ \circ \varphi_$ is the identity map, and that $\varphi_ \circ \varphi_ \circ \varphi_$ 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 $\varphi_\left(p\right) \in V_.$ 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 $\left(U,\varphi\right)$ is a differentiable chart where $U$ is an open set in $M$ containing ''p'' and $\varphi : U\to ^n$ is the map defining the chart, then ''f'' is differentiable at ''p''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
:$f\circ \varphi^ \colon \varphi\left(U\right)\subset ^n \to$ is differentiable at $\varphi\left(p\right)$, that is ''f'' is a differentiable function from the open set $\varphi\left(U\right)$, 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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... of a function on a differentiable manifold, the most fundamental of which is the
directional derivative In mathematics, the directional derivative of a multivariate differentiable function, differentiable (scalar) function along a given vector (mathematics), vector v at a given point x intuitively represents the instantaneous rate of change of the ...
. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable
affine Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Aff ...
structure with which to define
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
. 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 Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
in R''n''. Then the directional derivative of ''f'' at ''p'' along γ is :$\left.\fracf\left(\gamma\left(t\right)\right)\_.$ If ''γ''1 and ''γ''2 are two curves such that , and in any coordinate chart ''φ'', :$\left.\frac\varphi\circ\gamma_1\left(t\right)\_=\left.\frac\varphi\circ\gamma_2\left(t\right)\_$ 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 :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of differentiable curves ''γ'' with , modulo the equivalence relation of first-order
contact Contact may refer to: Interaction Physical interaction * Contact (geology)A geological contact is a boundary which separates one rock body from another. A contact can be formed during deposition, by the intrusion of magma, or through faulting ...
between the curves. Therefore, :$\gamma_1\equiv \gamma_2 \iff \left.\frac\varphi\circ\gamma_1\left(t\right)\_ = \left.\frac\varphi\circ\gamma_2\left(t\right)\_$ in every coordinate chart ''φ''. Therefore, the equivalence classes are curves through ''p'' with a prescribed
velocity vector The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ...
at ''p''. The collection of all tangent vectors at ''p'' forms a
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
: the
tangent space In , the tangent space of a 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 manifold at a point can ...
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'': :$Xf\left(p\right) := \left.\fracf\left(\gamma\left(t\right)\right)\_.$ 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 :$X\mapsto Xf\left(p\right)$ is a
linear functional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
on the tangent space. This linear functional is often denoted by ''df''(''p'') and is called the differential of ''f'' at ''p'': :$df\left(p\right) \colon T_pM \to .$

### 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 :$v_\alpha=D\Big, _\left(\varphi_\alpha\circ\varphi_\beta^\right)\left(v_\beta\right)$ 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 :$D\Big, _\left(f\circ\varphi_\alpha^\right)\left(v_\alpha\right),$ 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. 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 A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
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'': $\sum_i \varphi_i(x) = 1.$ (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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s to be considered: for instance L''p'' spaces,
Sobolev spaces In mathematics, a Sobolev space is a vector space of functions equipped with a normed space, norm that is a combination of Lp norm, ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a s ...
, 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 , the tangent space of a 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 manifold at a point can ...
of a point consists of the possible directional derivatives at that point, and has the same
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ... ''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 basisIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of the tangent space. The collection of tangent spaces at all points can in turn be made into a manifold, the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ... , whose dimension is 2''n''. The tangent bundle is where
tangent vector :''For a more general — but much more technical — treatment of tangent vectors, see tangent space.'' In mathematics, a tangent vector is a Vector (geometry), vector that is tangent to a curve or Surface (mathematics), surface at a given point. T ... s lie, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
) 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of a vector space is the set of real valued linear functions on the vector space. The
cotangent space In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
at a point is the dual of the tangent space at that point, and 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 may ...
is the collection of all cotangent spaces. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The
total space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
of a cotangent bundle has the structure of a
symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...
. Cotangent vectors are sometimes called ''
covector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
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, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form ''dQ'', for some differentiable function In calculus Calculus, originally called infinitesimal calculu ...
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 from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of all
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... s of the tangent bundle and the cotangent bundle. Each element of the bundle is a
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, which can act as a
multilinear operator In mathematics, a tensor is an algebraic object that describes a (multilinear mapping, multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include Vector (mathematics and ph ...
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 (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 Contact may refer to: Interaction Physical interaction * Contact (geology)A geological contact is a boundary which separates one rock body from another. A contact can be formed during deposition, by the intrusion of magma, or through faulting ...
. 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 300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator. In mathematics, a differential operator is an Operator (mathe ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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
locallyIn 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'' neighbourhood (mathematics), ne ...
. For example, there are versions of the implicit and
inverse function theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s 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 In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
, 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 Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus and
differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
s. The fundamental theorems of integral calculus in several variables—namely
Green's theorem In vector calculus, Green's theorem relates a line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geome ...
, the
divergence theorem In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ... , and
Stokes' theorem Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
—generalize to a theorem (also called Stokes' theorem) relating the
exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
and integration over
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s.

## Differential calculus of functions

Differentiable functions between two manifolds are needed in order to formulate suitable notions of
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, 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: :$df\left(p\right)\colon T_p M \to T_ N.$ The rank of ''f'' at ''p'' is the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
onto its image, then ''f'' is an
embedding In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Embeddings formalize the notion of ''M'' being a
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
s and
fibre bundle In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
s.

## Lie derivative

A
Lie derivative In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
, named after
Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Biography Marius Sop ... , is a
derivation Derivation may refer to: * Derivation (differential algebra), a unary function satisfying the Leibniz product law * Derivation (linguistics) * Formal proof or derivation, a sequence of sentences each of which is an axiom or follows from the precedi ...
on the
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
of
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s over a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ... ''M''. The
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of all Lie derivatives on ''M'' forms an infinite dimensional
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
with respect to the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightar ...
defined by : The Lie derivatives are represented by
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ... s, as infinitesimal generators of flows (
active Active may refer to: Music * Active (album), ''Active'' (album), a 1992 album by Casiopea * Active Records, a record label Ships * Active (ship), ''Active'' (ship), several commercial ships by that name * HMS Active, HMS ''Active'', the nam ...
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s) on ''M''. Looking at it the other way around, the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of diffeomorphisms of ''M'' has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
theory.

## Exterior calculus

The exterior calculus allows for a generalization of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ... ,
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ... and
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
operators. The bundle of
differential form In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
s, at each point, consists of all totally antisymmetric tensor, antisymmetric Multilinear map, 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

There is a map from scalars to covectors called the
exterior derivative On a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-interse ...
:$\mathrm \colon \mathcal\left(M\right) \to \mathrm^*\left(M\right) : f \mapsto \mathrmf$ such that :$\mathrmf \colon \mathrm\left(M\right) \to \mathcal\left(M\right) : V \mapsto V\left(f\right).$ This map is the one that 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 that 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 -form. The exterior derivative extends to this algebra, and satisfies a version of the product rule: $\mathrm(\omega \wedge \eta) = \mathrm \omega \wedge \eta+(-1)^(\omega \wedge \mathrm \eta).$ 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.

# 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 $\varphi_\alpha\circ\Phi\circ\varphi_\beta^$ and $\varphi_\alpha\circ\Phi^\circ\varphi_\beta^$ 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 Kervaire manifold, ten-dimensional example by . A Donaldson's theorem, major application of partial differential equations in differential geometry due to Simon Donaldson, 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, 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 sphere, 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 (topology), surface. 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 Teichmüller space, 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, is the Thurston's geometrization conjecture, geometrization conjecture, 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 presentation of a group, presented group, one can construct a closed 4-manifold having that group as fundamental group. Since there is no algorithm to decision problem, 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 decision problem, 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-manifolds have been classified up to homeomorphism by Michael Freedman, Freedman using the intersection theory, intersection form and Kirby–Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as the exotic R4, 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 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-manifolds by Dennis Barden.

# Structures on smooth manifolds

## (Pseudo-)Riemannian manifolds

A Riemannian manifold consists of a smooth manifold together with a positive-definite inner product space, inner product on each of the individual tangent spaces. This collection of inner products is called the Riemannian metric, 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 is a generalization of the notion of Riemannian manifold where the inner products are allowed to have an metric signature, indefinite signature, as opposed to being definite bilinear form, 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. Pseudo-Riemannian manifolds of signature are fundamental in
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. 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; 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 Differentiable manifold#Definition, smooth manifold, M , equipped with a Closed and exact differential forms, closed nondegenerate form, nondegenerate Differential form ...
is a manifold equipped with a closed form (calculus), closed, nondegenerate form, 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, are a motivating example, since they admit a Tautological one-form, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
consists of a ''C'' manifold $G$ together with a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 Hairy ball theorem, even-dimensional sphere can support a Lie group structure. The same argument shows, more generally, that every Lie group must be Parallelizable manifold, parallelizable.

# 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 A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
. 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 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'', symplectomorphisms, Möbius transformations, affine transformations, 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 sheaf (mathematics), structure sheaf of ''M'', denoted C''k'', is a sort of functor 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 argument, ''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 scheme (mathematics), schemes in algebraic geometry, but uses local rings of the germ (mathematics), 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 germ (mathematics), 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 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 In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
Hausdorff space, 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 scheme (mathematics), schemes modelled 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), one could take this to be the space of complex numbers C''n'' 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). In broader terms, this concept can be adapted for any suitable notion of a scheme (see topos, topos theory). 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). 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 In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...
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 may ...
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 may ...
for details). * Taylor series (and jets) can be approached in a coordinate-independent manner using the Completion (algebra)#Krull topology, ''I''''p''-adic filtration on O''M'',''p''. * The
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ... (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 theory, category 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 orbifolds 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 Convenient vector space#Application: Manifolds of mappings between finite dimensional manifolds, manifolds of mappings are infinite dimensional differentiable manifolds.

## Non-commutative geometry

For a ''Ck'' manifold ''M'', the Set (mathematics), set of real-valued ''Ck'' functions on the manifold forms an algebra over a field, 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 the multiplicative identity, and is a differentiable analog of the ring of regular functions 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 – 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.