∞-structure (every C^k-structure is uniquely smoothable) – a result of Whitney

Hassler Whitney

Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...

 (and further, two C^k atlases that are equivalent to a single C^∞ atlas are equivalent as C^k atlases, so two distinct C^k atlases do not collide); see Differential structure: Existence and uniqueness theorems for details. Thus one uses the terms "differentiable manifold" and "smooth manifold" interchangeably. This is in stark contrast to C^k maps, where there are meaningful differences for different k. For example, the Nash embedding theorem

Nash embedding theorem

The Nash embedding theorems , named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path...

 states that any manifold can be C^k isometrically embedded in Euclidean space R^N – for any 1 ≤ k ≤ ∞ there is a sufficiently large N, but N depends on k.

On the other hand, complex manifolds are significantly more restrictive. As an example, Chow's theorem states that any projective complex manifold is in fact a projective variety – it has an algebraic structure.

Atlases

An atlas

Atlas (topology)

In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

 on a topological space X is a collection of pairs {(U_α,φ_α)} called charts, where the U_α are open sets that cover X, and for each index α
is a homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 of U_α onto an open subset of n-dimensional real space. The transition maps of the atlas are the functions

Every topological manifold has an atlas. A C^k-atlas is an atlas whose transition maps are C^k. A topological manifold has a C⁰-atlas and in general a C^k-manifold has a C^k-atlas. A continuous atlas is a C⁰ atlas, a smooth atlas is a C^∞ atlas and an analytic atlas is a C^ω atlas. If the atlas is at least C¹, it is also called a differential structure or differentiable structure. A holomorphic atlas is an atlas whose underlying Euclidean space is defined on the complex field and whose transition maps are biholomorphic.

Compatible atlases

Different atlases can give rise to, in essence, the same manifold. The circle can be mapped by two coordinate charts, but if the domains of these charts are changed slightly a different atlas for the same manifold is obtained. These different atlases can be combined into a bigger atlas. It can happen that the transition maps of such a combined atlas are not as smooth as those of the constituent atlases. If C^k atlases can be combined to form a C^k atlas, then they are called compatible. Compatibility of atlases is an equivalence relation

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

; by combining all the atlases in an equivalence class, a maximal atlas can be constructed. Each C^k atlas belongs to a unique maximal C^k atlas.

Pseudogroups

The notion of a pseudogroup

Pseudogroup

In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra...

 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 consisting of homeomorphisms from open subsets of S to other open subsets of S such that

1. If , and is an open subset of the domain of , then the restriction is also in .
2. If is a homeomorphism from a union of open subsets of , , to an open subset of , then provided for every .
3. For every open , the identity transformation of is in .
4. If , then .
5. The composition of two elements of is in .

These last three conditions are analogous to the definition of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

. Note that need not be a group, however, since the functions are not globally defined on . For example, the collection of all local C^k diffeomorphisms on Rⁿ form a pseudogroup. All biholomorphisms between open sets in Cⁿ form a pseudogroup. More examples include: orientation preserving maps of Rⁿ, symplectomorphism

Symplectomorphism

In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...

s, Moebius transformations, affine transformation

Affine transformation

In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

s, and so on. Thus a wide variety of function classes determine pseudogroups.

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

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

Structure sheaf

Sometimes it can be useful to use an 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

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

 of M, denoted C^k, is a sort of functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

 that defines, for each open set U ⊂ M, an algebra C^k(U) of continuous functions U → R. A structure sheaf C^k is said to give M the structure of a C^k manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x¹,...,xⁿ ∈ C^k(U) such that the map f = (x¹, ..., xⁿ) : U → Rⁿ is a homeomorphism onto an open set in Rⁿ, and such that C^k|_U is the pullback

Pullback

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...

 of the sheaf of k-times continuously differentiable functions on Rⁿ.

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

A fortiori argument

The Latin phrase ' denotes "argument 'from [the] stronger [reason]'." For example, if it has been established that a person is deceased, then one can, with equal or greater certainty, argue that the person is not breathing.-Usage:...

 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

Ringed space

In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...

. This approach is strongly influenced by the theory of schemes

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

 in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, but uses local ring

Local ring

In abstract algebra, more particularly in 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 number fields examined at a particular place, or...

s of the germs

Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...

 of differentiable functions. It is especially popular in the context of complex manifolds.

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

O(U) = C^k(U,R)

consist of all real-valued k-times continuously differentiable functions on U. As U varies, this determines a sheaf of rings on Rⁿ. The stalk O_p for p ∈ Rⁿ consists of germs

Germ (mathematics)

In mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...

 of functions near p, and is an algebra over R. In particular, this is a local ring

Local ring

In abstract algebra, more particularly in 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 number fields examined at a particular place, or...

 whose unique maximal ideal

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...

 consists of those functions that vanish at p. The pair (Rⁿ, O) is an example of a locally ringed space: it is a topological space equipped with a sheaf whose stalks are each local rings.

A differentiable manifold (of class C^k) consists of a pair (M, O_M) where M is a topological space, and O_M is a sheaf of local R-algebras defined on M, such that the locally ringed space (M,O_M) is locally isomorphic to (Rⁿ, O). In this way, differentiable manifolds can be thought of as schemes

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

 modelled on Rⁿ. This means that, for each point p ∈ M, there is a neighborhood U of p, and a pair of functions (f,f^#) where

1. f : U → f(U) ⊂ Rⁿ is a homeomorphism onto an open set in Rⁿ.
2. f^# : O|_f(U) → f_* (O_M|_U) is an isomorphism of sheaves.
3. The localization of f^# is an isomorphism of local rings

f^#_p : O_f(p) → O_{M, p}.

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be Rⁿ. For example (in particular in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

), one could take this to be the space of complex numbers Cⁿ equipped with the sheaf of holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s (thus arriving at the spaces of complex analytic geometry

Complex analytic geometry

In mathematics, complex analytic geometry sometimesdenotes the application of complex numbers to plane geometry.Rather than represent a point in the plane as a pair of Cartesian coordinates, it can be represented as a single complex number, which can be written at will in either rectangular or...

), or the sheaf of polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s (thus arriving at the spaces of interest in complex algebraic geometry). In broad 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 (f, f^#), but these merely quantify the idea of local isomorphism rather than being central to the discussion (as in the case of charts and atlases). 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

Synthetic differential geometry

In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...

).

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
  Cotangent space
  In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
  
   at a point is I_p/I_p², where I_p is the maximal ideal of the stalk O_M,p.
• In general, the entire cotangent bundle
  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...
  
   can be obtained by a related technique (see cotangent bundle
  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...
  
   for details).
• Taylor series
  Taylor series
  In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
  
   (and jets
  Jet (mathematics)
  In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
  
  ) can be approached in a coordinate-independent manner using the I_p-adic filtration
  Completion (ring theory)
  In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
  
   on O_M,p.
• The tangent bundle
  Tangent bundle
  In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
  
   (or more precisely its sheaf of sections) can be identified with the sheaf of morphisms of O_M into the ring of dual numbers.

Differentiable functions

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

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

is differentiable at φ(p). The definition of differentiability depends on the choice of chart at p; in general there will be many available charts. However, it follows from the chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

 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

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 of a function on a differentiable manifold, the most fundamental of which is the directional derivative

Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...

. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

 structure with which to define vectors. The directional derivative therefore looks at curves in the manifold instead of vectors.

Directional differentiation

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

If γ₁ and γ₂ are two curves such that γ₁(0) = γ₂(0) = p, and in any coordinate chart φ,
then, by the chain rule, f has the same directional derivative at p along γ₁ as along γ₂. This means that the directional derivative depends only on the tangent vector

Tangent vector

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 Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

 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 vectors and the differential

A tangent vector at p ∈ M is an equivalence class of differentiable curves γ with γ(0) = p, modulo the equivalence relation of first-order contact

Contact (mathematics)

In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets...

 between the curves. Therefore,

in any (and hence all) coordinate charts 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

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

: the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 to M at p, denoted T_pM.

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

Linear functional

In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars.  In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...

 on the tangent space. This linear functional is often denoted by df(p) and is called the differential of f at p:

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

• The supports
  Support (mathematics)
  In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
  
   of the φ_i are compact
  Compact space
  In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
  
   and locally finite
  Locally finite collection
  In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension....
  
  ;
• 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 Rⁿ to be carried over to the category of differentiable manifolds. In particular, it is possible to discuss integration by choosing a partition of unity subordinate to a particular coordinate atlas, and carrying out the integration in each chart of Rⁿ. Partitions of unity therefore allow for certain other kinds of function space

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

s to be considered: for instance L^p spaces

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector 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 C^k(M, N)" mean for k≥1? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be C^k(R^m, Rⁿ). We define "f is C^k(M, N)" to mean that all such compositions of f with charts are C^k(R^m, Rⁿ). Once again the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already a Euclidean space, then we don't need a chart to map it to one.

Algebra of scalars

For a C^k manifold M, the set of real-valued C^k functions on the manifold forms an algebra

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

 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

Regular function

In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic 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 φ : C^k(M) → R, as such a homomorphism φ corresponds a codimension one ideal in C_k

Tangent bundle

Cotangent bundle

Tensor bundle

Frame bundle

Jet bundles

Calculus on manifolds

• The Lie derivative
  Lie derivative
  In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
  
  , which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
• An affine connection
  Affine connection
  In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
  
  , 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.

Differential calculus of functions

• If m ≤ n, and f : M → N has rank m at p ∈ M, then f is called an immersion at p. If f is an immersion at all points of M and is a homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
   onto its image, then f is an embedding
  Embedding
  In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
  
  . Embeddings formalize the notion of M being a submanifold
  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 S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...
  
   of N. In general, an embedding is an immersion without self-intersections and other sorts of non-local topological irregularities.

• If m ≥ n, and f : M → N has rank n at p ∈ M, 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 (y₁,...,y_n) in a neighborhood of f(p) in N, and m − n functions x₁,...,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

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

s and fibre bundles.

Lie derivative

Exterior calculus

Exterior derivative

Relationship with topological manifolds

Classification

(Pseudo-)Riemannian manifolds

Symplectic manifolds

This form is clearly nondegenerate, and it must be closed because it is top-dimensional with respect to the surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

; this reflects the exceptional isomorphism

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families of mathematical objects, that is not an example of a pattern of such isomorphisms.Because these series of objects are presented differently, they are not...

 of Lie groups between the symplectic group

Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...

 (corresponding to symplectic structure) and the special linear group

Special linear group

In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

 (corresponding to orientable structure). Note that a symplectic structure requires an additional integrability condition, beyond this isomorphism of groups: it is not just a G-structure

G-structure

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM of M....

.

Lie groups

Generalizations

See also

• Affine connection
  Affine connection
  In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
  
  
• Atlas (topology)
  Atlas (topology)
  In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
  
  
• Christoffel symbols
  Christoffel symbols
  In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
  
  
• Differential geometry
• Introduction to mathematics of general relativity
• List of formulas in Riemannian geometry
• 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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
  
  
• Space (mathematics)