| Differentiable Manifold |
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Informally, a differentiable manifold is a type of Manifold (which is in turn a kind of Topological Space ) that is similar enough to Euclidean Space to allow one to do Calculus . It is important to note that '''differentiable''' can mean slightly different things in different contexts, such as Continuously Differentiable , ''k'' times Differentiable , infinitely differentiable (also known as Smooth ), or Complex Differentiable (also known as Holomorphic ). Any manifold can be described by a collection (or Atlas ) of Charts . Each chart specifies a Coordinate System on a piece of the manifold, which is a function from that piece of the manifold into a Euclidean space. One may then apply ideas from calculus while working within the individual charts, since these lie in Euclidean spaces to which the usual rules of calculus apply. Problems can arise, however, when passing from one chart to the other. Consider the image to the right. Here the results of calculus clearly do not carry over from one chart to the other. In the middle chart, for instance, the Tropic of Cancer is a smooth curve, whereas in the first chart it has a sharp corner. The notion of a differentiable manifold refines the notion of a manifold by requiring the transition from one chart to another to be differentiable. More formally, 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 Homeomorphism s in its atlas, combined with the standard differentiable 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, the Compositions of the homeomorphisms on overlaps between charts in the atlas must be differentiable functions on Euclidean 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 other chart. These maps that relate the coordinates defined by the various charts to each other in areas of intersection are called ''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 Carl Friedrich Gauss and Bernhard Riemann . Riemann first described manifolds in his famous habilitation lectureB. Riemann (1867). 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 , and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita See G. Ricci (1888), G. Ricci and T. Levi-Civita (1901), T. Levi-Civita (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 Transformation s. 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 Surface sSee H. Weyl (1955).. The widely accepted general definition of a manifold in terms of an Atlas is due to Hassler Whitney H. Whitney (1936).. For more on the history of manifolds see the History section of the primary Manifold entry. DEFINITION A topological manifold is a Second Countable Hausdorff Space which is locally homeomorphic to Euclidean space, by a collection (called an ''atlas'') of Homeomorphism s 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 Euclidean space onto another open subset of Euclidean space. 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.
These definitions, however, leave out an important feature. They each still involve a preferred choice of atlas. Given a differentiable manifold (in any of the above senses), there is a notion of when two atlases are ''equivalent''. Then, strictly speaking, a differentiable manifold is an Equivalence Class of such atlases. (See below.) Atlases An Atlas on a topological space ''X'' is a collection of pairs {(''U''α,φα)} called ''charts'', where the ''U''α are open sets which Cover ''X'', and for each index α : is a Homeomorphism of ''U''α onto an open subset of ''n''-dimensional Euclidean space. The transition maps of the atlas are the functions | |||||
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| \left Rac{d}{dt}\phi\circ\gamma 1(t) Ight {t | 0} = \left rac{d}{dt}\phi\circ\gamma_2(t)
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