Differentiable manifold

A differentiable manifold is a topological manifold with a globally defined differentiable structure. Any topological manifold can be given a differentiable structure locally by using the homeomorphisms in its atlas, combined with the standard differerentiable structure on the Euclidean space. In other words, the homeomorphism can be used to give a local coordinate system. To induce a global differentiable structure, one can show that the natural compositions of the homeomorphisms on overlaps between charts in the atlas produce differentiable functions on Euclidean space. In other words, the coordinates defined by each homeomorphism are differentiable with respect to the coordinates defined by each other homeomorphism, provided the two charts have overlapping domains. This idea is often presented formally using transition maps.

This allows one to extend the meaning of differentiability to spaces without global coordinate systems. Specifically, a differentiable structure allows one to define a global differentiable tangent space, and consequently, differentiable functions, and differentiable tensor fields (including vector fields). Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the arena for physical theories such as classical mechanics (Hamiltonian mechanics, Lagrangian mechanics), general relativity and Yang-Mills theory (gauge theory). It is possible to develop calculus on differentiable manifolds, leading 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 C. F. Gauss and his student, Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture B. Riemann, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses which lie at the Bases of Geometry), Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, vol. 13, 1867. Available online ^* 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 like J. C. Maxwell, and mathematicians Gregorio Ricci-CurbastroG. Ricci, Delle derivazioni covarianti e controvarianti e del loro uso nella analisi applicata. (Italian) , (1888)Ricci, G.; Levi-Civita, T. Die Methoden des absoluten Differentialkalkuls (1901) and Tullio Levi-Civita The absolute differential calculus (calculus of tensors) (1927) lead 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 Einsteins theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann WeylDie Idee der Riemannschen Fläche, Teubner, 1955. in his 1913 book on Riemann surfaces. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler WhitneyHassler Whitney, Differentiable Manifolds, Annals of Mathematics 37 (1936), 645-680..

For more on the history of manifolds see the history section of the primary manifold entry.

Definition

A differentiable manifold is a topological manifold (with or without boundary) whose transition maps are all differentiable. A topological manifold without boundary is a topological space which is locally homeomorphic to Euclidean space, by homeomorphisms called charts. By composing two charts we can get a real function, called a transition map.

For instance, if φ _α and φ_β represent homeomorphisms of the topological manifold on charts U_α and U_β which overlap, then φ_α ο φ_β ^-1 must be a differentiable function on the open sets in Euclidean space corresponding to the map. In particular,

φ_α ο φ_β^-1 : φ_β(U_α ʌ U_β) -> φ_α (U_α ʌ U_β) is a differentiable bijective map from the function domain φ_β(U_α ʌ U_β), where ʌ denotes intersection.

A C^k n-manifold is a topological n-manifold for which all transition maps are C^k(Rⁿ). Thus a C⁰ n-manifold is a topological n-manifold and for k>0 we speak of differentiable manifolds.

A smooth manifold or C^∞-manifold is a differentiable manifold for which all the transitions maps are smooth. That is derivatives of all orders exist; so it is a C^k-manifold for all k.

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 on some open ball.

Atlas

An atlas for a topological space is a collection of open sets (i.e. charts) in the topology of the space such that the union is the topological space and each open set is homeomorphic (i.e. there is a continuous bijection) to an open set in Euclidean space. Every topological manifold has an atlas. A C^k-atlas is an atlas for which all transition maps are C^k. A topological manifold has a C⁰-atlas and generally 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 differentiable structure.

Compatible atlases

Different atlases can give rise to essentially 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. By combining all compatible atlases of a manifold, a so-called maximal atlas can be constructed; it is unique (for a fixed value of k).

Subatlases

A subatlas of an atlas, is a subset of its charts which still covers the manifold. It is possible for an atlas to have a subatlas which is smoother than itself. It turns out that every C¹ atlas admits a smooth atlas; thus it is not useful to distinguish differentiable and smooth manifolds. This is not true for an atlas which is merely continuous.

Sheaf

An alternate definition of a differentiable manifold is a topological space with a sheaf of functions, which is locally isomorphic to Euclidean space with the sheaf of differentiable functions. That means that the section associated with any open subset possessing a chart is isomorphic to the section of differentiable functions on the corresponding subset of Euclidean space.

Differentiable functions

Given a real valued function f on an m dimensional differentiable manifold M, the directional derivative at a point x in M is defined using the coordinate system of any chart containing x. The differentiable structure ensures that the directional derivative is the same for any choice of coordinate system via a chart. The directional derivatives for f define a linear transformation df on the tangent space, often called the differential of f.

Equivalently, the tangent vectors are sometimes defined as a set of linear functions on the space of differentiable real valued functions such that the linear functions obey the product rule. Given a differentiable parametric curve in the manifold, we may restrict the function to the curve and differentiate the function with respect to the parameter of the curve. The curve has a tangent vector at each point and the derivative at a point is then related to a directional derivative for that tangent vector. Note that the usual notion of directional derivative is given for unit vectors. This is not assumed in this case, since we do not assume a metric

Since differentiability is defined locally we may extend the idea of the differential to maps between differentiable manifolds. In other words, if f is a map between the manifolds M₁ and M₂, then we may use the local coordinate charts to define the differential df as if the mapping were between two Euclidean spaces. The differentiable structure of the manifolds ensures that the differential (which is a linear transformation on the respective tangent spaces) is independent of the choice of coordinates.

It is also possible to define differentiability in terms of the transition functions. This is particularly important from a theoretical point of view.

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 which 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). Of course if M or N is a Euclidean space we can forget about one of the charts.

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 unit. It is possible to reconstruct a differentiable manifold from its algebra of scalars.

Tangent bundle

The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension n as the manifold does. The collection of all tangent spaces can in turn be made into a manifold, the tangent bundle, whose dimension is 2n. The tangent bundle is where tangent vectors live, and is itself a differentiable manifold. The Lagrangian is a function on the tangent bundle.

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. In particular, if the vector space is finite and has an inner product then the linear functionals can be realized by the functions f_v(w) = .

The cotangent bundle is the dual tangent bundle in the sense that at each point, the cotangent space is the dual of the tangent space. The cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. The total space of a cotangent bundle naturally has the structure of a symplectic manifold. Cotangent vectors are sometimes called covectors.

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.

Jet bundle

The jet bundle is a generalization of both the tangent bundle and the cotangent bundle; where these bundles represent first derivatives and differentials, the jet bundle represents higher derivatives and differentials. A connection is a tensor on the jet bundle.

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 cannot be a differentiable manifold, 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.

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

^* := \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. More generally, an n-form is a tensor with contangent 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} : \mathcal{C}(M) \to \mathrm{T}^*(M) : f \mapsto \mathrm{d}f

such that

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

This map is the one which 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 which 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. This product interacts with the exterior derivative in accordance with a modified product rule:

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

Interior product

The interior product on the differential forms will send an n-form F and a vector field X to the n-1-form ι_X = n F(X,...), where the remaining arguments to F are filled in the arguments to ι_X.

Relationship with Lie derivative

For a general differential form, the Lie derivative is a contraction, taking into account the variation in X:

\mathcal{L}_X\omega = i_Xd\omega + d(i_X \omega)

Relationship with topological manifolds

Every topological manifold in dimension 1, 2, or 3 has a unique differentiable structure; 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 differentiable structure Donaldson, Simon. An Application of Gauge Theory to Four Dimensional Topology. Journal of Differential Geometry vol. 18, 1983, 279-315. , and some with multiple incompatible structures. The classic example of a manifold with multiple incompatible structures is the exotic sphere of John Milnor On Manifolds Homeomorphic to the 7-Sphere, John Milnor, The Annals of Mathematics, 2nd Ser., Vol. 64, No. 2. (Sep., 1956), pp. 399-405. This gives the first examples of exotic spheres..

Classification

Every connected second-countable 1-manifold without boundary is homeomorphic to R or to S(the circle). The unconnected ones are disjoint unions of these two.

For a classification of 2-manifolds, see surface.

The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; his work is currently being evaluated, as of June 14, 2003.

The classification of n-manifolds for n greater than three is known to be impossible, even up to homotopy equivalence; it is equivalent to the so-called word problem in group theory, which has been shown to be undecidable. In other words, there is no algorithm for deciding whether a given manifold (differentiable or topological) is simply connected. However, there is a classification of simply connected differentiable manifolds of dimension ≥ 5, using cobordism and surgery. Andrew Ranicki, Algebraic and Geometric Surgery, ISBN-10: 0-19-850924-3, ISBN-13: 978-0-19-850924-0, 26 September 2002, Clarendon Press, Oxford Mathematical Monographs

(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 to 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 surfaces have a symplectic structure, since a symplectic structure is simply a volume form.

Lie groups

A Lie group is C^∞ manifold which 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. The diffeological spaces, (differential spaces) use a different notion of chart known as "plot". Frölicher spaces are another attempt.

References

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