Differentiable manifold

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A differentiable manifold is a generalization of Euclidean space to extend the meaning of differentiabillity. A differentiable manifold is a special kind of topological manifold, in which we know what it means for a function to be differentiable. 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.

Contents 1 History 2 Definition 2.1 Atlas 2.1.1 Compatible atlases 2.1.2 Subatlases 2.2 Sheaf 3 Differentiable functions 3.1 Algebra of scalars 4 Tangent bundle 5 Cotangent bundle 6 Jet bundle 7 Tensor bundle 8 Exterior calculus 8.1 Exterior derivative 8.2 Interior product 8.3 Lie derivative 9 Classification 10 Subtypes 11 (pseudo-)Riemannian manifolds 12 Symplectic manifolds 13 Lie groups 14 Generalizations

History

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.

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 is a C^∞-manifold and an analytic manifold is a C^ω-manifold.

Atlas

An atlas for a topological space is a collection of charts which cover it. 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 a analytic atlas is a C^ω atlas

Compatible atlases

Different atlases can give rise to essentially the same manifold. The circle can be mapped by two coordinate charts, but if you change the domains of these charts slightly you have a different atlas for the same manifold. We can combine these different atlases 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 which is unique.

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. 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 every open subset together with the restriction of the sheaf of functions to that subset is isomorphic to Euclidean space with the sheaf of differentiable functions.

Differentiable functions

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

For more details on this topic, see tangent bundle.

The tangent bundle is where vector fields live. It is a manifold. The Lagrangian is a scalar on the tangent bundle.

Associated with every point on a differentiable manifold is a tangent space and its dual, the cotangent space. The former consists of the possible directional derivatives, and the latter of the differentials, which can be thought of as infinitesimal elements of the manifold. These spaces always have 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.

Cotangent bundle

For more details on this topic, see cotangent bundle.

The cotangent bundle is the dual space of the tangent bundle. It is a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle. Symplectic manifold.

Jet bundle

For more details on this topic, see jet bundle.

The jet bundle is a generalization of both the tangent bundle and the cotangent bundle. A connection is a tensor on the jet bundle.

Tensor bundle

For more details on this topic, see tensor bundle.

The tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle.

Exterior calculus

For more details on this topic, see exterior calculus.

The exterior calculus allows for a generalization of the gradient, divergence and curl operators.

Exterior derivative

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

<math>\mathrm{d} : \mathcal{C}(M) \to \mathrm{T}^*(M) : f \mapsto \mathrm{d}f</math>

such that

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

Interior product

Please see exterior algebra for details

Lie derivative

Classification

Every connected second-countable topological 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; 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 given manifold is simply connected. However, there is a classification of simply connected manifolds of dimension ≥ 5.

manifold needs to disambiguated to either top. or diff.

Subtypes

(pseudo-)Riemannian manifolds

For more details on this topic, see Riemannian manifold.

In order to measure lengths and angles, even more structure is needed: one defines Riemannian manifolds to recover these geometrical ideas.

...

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. On a Riemannian manifold one has notions of length, volume, and angle.

• 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.

• A Finsler manifold is a generalization of a Riemannian manifold.

Symplectic manifolds

For more details on this topic, see symplectic manifold.

A symplectic manifold is a manifold equipped with a closed, nondegenerate, alternating 2-form. Such manifolds arise in the study of Hamiltonian mechanics.

Lie groups

For more details on this topic, see Lie group.

A Lie group is C^∞ manifold which also carries a smooth group structure. These are the proper objects for describing symmetries of analytical structures.

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.