In mathematics, a differentiable manifold (also differential manifold) is a type of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
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 can ...
to allow one to apply
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 generalizati ...
. Any manifold can be described by a collection of charts (
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
). 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 its ...
), 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 dif ...
. 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 isom ...
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
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
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 tensor ...
and
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
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 ...
. 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 classi ...
,
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
In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(''N''), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using t ...
. 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 ...
, ''k''-times differentiable,
smooth (which itself has many meanings), and
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
.
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 and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
. 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
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus.
With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on th ...
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
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the 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 le ...
, 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 of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
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 and most influential physicists of all time. Einstein is best known for developing the theor ...
'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 (su ...
. 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 mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
in his 1913 book on
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. The widely accepted general definition of a manifold in terms of an
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geogra ...
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, immersions, characteristic classes, and geometric integratio ...
.
Definition
Atlases
Let be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
. 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 isom ...
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 Euclidea ...
. 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:
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
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 In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.
Definition
A smooth structure on a manifold M is ...
; 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, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
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 bum ...
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 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
let
be a collection of open subsets of
and for each
let
be an open (possibly empty) subset of
and let
be a smooth map. Suppose that
is the identity map, that
is the identity map, and that
is the identity map. Then define an equivalence relation on the disjoint union
by declaring
to be equivalent to
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
is a differentiable chart where
is an open set in
containing ''p'' and
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 bic ...
is differentiable at
, that is
is a differentiable function from the open set
, considered as a subset of
, to
. 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 ''C
k'' 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
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
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 in R
''n''. Then the directional derivative of ''f'' at ''p'' along γ is
If ''γ''
1 and ''γ''
2 are two curves such that , and in any coordinate chart
,
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 e ...
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
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* C ...
between the curves. Therefore,
in every coordinate chart
. Therefore, the equivalence classes are curves through ''p'' with a prescribed
velocity vector at ''p''. The collection of all tangent vectors at ''p'' forms a
vector space
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 can ...
: 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'':
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
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'':
Definition of tangent space and differentiation in local coordinates
Let
be a topological
-manifold with a smooth atlas
Given
let
denote
A "tangent vector at
" is a mapping
here denoted
such that
for all
Let the collection of tangent vectors at
be denoted by
Given a smooth function
, define
by sending a tangent vector
to the number given by
which due to the chain rule and the constraint in the definition of a tangent vector does not depend on the choice of
One can check that
naturally has the structure of a
-dimensional real vector space, and that with this structure,
is a linear map. The key observation is that, due to the constraint appearing in the definition of a tangent vector, the value of
for a single element
of
automatically determines
for all
The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically
:
and
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 ''C
k'', where . Let be an open covering of ''M''. Then a partition of unity subordinate to the cover is a collection of real-valued ''C
k'' 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 Britis ...
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'':
(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 ''C
k'' manifold ''M'' has a ''C
k'' partition of unity. This allows for certain constructions from the topology of ''C
k'' 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
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
, 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 coord ...
''n'' as does the manifold. For a set of (non-singular) coordinates ''x
k'' local to the point, the coordinate derivatives
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 of ...
, 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 e ...
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 elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
) 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 cons ...
of a vector space is the set of real valued linear functions on the vector space. The
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
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. Th ...
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
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 an ...
of a cotangent bundle has the structure of a
symplectic manifold. Cotangent vectors are sometimes called ''
covectors''. One can also define the cotangent bundle as the bundle of 1-
jets of functions from ''M'' to R.
Elements of the cotangent space can be thought of as
infinitesimal displacements: if ''f'' is a differentiable function we can define at each point ''p'' a cotangent vector ''df
p'', which sends a tangent vector ''X
p'' to the derivative of ''f'' associated with ''X
p''. However, not every covector field can be expressed this way. 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  ...
s. For a given set of local coordinates ''x
k,'' the differentials ''dx'' form a basis of the cotangent space at ''p''.
Tensor bundle
The tensor bundle is the
direct sum 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 \otime ...
s of the tangent bundle and the cotangent bundle. Each element of the bundle is a
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
, which can act as a
multilinear operator
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 tensor ...
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 ...
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
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
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 common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* C ...
. 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 operators 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, ''F
k''(''M''), is a principal ''G
k'' bundle over ''M'', where ''G
k'' is the
group of ''k''-jets; i.e., the group made up of
''k''-jets of diffeomorphisms of R
''n'' that fix the origin. Note that is naturally isomorphic to ''G''
1, and a subgroup of every ''G
k'', . 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
''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used ...
'', 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 theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its ''derivative is continuous and non-zero at ...
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, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
, 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 Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
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 forms. The fundamental theorems of integral calculus in several variables—namely
Green's theorem, the
divergence theorem, and
Stokes' theorem—generalize to a theorem (also called Stokes' theorem) relating the
exterior derivative and integration over
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s.
Differential calculus of functions
Differentiable functions between two manifolds are needed in order to formulate suitable notions of
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
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:
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, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
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 isom ...
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 g ...
. Embeddings formalize the notion of ''M'' being a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
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
is a system of local coordinates of ''M'' in a neighborhood of ''p''. Submersions form the foundation of the theory of
fibrations 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
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
, 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 Soph ...
, 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 proc ...
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 ...
of
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s over a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
''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 can ...
of all Lie derivatives on ''M'' forms an infinite dimensional
Lie algebra with respect to the
Lie bracket
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 identi ...
defined by
The Lie derivatives are represented by
vector fields, as
infinitesimal generators of flows (
active
Active may refer to:
Music
* ''Active'' (album), a 1992 album by Casiopea
* Active Records, a record label
Ships
* ''Active'' (ship), several commercial ships by that name
* HMS ''Active'', the name of various ships of the British Royal ...
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of diffeomorphisms of ''M'' has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the
Lie group 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 gr ...
,
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
and
curl operators.
The bundle of
differential forms, at each point, consists of all totally
antisymmetric multilinear maps on the tangent space at that point. It is naturally divided into ''n''-forms for each ''n'' at most equal to the dimension of the manifold; an ''n''-form is an ''n''-variable form, also called a form of degree ''n''. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. 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 of all smooth differential forms on a smooth manifold
. It is usually denoted by
. More precisely, if
, for
the operator
maps the space
of
-forms on
into the space
of
-forms (if
there are no non-zero
-forms on
so the map
is identically zero on
-forms).
For example, the exterior differential of a smooth function
is given in local coordinates
, with associated local co-frame
by the formula :
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:
The exterior derivative also satisfies the identity
. That is, if
is a
-form then the
-form
is identically vanishing. A form
such that
is called ''closed'', while a form
such that
for some other form
is called ''exact''. Another formulation of the identity
is that an exact form is closed. This allows one to define
de Rham cohomology of the manifold
, where the
th cohomology group is the
quotient group of the closed forms on
by the exact forms on
.
Topology of differentiable manifolds
Relationship with topological manifolds
Suppose that
is a topological
-manifold.
If given any smooth atlas
, it is easy to find a smooth atlas which defines a different smooth manifold structure on
consider a homeomorphism
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
and
are smooth for any
and
with these conditions being exactly the definition that both
and
are smooth, in contradiction to how
was selected.
With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on
by declaring that smooth atlases
and
are equivalent if there is a homeomorphism
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
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
to be the identity map.
If the dimension of
is 1, 2, or 3, then there exists a smooth structure on
, 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 ...
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
Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional gene ...
, 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
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
in the form of the
exotic 7-spheres.
Classification
Every one-dimensional connected smooth manifold is diffeomorphic to either
or
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:
or
or
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 ...
, 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 Mostow may refer to: People
* George Mostow (1923–2017), American mathematician
** Mostow rigidity theorem
* Jonathan Mostow
Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...
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
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
. 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 mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a ...
s have been classified up to homeomorphism by
Freedman using the
intersection form and
Kirby–Siebenmann invariant. Smooth 4-manifold theory is known to be much more complicated, as the
exotic smooth structures on R
4 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
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
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 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
for each
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 where the inner products are allowed to have an
indefinite signature, as opposed to being
positive-definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
; 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. ...
. 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
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth c ...
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 ze ...
; as such, this allows the definition of length, but not angle.
Symplectic manifolds
A
symplectic manifold is a manifold equipped with a
closed,
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
2-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, ...
. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric
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) ''momenta ...
, are a motivating example, since they admit a
natural symplectic form.
* All oriented two-dimensional Riemannian manifolds
are, in a natural way, symplectic, by defining the form
where, for any
denotes the vector such that
is an oriented
-orthonormal basis of
Lie groups
A
Lie group consists of a ''C''
∞ manifold
together with a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
structure on
such that the product and inversion maps
and
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
and any
, one could consider the map
which sends the identity element
to
and hence, by considering the differential
gives a natural identification between any two tangent spaces of a Lie group. In particular, by considering an arbitrary nonzero vector in
one can use these identifications to give a smooth non-vanishing vector field on
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. Equi ...
.
Alternative definitions
Pseudogroups
The notion of a
pseudogroup In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
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'',
, to an open subset of ''S'', then provided
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 of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
. Note that Γ need not be a group, however, since the functions are not globally defined on ''S''. For example, the collection of all local ''C
k''
diffeomorphisms
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 man ...
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 transformations,
affine transformations, and so on. Thus, a wide variety of function classes determine pseudogroups.
An atlas (''U
i'', ''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 ''C
k''-structure. Here ''k'' = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The
structure sheaf of ''M'', denoted C
''k'', is a sort of
functor
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 m ...
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
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
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
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
. This approach is strongly influenced by the theory of
schemes in
algebraic geometry, 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 n ...
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 R
n. 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 n ...
whose unique
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
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 ''C
k'') consists of a pair where ''M'' is a
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
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 m ...
, 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 R
n. 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
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 example ...
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 notio ...
). 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, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
at a point is ''I
p''/''I
p''
2, where ''I
p'' 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. Th ...
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. Th ...
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 ser ...
(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 of ...
(or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of O
''M'' into the ring of
dual numbers
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
.
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 space
In mathematics, a diffeology on a set generalizes the concept of smooth charts in a differentiable manifold, declaring what the "smooth parametrizations" in the set are.
The concept was first introduced by Jean-Marie Souriau in the 1980s under t ...
s 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 manifold
In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below) ...
s and
Fréchet manifolds, in particular
manifolds of mappings
are infinite dimensional differentiable manifolds.
Non-commutative geometry
For a ''C
k'' 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 ''C
k'' 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 ...
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 In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
In brief, the Banach–Stone theorem allows one to recove ...
, and is more formally known as the
spectrum of a C*-algebra In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra ''A'', denoted ''Â'', is the set of unitary equivalence classes of irreducible *-representations of ''A''. A *-representation π of ''A'' on a Hilbert space ''H'' is irreduc ...
. 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 ''C
k''(''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 ''C
k''(''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
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
(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
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
.
See also
*
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)
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
*
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 distanc ...
*
Introduction to the mathematics of general relativity
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved ...
*
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)
References
Bibliography
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