Differentiable manifold - Wikipedia, the free encyclopedia Page 1 of 17 Differentiable manifold From Wikipedia, the free encyclopedia 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 A nondifferentiable atlas of charts for the globe. The differentiable), then computations done in one chart are valid in any other differentiable chart. results of calculus may not be compatible between charts Note that a differentiable manifold as it stands if the atlas is not differentiable. In the middle chart the does not have any metric structure or any notion Tropic of Cancer is a smooth curve, whereas in the first of orthogonality. The addition of metric (or it has a sharp corner. The notion of a differentiable pseudo-metric) structure corresponds to the manifold refines that of a manifold by requiring the linear space mentioned above actually being functions which transform between charts to be Euclidean space (or pseudo-Euclidean space). differentiable. More formally, 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, 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. Contents 1 History 2 Definition 2.1 Atlases 2.2 Compatible atlases 3 Alternative definitions http://en.wikipedia.org/wiki/Differentiable_manifold 5/23/2011 Differentiable manifold - Wikipedia, the free encyclopedia Page 2 of 17 3.1 Pseudogroups 3.2 Structure sheaf 3.2.1 Sheaves of local rings 4 Differentiable functions 4.1 Differentiation of functions 4.1.1 Directional differentiation 4.1.2 Tangent vectors and the differential 4.2 Partitions of unity 4.3 Differentiability of mappings between manifolds 4.4 Algebra of scalars 5 Bundles 5.1 Tangent bundle 5.2 Cotangent bundle 5.3 Tensor bundle 5.4 Frame bundle 5.5 Jet bundles 6 Calculus on manifolds 6.1 Differential calculus of functions 6.2 Lie derivative 6.3 Exterior calculus 6.3.1 Exterior derivative 7 Topology of differentiable manifolds 7.1 Relationship with topological manifolds 7.2 Classification 8 Structures on manifolds 8.1 (Pseudo-)Riemannian manifolds 8.2 Symplectic manifolds 8.3 Lie groups 9 Generalizations 10 See also 11 Notes 12 References 13 Bibliography History Main article: History of manifolds and varieties 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 http://en.wikipedia.org/wiki/Differentiable_manifold 5/23/2011 Differentiable manifold - Wikipedia, the free encyclopedia Page 3 of 17 The works of physicists such as James Clerk Maxwell[citation needed ], and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita [2] 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.[3] The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.[4] Definition A presentation of a topological manifold is a second countable Hausdorff space which 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, and conversely 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. More generally a Ck-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. ω An analytic manifold , or C -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. 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 Ck atlas, there is no distinct notion of a Ck manifold other than C0 (continuous maps: a topological manifold) and C∞ (smooth maps: a smooth manifold), because every Ck-structure with k > 0, there is a unique Ck-equivalent C∞-structure (every Ck-structure is uniquely smoothable ) – a result of Whitney (and further, two Ck atlases that are equivalent to a single C∞ atlas are equivalent as Ck atlases, so two distinct Ck 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 Ck maps, where there are meaningful differences for different k. For example, the Nash embedding theorem states that any manifold can be Ck isometrically embedded in Euclidean space RN – for any 1 ≤ k ≤ ∞ there is a sufficiently large N, but N http://en.wikipedia.org/wiki/Differentiable_manifold 5/23/2011 Differentiable manifold - Wikipedia, the free encyclopedia Page 4 of 17 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 on a topological space X is a collection of pairs {( Uα,φα)} called charts , where the Uα are open sets which cover X, and for each index α is a homeomorphism of Uα 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 C 0-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 C 1, 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 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.