In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
that is locally similar enough to a linear space
to allow one to do calculus
. Any manifold can be described by a collection of charts, also known as an atlas
. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable
), then computations done in one chart are valid in any other differentiable chart.
In formal terms, a differentiable manifold is a topological manifold
with a globally defined differential structure
. Any topological manifold can be given a differential structure ''locally'' by using the homeomorphism
s in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition
on chart intersections in the atlas must be differentiable functions on the corresponding linear space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called ''transition maps.''
Differentiability means different things in different contexts including: continuously differentiable
, ''k'' times differentiable, smooth
, and holomorphic
. Furthermore, the ability to induce such a differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A differential structure allows one to define the globally differentiable tangent space
, differentiable functions, and differentiable tensor
fields. Differentiable manifolds are very important in physics
. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics
, general relativity
, and Yang–Mills theory
. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus.
The study of calculus on differentiable manifolds is known as differential geometry.
The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss
and Bernhard Riemann
. Riemann first described manifolds in his famous habilitation
lecture before the faculty at Göttingen
. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:
: ''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
, and mathematicians Gregorio Ricci-Curbastro
and Tullio Levi-Civita
led to the development of tensor analysis
and the notion of covariance
, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformation
s. These ideas found a key application in Albert Einstein
's theory of general relativity
and its underlying equivalence principle
. A modern definition of a 2-dimensional manifold was given by Hermann Weyl
in his 1913 book on Riemann surface
s. The widely accepted general definition of a manifold in terms of an atlas
is due to Hassler Whitney
Let be a topological space
. A chart on consists of an open subset of , and a homeomorphism
from to an open subset of some Euclidean space
. 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
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
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
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 map
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
; 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.
A differentiable manifold is a Hausdorff
and second countable
topological space , together with a maximal differentiable atlas on . Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of bump function
s and partitions of unity
, both of which are used ubiquitously.
The notion of a manifold is identical to that of a topological manifold
. 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
be a collection of open subsets of
and for each
be an open (possibly empty) subset of
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
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.
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
is differentiable at
, that is ''f'' is a differentiable function from the open set
, considered as a subset of
. 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
of a function on a differentiable manifold, the most fundamental of which is the directional derivative
. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine
structure with which to define vectors
. Therefore, the directional derivative looks at curves in the manifold instead of vectors.
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
. Then the directional derivative of ''f'' at ''p'' along γ is
are two curves such that , and in any coordinate chart ''φ'',
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
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
of differentiable curves ''γ'' with , modulo the equivalence relation of first-order contact
between the curves. Therefore,
in every coordinate chart ''φ''. 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
: the tangent space
to ''M'' at ''p'', denoted ''T''''p''
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
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
be a topological
-manifold with a smooth atlas
A "tangent vector at
" is a mapping
Let the collection of tangent vectors at
be denoted by
Given a smooth function
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
The above formal definitions correspond precisely to a more informal notation which appears often in textbooks, specifically
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''
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 ''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
s to be considered: for instance L''p'' spaces
, Sobolev spaces
, and other kinds of spaces that require integration.
Differentiability of mappings between manifolds
Suppose ''M'' and ''N'' are two differentiable manifolds with dimensions ''m'' and ''n'', respectively, and ''f'' is a function from ''M'' to ''N''. Since differentiable manifolds are topological spaces we know what it means for ''f'' to be continuous. But what does "''f'' is " mean for ? We know what that means when ''f'' is a function between Euclidean spaces, so if we compose ''f'' with a chart of ''M'' and a chart of ''N'' such that we get a map that goes from Euclidean space to ''M'' to ''N'' to Euclidean space we know what it means for that map to be . We define "''f'' is " to mean that all such compositions of ''f'' with charts are . Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on ''M'' and ''N'' are selected. However, defining the derivative itself is more subtle. If ''M'' or ''N'' is itself already a Euclidean space, then we don't need a chart to map it to one.
The tangent space
of a point consists of the possible directional derivatives at that point, and has the same dimension
''n'' as does the manifold. For a set of (non-singular) coordinates ''xk
'' 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
, whose dimension is 2''n''. The tangent bundle is where tangent vector
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
) 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.
The dual space
of a vector space is the set of real valued linear functions on the vector space. The cotangent space
at a point is the dual of the tangent space at that point, and the cotangent bundle
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
of a cotangent bundle has the structure of a symplectic manifold
. Cotangent vectors are sometimes called ''covector
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
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
s. For a given set of local coordinates ''xk
,'' the differentials ''dx'' form a basis of the cotangent space at ''p''.
The tensor bundle is the direct sum
of all tensor product
s of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field
, which can act as a multilinear operator
on vector fields, or on other tensor fields.
The tensor bundle is not a differentiable manifold in the traditional sense, since it is infinite dimensional. It is however an algebra
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.
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
made up of the set of all frames over ''M''. The frame bundle is useful because tensor fields on ''M'' can be regarded as equivariant
vector-valued functions on F(''M'').
On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact
. By analogy, the ''k''-th order tangent bundle is the collection of curves modulo the relation of ''k''-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the ''k''-jet bundle is the bundle of their ''k''-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential operator
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
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
of a function: the differential. From the perspective of calculus, the derivative of a function on a manifold behaves in much the same way as the ordinary derivative of a function defined on a Euclidean space, at least locally
. For example, there are versions of the implicit
and inverse function theorem
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
, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
* An affine connection
, 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
also carry over to differential manifolds. These are naturally expressed in the language of exterior calculus
and differential form
s. 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
Differential calculus of functions
Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifold
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
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
* 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
onto its image, then ''f'' is an embedding
. Embeddings formalize the notion of ''M'' being a submanifold
of ''N''. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.
* If , and has rank ''n'' at , then ''f'' is called a submersion at ''p''. The implicit function theorem states that if ''f'' is a submersion at ''p'', then ''M'' is locally a product of ''N'' and R''m''−''n''
near ''p''. In formal terms, there exist coordinates in a neighborhood of ''f''(''p'') in ''N'', and functions ''x''1
, ..., ''x''''m''−''n''
defined in a neighborhood of ''p'' in ''M'' such that
:is a system of local coordinates of ''M'' in a neighborhood of ''p''. Submersions form the foundation of the theory of fibration
s and fibre bundle
A Lie derivative
, named after Sophus Lie
, is a derivation
on the algebra
of tensor field
s over a manifold
''M''. The vector space
of all Lie derivatives on ''M'' forms an infinite dimensional Lie algebra
with respect to the Lie bracket
The Lie derivatives are represented by vector field
s, as infinitesimal generator
s of flows (active diffeomorphism
s) on ''M''. Looking at it the other way around, the group
of diffeomorphisms of ''M'' has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the Lie group
The exterior calculus allows for a generalization of the gradient
The bundle of differential form
s, at each point, consists of all totally antisymmetric multilinear
maps on the tangent space at that point. It is naturally divided into ''n''-forms for each ''n'' at most equal to the dimension of the manifold; an ''n''-form is an ''n''-variable form, also called a form of degree ''n''. The 1-forms are the cotangent vectors, while the 0-forms are just scalar functions. In general, an ''n''-form is a tensor with cotangent rank ''n'' and tangent rank 0. But not every such tensor is a form, as a form must be antisymmetric.
There is a map from scalars to covectors called the exterior derivative
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
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
is a topological
If given any smooth atlas
, it is easy to find a smooth atlas which defines a different smooth manifold structure on
consider a homemorphism
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
are smooth for any
with these conditions being exactly the definition that both
are smooth, in contradiction to how
With this observation as motivation, one can define an equivalence relation on the space of smooth atlases on
by declaring that smooth atlases
are equivalent if there is a homeomorphism
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 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
*Some topological manifolds admit many smooth structures which are not equivalent in the sense given above. This was originally discovered by John Milnor
in the form of the exotic 7-spheres
Every one-dimensional connected smooth manifold is diffeomorphic to either
each with their standard smooth structures.
For a classification of smooth 2-manifolds, see surface
. A particular result is that every two-dimensional connected compact smooth manifold is diffeomorphic to one of the following:
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
, is the 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 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
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 structure
s on R4
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-manifold
s by Dennis Barden.
Structures on smooth manifolds
A Riemannian manifold
consists of a smooth manifold together with a positive-definite 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
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 indefinite signature
, as opposed to being positive-definite
; they are still required to be non-degenerate. Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as the Riemann curvature tensor
. Pseudo-Riemannian manifolds of signature are fundamental in general relativity
. 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.
A symplectic manifold
is a manifold equipped with a closed
, nondegenerate 2-form
. This condition forces symplectic manifolds to be even-dimensional, due to the fact that skew-symmetric
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 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
A Lie group
consists of a ''C''∞
together with a group
such that the product and inversion maps
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
, one could consider the map
which sends the identity element
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
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'',
, 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
. 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
form a pseudogroup. All biholomorphism
s between open sets in C''n''
form a pseudogroup. More examples include: orientation preserving maps of R''n''
s, Möbius transformation
s, affine transformation
s, and so on. Thus, a wide variety of function classes determine pseudogroups.
An atlas (''Ui
) 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.
Sometimes, it can be useful to use an alternative approach to endow a manifold with a ''Ck
''-structure. Here ''k'' = 1, 2, ..., ∞, or ω for real analytic manifolds. Instead of considering coordinate charts, it is possible to start with functions defined on the manifold itself. The structure sheaf
of ''M'', denoted C''k''
, is a sort of functor
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''
is the pullback
of the sheaf of ''k''-times continuously differentiable functions on R''n''
In particular, this latter condition means that any function ''h'' in C''k''
(''V''), for ''V'', can be written uniquely as , where ''H'' is a ''k''-times differentiable function on ''f''(''V'') (an open set in R''n''
). Thus, the sheaf-theoretic viewpoint is that the functions on a differentiable manifold can be expressed in local coordinates as differentiable functions on R''n''
, and ''a fortiori''
this is sufficient to characterize the differential structure on the manifold.
Sheaves of local rings
A similar, but more technical, approach to defining differentiable manifolds can be formulated using the notion of a ringed space
. This approach is strongly influenced by the theory of schemes
in algebraic geometry
, but uses local ring
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''
:O(''U'') = ''C''''k''
consist of all real-valued ''k''-times continuously differentiable functions on ''U''. As ''U'' varies, this determines a sheaf of rings on Rn
. The stalk O''p''
for consists of germs
of functions near ''p'', and is an algebra over R. In particular, this is a local ring
whose unique maximal ideal
consists of those functions that vanish at ''p''. The pair is an example of a locally ringed space
: it is a topological space equipped with a sheaf whose stalks are each local rings.
A differentiable manifold (of class ''Ck
'') consists of a pair where ''M'' is a second countable Hausdorff space
, 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
modelled on R''n''
. This means that
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''
) is an isomorphism of sheaves.
# The localization of ''f''#
is an isomorphism of local rings
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 function
s (thus arriving at the spaces of complex analytic geometry
), or the sheaf of polynomial
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
). 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
at a point is ''Ip
, where ''Ip
'' is the maximal ideal of the stalk O''M'',''p''
* In general, the entire cotangent bundle
can be obtained by a related technique (see cotangent bundle
* Taylor series
) can be approached in a coordinate-independent manner using the ''I''''p''-adic filtration
* The tangent bundle
(or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of O''M''
into the ring of dual numbers
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
s use a different notion of chart known as a "plot". Frölicher space
s and orbifold
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.
s and Fréchet manifold
s, in particular
manifolds of mappings
are infinite dimensional differentiable manifolds.
For a ''Ck
'' manifold ''M'', the set
of real-valued ''Ck
'' functions on the manifold forms an algebra
under pointwise addition and multiplication, called the ''algebra of scalar fields'' or simply the ''algebra of scalars''. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of regular function
s in algebraic geometry.
It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the Banach–Stone theorem
, and is more formally known as the spectrum of a C*-algebra
. First, there is a one-to-one correspondence between the points of ''M'' and the algebra homomorphisms , as such a homomorphism ''φ'' corresponds to a codimension one ideal in ''Ck
''(''M'') (namely the kernel of ''φ''), which is necessarily a maximal ideal. On the converse, every maximal ideal in this algebra is an ideal of functions vanishing at a single point, which demonstrates that MSpec (the Max Spec) of ''Ck
''(''M'') recovers ''M'' as a point set, though in fact it recovers ''M'' as a topological space.
One can define various geometric structures algebraically in terms of the algebra of scalars, and these definitions often generalize to algebraic geometry (interpreting rings geometrically) and operator theory
(interpreting Banach spaces geometrically). For example, the tangent bundle to ''M'' can be defined as the derivations of the algebra of smooth functions on ''M''.
This "algebraization" of a manifold (replacing a geometric object with an algebra) leads to the notion of a C*-algebra
– 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
*Introduction to the mathematics of general relativity
*List of formulas in Riemannian geometry