Differentiable manifold

In mathematics, a differentiable 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 homeomorphisms 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 and vector 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.

History

The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture^[1] 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,^[2] and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita^[3] 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 transformations. These ideas found a key application in 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 surfaces.^[4] The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.^[5]

Definition

A presentation of a topological manifold is a second countable Hausdorff space that is locally homeomorphic to a linear space, by a collection (called an atlas) of homeomorphisms called charts. The composition of one chart with the inverse of another chart is a function called a transition map, and defines a homeomorphism of an open subset of the linear space onto another open subset of the linear space. This formalizes the notion of "patching together pieces of a space to make a manifold" – the manifold produced also contains the data of how it has been patched together. However, different atlases (patchings) may produce "the same" manifold; a manifold does not come with a preferred atlas. And, thus, one defines a topological manifold to be a space as above with an equivalence class of atlases, where one defines equivalence of atlases below.

There are a number of different types of differentiable manifolds, depending on the precise differentiability requirements on the transition functions. Some common examples include the following.

• A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. In broader terms, a C^k-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.
• A smooth manifold or C^∞-manifold is a differentiable manifold for which all the transition maps are smooth. That is, derivatives of all orders exist; so it is a C^k-manifold for all k. An equivalence class of such atlases is said to be a smooth structure.
• An analytic manifold, or C^ω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is absolutely convergent and equals the function on some open ball.
• A complex manifold is a topological space modeled on a Euclidean space over the complex field and for which all the transition maps are holomorphic.

While there is a meaningful notion of a C^k atlas, there is no distinct notion of a C^k manifold other than C⁰ (continuous maps: a topological manifold) and C^∞ (smooth maps: a smooth manifold), because for every C^k-structure with k > 0, there is a unique C^k-equivalent C^∞-structure (every C^k-structure is uniquely smoothable to a C^∞-structure) – a result of Whitney.^[6] In fact, every C^k-structure is uniquely smoothable to a C^ω-structure. Furthermore, 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 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

$X$

$U_{\alpha }$

$U_{\beta }$

$\varphi _{\alpha }$

$\varphi _{\beta }$

$\varphi _{\alpha \beta }$

$\varphi _{\beta \alpha }$

$\mathbf {R} ^{n}$

$\mathbf {R} ^{n}$

Charts on a manifold

An atlas 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

$\varphi _{\alpha }\colon U_{\alpha }\to {\mathbf {R} }^{n}$

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

$\varphi _{\alpha \beta }=\varphi _{\beta }\circ \varphi _{\alpha }^{-1}|_{\varphi _{\alpha }(U_{\alpha }\cap U_{\beta })}\colon \varphi _{\alpha }(U_{\alpha }\cap U_{\beta })\to \varphi _{\beta }(U_{\alpha }\cap U_{\beta }).$

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; 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.

Alternative definitions

Pseudogroups

The notion of a pseudogroup^[7] 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

1. If f ∈ Γ, and U is an open subset of the domain of f, then the restriction f|_U is also in Γ.
2. If f is a homeomorphism from a union of open subsets of S, $\cup _{i}\,U_{i}$, to an open subset of S, then f ∈ Γ provided $f|_{U_{i}}\in \Gamma$ for every i.
3. For every open U ⊂ S, the identity transformation of U is in Γ.
4. If f ∈ Γ, then f⁻¹ ∈ Γ.
5. 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 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, symplectomorphisms, 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 U_i ⊂ M to open subsets of a topological space S is said to be compatible with a pseudogroup Γ 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 Γ.

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 of M, denoted C^k, is a sort of functor 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 of the sheaf of k-times continuously differentiable functions on Rⁿ.^[8]

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 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 rings 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 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 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 (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 second countable Hausdorff 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 modelled on Rⁿ. This means that,^[9] 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^#_f(p): O_f(p) → O_{M, p}.

There are a number of important motivations for studying differentiable manifolds within this abstract framework. First, there is no a priori reason that the model space needs to be Rⁿ. For example, (in particular in algebraic geometry), one could take this to be the space of complex numbers Cⁿ equipped with the sheaf of holomorphic functions (thus arriving at the spaces of complex analytic geometry), or the sheaf of polynomials (thus arriving at the spaces of interest in complex algebraic geometry). In 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).

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 I_p/I_p², where I_p 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 for details).
• Taylor series (and jets) can be approached in a coordinate-independent manner using the I_p-adic filtration on O_{M, p}.
• 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.

Differentiable functions

A real valued function f on an n-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

$f\circ \phi ^{-1}\colon \phi (U)\subset {\mathbf {R} }^{n}\to {\mathbf {R} }$

is differentiable at φ(p). 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 C^k 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. 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

$\left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.$

If γ₁ and γ₂ are two curves such that γ₁(0) = γ₂(0) = p, and in any coordinate chart φ,

$\left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}$

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 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 p ∈ M is an equivalence class of differentiable curves γ with γ(0) = p, modulo the equivalence relation of first-order contact between the curves. Therefore,

$\gamma _{1}\equiv \gamma _{2}\iff \left.{\frac {d}{dt}}\phi \circ \gamma _{1}(t)\right|_{t=0}=\left.{\frac {d}{dt}}\phi \circ \gamma _{2}(t)\right|_{t=0}$

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_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:

$Xf(p):=\left.{\frac {d}{dt}}f(\gamma (t))\right|_{t=0}.$

Once again, the chain rule establishes that this is independent of the freedom in selecting γ from the equivalence class, since any curve with the same first order contact will yield the same directional derivative.

If the function f is fixed, then the mapping

$X\mapsto Xf(p)$

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

$df(p)\colon T_{p}M\to {\mathbf {R} }.$

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 of the φ_i are compact and locally finite;
• The support of φ_i is completely contained in U_α for some α;
• The φ_i sum to one at each point of M:

$\sum _{i}\phi _{i}(x)=1.\,$

(Note that this last condition is actually a finite sum at each point because of the local finiteness of the supports of the φ_i.)

Every open covering of a 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 spaces 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 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 under pointwise addition and multiplication, called the algebra of scalar fields or simply the algebra of scalars. This algebra has the constant function 1 as the multiplicative identity, and is a differentiable analog of the ring of regular functions in algebraic geometry.

It is possible to reconstruct a manifold from its algebra of scalars, first as a set, but also as a topological space – this is an application of the Banach–Stone theorem, and is more formally known as the spectrum of a C*-algebra. First, there is a one-to-one correspondence between the points of M and the algebra homomorphisms φ: C^k(M) → R, as such a homomorphism φ corresponds 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 (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 noncommutative C*-algebras as non-commutative generalizations of manifolds. This is the basis of the field of noncommutative geometry.

Bundles

Tangent bundle

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 x_k local to the point, the coordinate derivatives $\partial _{k}={\frac {\partial }{\partial x_{k}}}$ typically define a 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 2n. The tangent bundle is where tangent vectors 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 U_α × Rⁿ, 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 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 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 differentials. For a given set of local coordinates x^k the differentials dxk
p form a basis of the cotangent space at p.

Tensor bundle

The tensor bundle is the direct sum of all tensor products 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.

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 Rⁿ 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 GL(n, R) 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).

Jet bundles

On a manifold that is sufficiently smooth, various kinds of jet bundles can also be considered. The (first-order) tangent bundle of a manifold is the collection of curves in the manifold modulo the equivalence relation of first-order contact. By analogy, the k-th order tangent bundle is the collection of curves modulo the relation of k-th order contact. Likewise, the cotangent bundle is the bundle of 1-jets of functions on the manifold: the k-jet bundle is the bundle of their k-jets. These and other examples of the general idea of jet bundles play a significant role in the study of differential 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 from Rⁿ to M.^[10] 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 Rⁿ that fix the origin. Note that GL(n, R) is naturally isomorphic to G¹, and a subgroup of every G^k, k ≥ 2. In particular, a section of F²(M) gives the frame components of a connection on M. Thus, the quotient bundle F²(M)/ GL(n, R) is the bundle of 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 theorems for such functions.

There are, however, important differences in the calculus of vector fields (and tensor fields in general). In brief, the directional derivative of a vector field is not well-defined, or at least not defined in a straightforward manner. Several generalizations of the derivative of a vector field (or tensor field) do exist, and capture certain formal features of differentiation in Euclidean spaces. The chief among these are:

• The Lie derivative, which is uniquely defined by the differential structure, but fails to satisfy some of the usual features of directional differentiation.
• An affine connection, 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 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 submanifolds.

Differential calculus of functions

Differentiable functions between two manifolds are needed in order to formulate suitable notions of submanifolds, and other related concepts. If f: M → N 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 df: TM → TN. It is also denoted by Tf and called the tangent map. At each point of M, this is a linear transformation from one tangent space to another:

$df(p)\colon T_{p}M\to T_{f(p)}N.$

The rank of f at p is the rank of this linear transformation.

Usually the rank of a function is a pointwise property. However, if the function has maximal rank, then the rank will remain constant in a neighborhood of a point. A differentiable function "usually" has maximal rank, in a precise sense given by Sard's theorem. Functions of maximal rank at a point are called immersions and submersions:

• If 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 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 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

$(y_{1}\circ f,\dotsc ,y_{n}\circ f,x_{1},\dotsc ,x_{m-n})$

is a system of local coordinates of M in a neighborhood of p. Submersions form the foundation of the theory of fibrations and fibre bundles.

Lie derivative

A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields 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 defined by

$[A,B]:={\mathcal {L}}_{A}B=-{\mathcal {L}}_{B}A.$

The Lie derivatives are represented by vector fields, as infinitesimal generators of flows (active diffeomorphisms) on M. Looking at it the other way round, the group 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, divergence 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

There is a map from scalars to covectors called the exterior derivative

$\mathrm {d} \colon {\mathcal {C}}(M)\to \mathrm {T} ^{*}(M):f\mapsto \mathrm {d} f$

such that

$\mathrm {d} f\colon \mathrm {T} (M)\to {\mathcal {C}}(M):V\mapsto V(f).$

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 (k+l)-form. The exterior derivative extends to this algebra, and satisfies a version of the product rule:

$\mathrm {d} (\omega \wedge \eta )=\mathrm {d} \omega \wedge \eta +(-1)^{{\rm {deg\,}}\omega }(\omega \wedge \mathrm {d} \eta ).$

From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold. The rank n cohomology group is the quotient group of the closed forms by the exact forms.

Topology of differentiable manifolds

Relationship with topological manifolds

Every topological manifold in dimension 1, 2, or 3 has a unique differential structure (up to diffeomorphism); thus the concepts of topological and differentiable manifold are distinct only in higher dimensions. It is known that in each higher dimension, there are some topological manifolds with no smooth structure, and some with multiple non-diffeomorphic structures.

The existence of non-smoothable manifolds was proven by Kervaire (1960), see Kervaire manifold, and later explained in the context of Donaldson's theorem (compare Hilbert's fifth problem);^[11] a good example of a non-smoothable manifold is the E₈ manifold.

The classic example of manifolds with multiple incompatible structures are the exotic 7-spheres of John Milnor.^[12]

Classification

Every second-countable 1-manifold without boundary is homeomorphic to a disjoint union of countably many copies of R (the real line) and S (the circle); the only connected examples are R and S, and of these only S is compact. In higher dimensions, classification theory normally focuses only on compact connected manifolds.

For a classification of 2-manifolds, see surface: in particular compact connected oriented 2-manifolds are classified by their genus, which is a nonnegative integer.

A classification of 3-manifolds follows in principle from the geometrization of 3-manifolds and various recognition results for geometrizable 3-manifolds, such as Mostow rigidity and Sela's algorithm for the isomorphism problem for hyperbolic groups.^[13]

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-manifolds 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 R⁴ demonstrate.

However, the situation becomes more tractable for simply connected smooth manifolds of dimension ≥ 5, where the h-cobordism theorem can be used to reduce the classification to a classification up to homotopy equivalence, and surgery theory can be applied.^[14] This has been carried out to provide an explicit classification of simply connected 5-manifolds by Dennis Barden.

Structures on manifolds

(Pseudo-)Riemannian manifolds

A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. This metric can be used to interconvert vectors and covectors, and to define a rank 4 Riemann curvature tensor. On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure.

A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one). Pseudo-Riemannian manifolds of signature (3, 1) are important in general relativity. Not every differentiable manifold can be given a pseudo-Riemannian structure; there are topological restrictions on doing so.

A Finsler manifold is a generalization of a Riemannian manifold, in which the inner product is replaced with a vector norm; this allows the definition of length, but not angle.

Symplectic manifolds

A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional. Cotangent bundles, which arise as phase spaces in Hamiltonian mechanics, are the motivating example, but many compact manifolds also have symplectic structure. All orientable surfaces embedded in Euclidean space have a symplectic structure, the signed area form on each tangent space induced by the ambient Euclidean inner product.^{[note 1]} Every Riemann surface is an example of such a surface, and hence a symplectic manifold, when considered as a real manifold.

Lie groups

A Lie group is C^∞ manifold that also carries a group structure whose product and inversion operations are smooth as maps of manifolds. These objects arise naturally in describing symmetries.

Generalizations

The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. Diffeological spaces use a different notion of chart known as a "plot". Frölicher spaces and orbifolds are other attempts.

A rectifiable set generalizes the idea of a piece-wise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds.

Banach manifolds and Fréchet manifolds, in particular manifolds of mappings are infinite dimensional differentiable manifolds.

See also

• Affine connection
• Atlas (topology)
• Christoffel symbols
• Introduction to mathematics of general relativity
• List of formulas in Riemannian geometry
• Riemannian geometry
• Space (mathematics)

Notes

1. This form is clearly nondegenerate, and it must be closed because it is top-dimensional with respect to the surface; this reflects the exceptional isomorphism of Lie groups Sp(2, R) ≅ SL(2, R) between the symplectic group (corresponding to symplectic structure) and the special linear group (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.

References

1. B. Riemann (1867).
2. Maxwell himself worked with quaternions rather than tensors, but his equations for electromagnetism were used as an early example of the tensor formalism; see Dimitrienko, Yuriy I. (2002), Tensor Analysis and Nonlinear Tensor Functions, Springer, p. xi, ISBN 9781402010156.
3. See G. Ricci (1888), G. Ricci and T. Levi-Civita (1901), T. Levi-Civita (1927).
4. See H. Weyl (1955).
5. H. Whitney (1936).
6. H. Whitney (1936).
7. Kobayashi and Nomizu (1963), Volume 1.
8. This definition can be found in MacLane and Moerdijk (1992). For an equivalent, ad hoc definition, see Sternberg (1964) Chapter II.
9. Hartshorne (1997)
10. See S. Kobayashi (1972).
11. S. Donaldson (1983).
12. J. Milnor (1956). These are the first examples of exotic spheres.
13. Z. Sela (1995). However, 3-manifolds are only classified in the sense that there is an (impractical) algorithm for generating a non-redundant list of all compact 3-manifolds.
14. See A. Ranicki (2002).

Bibliography

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