Information AboutManifold |
| CATEGORIES ABOUT MANIFOLD | |
| topology | |
| differential topology | |
| differential geometry | |
| geometric topology | |
| manifolds | |
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A manifold is an abstract Mathematical Space in which every point has a Neighborhood which resembles Euclidean Space , but in which the global structure may be more complicated. In discussing manifolds, the idea of Dimension is important. For example, Lines are one-dimensional, and Planes two-dimensional. In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a Circle , and two separate circles. In a two-manifold, every point has a neighborhood that looks like a Disk . Examples include a plane, the surface of a Sphere , and the surface of a Torus . Manifolds are important objects in mathematics and Physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. Additional structures are often defined on manifolds. Examples of manifolds with additional structure include Differentiable Manifold s on which one can do Calculus , Riemannian Manifold s on which distances and angles can be defined, Symplectic Manifold s which serve as the Phase Space in Classical Mechanics , and the four-dimensional Pseudo-Riemannian Manifold s which model Space-time in General Relativity . A technical mathematical definition of a manifold is given below. To fully understand the mathematics behind manifolds, it is necessary to know elementary concepts regarding Set s and Function s, and helpful to have a working knowledge of Calculus and Topology . MOTIVATIONAL EXAMPLES Circle The Circle is the simplest example of a topological manifold after a line. Topology ignores bending, so a small piece of a circle is exactly the same as a small piece of a line. Consider, for instance, the top half of the Unit Circle , ''x''2 + ''y''2 = 1, where the ''y''-coordinate is positive (indicated by the yellow arc in ''Figure 1''). Any point of this semicircle can be uniquely described by its ''x''-coordinate. So, by Projecting onto the first coordinate, one obtains a Continuous Mapping between the semicircle and the Open Interval (−1, 1): : and similarly : Such a function is called a ''chart''. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle. Together, these parts cover the whole circle and the four charts form an Atlas for the circle. The top and right charts overlap: their intersection lies in the quarter of the circle where both the ''x''- and the ''y''-coordinates are positive. The two charts χtop and χright each map this part bijectively to the interval (0, 1). Thus a function ''T'' from (0, 1) to itself can be constructed, which first inverts the yellow chart to reach the circle and then follows the green chart back to the interval. Let ''a'' be any number in (0, 1), then: : Such a function is called a ''transition map''. The top, bottom, left, and right charts show that the circle is a manifold, but they do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of some choice. Consider the charts : and : Here ''s'' is the slope of the line through the point at coordinates (''x'',''y'') and the fixed pivot point (−1,0); ''t'' is the mirror image, with pivot point (+1,0). The inverse mapping from ''s'' to (''x'',''y'') is given by : it can easily be confirmed that ''x''2+''y''2 = 1 for all values of the slope ''s''. These two charts provide a second atlas for the circle, with : Each chart omits a single point, either (−1,0) for ''s'' or (+1,0) for ''t'', so neither chart alone is sufficient to cover the whole circle. It is not possible to cover the full circle with a single chart, since the circle is doubly connected and the line is only Simply Connected . Note that it is possible to construct a circle by "gluing" together a single piece of the line; this does not produce a chart, since a portion of the circle will be mapped to both "glued" regions at once. Other curves Manifolds need not be Connected (all in "one piece"); thus a pair of separate circles is also a manifold. They need not be Closed ; thus a line segment without its ends is a manifold. And they need not be finite; thus a Parabola is a manifold. Putting these freedoms together, two other example manifolds are a Hyperbola (two open, infinite pieces) and the Locus of points on the Cubic Curve ''y''2 = ''x''3−''x'' (a closed loop piece and an open, infinite piece). However, we exclude examples like two touching circles that share a point to form a figure-8; at the shared point we cannot create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. Enriched circle Viewed using Calculus , the circle transition function ''T'' is simply a function between open intervals, which gives a meaning to the statement that ''T'' is Differentiable . The transition map ''T'', and all the others, are differentiable on (0, 1); therefore, with this atlas the circle is a '' Differentiable Manifold ''. It is also ''smooth'' and ''analytic'' because the transition functions have these properties as well. Other circle properties allow it to meet the requirements of more specialized types of manifold. For example, the circle has a notion of distance between two points, the arc-length between the points; hence it is a '' Riemannian Manifold ''. HISTORY The study of manifolds combines many important areas of mathematics: it generalizes concepts such as Curve s and Surfaces as well as ideas from Linear Algebra and Topology . Prehistory Before the modern concept of a manifold there were several important results. Non-Euclidean Geometry considers spaces where Euclid 's Parallel Postulate fails. Saccheri first studied them in 1733 . Lobachevsky , Bolyai , and Riemann developed them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean Space ; these gave rise to Hyperbolic Geometry and Elliptic Geometry . In the modern theory of manifolds, these notions correspond to Riemannian Manifold s with constant negative and positive Curvature , respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His Theorema Egregium gives a method for computing the Curvature of a Surface without considering the Ambient Space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more Topological example of an intrinsic Property of a manifold is its Euler Characteristic . Leonhard Euler showed that for a convex Polytope in the three-dimensional Euclidean space with ''V'' vertices (or corners), ''E'' edges, and ''F'' faces, : ''V''-''E''+''F''= 2. The same formula will hold if we project the vertices and edges of the polytope onto a Sphere , creating a 'map' with ''V'' vertices, ''E'' edges, and ''F'' faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.The notion of a map can formalized as a Cell Decomposition . Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a Torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with ''V''=1 vertex, ''E''=2 edges, and ''F''=1 face. Thus the Euler characteristic of the torus is 1-2+1=0. The Euler characteristic of other surfaces is a useful Topological Invariant , which can be extended to higher dimensions using Betti Number s. In the mid nineteenth century, the Gauss–Bonnet Theorem linked the Euler characteristic to the Gaussian curvature. Synthesis Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of Elliptic Integral s in the first half of 19th century led them to consider special types of Complex Manifold s, now known as Jacobians . Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of Analytic Continuation of functions of complex variables, although these ideas were way ahead of their time. Another important source of manifolds in 19th century mathematics was Analytical Mechanics , as developed by Simeon Poisson , Jacobi, and William Rowan Hamilton . The possible states of a mechanical system are thought to be points of an abstract space, Phase Space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose Dimension corresponds to the degrees of freedom of the system and where the points are specified by their Generalized Coordinate s. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various Conservation Laws constrain it to more complicated formations, e.g. Liouville Tori . The theory of a rotating solid body, developed in 18th century by Leonhard Euler and Joseph Lagrange , gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré , one of the founders of Topology . Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name ''manifold'' comes from Riemann's original German term, ''Mannigfaltigkeit'', which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a ''Mannigfaltigkeit'', because the variable can have ''many'' values. He distinguishes between ''stetige Mannigfaltigkeit'' and ''diskrete'' ''Mannigfaltigkeit'' (''continuous manifoldness'' and ''discontinuous manifoldness''), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using Induction , Riemann constructs an ''n-fach ausgedehnte Mannigfaltigkeit'' (''n times extended manifoldness'' or ''n-dimensional manifoldness'') as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a ''Mannigfaltigkeit'' evolved into what is today formalized as a manifold. Riemannian Manifold s and Riemann Surface s are named after Bernhard Riemann. Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a Topological Space that followed shortly. During the 1930s Hassler Whitney and others clarified the Foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through Differential Geometry and Lie Group theory. Topology of manifolds: highlights Two-dimensional manifolds, also known as ''surfaces'', were considered by Riemann under the guise of Riemann Surface s, and rigorously classified in the beginning of the 20th century by Poul Heegard and Max Dehn . Henri Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré Conjecture . After nearly a century of effort by many mathematicians, starting with Poincaré himself, a consensus among experts (as of 2006) is that Grigori Perelman has proved the Poincaré conjecture (see the Hamilton-Perelman Solution Of The Poincaré Conjecture ). Bill Thurston 's Geometrization Program , formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson , who was motivated by the then recent progress in theoretical physics ( Yang-Mills Theory ), where they serve as a substitute for ordinary 'flat' Space-time . Important work on higher-dimensional manifolds, including analogues of Poincaré conjecture, had been done earlier by René Thom , John Milnor , Stephen Smale and Sergei Novikov . One of the most pervasive and flexible techniques underlying much work on the Topology Of Manifolds is Morse Theory . MATHEMATICAL DEFINITION In topology, an n-manifold is a Second Countable Hausdorff Space in which every point has a neighborhood Homeomorphic to an open Euclidean ''n''-ball, | ||
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| In Technical Language, A Manifold With Boundary Is A Space Containing Both Interior Points And Boundary Points Every Interior Point Has A Neighborhood Homeomorphic To The Open ''n''-ball {(x<sub>1</sub>, X<sub>2</sub>, …, X<sub>n</sub>) Σ X<sub>i</sub><sup>2</sup> < 1} Every Boundary Point Has A Neighborhood Homeomorphic To The "half" ''n''-ball {(x<sub>1</sub>, X<sub>2</sub>, …, X<sub>n</sub>) Σ X<sub>i</sub><sup>2</sup> < 1 And X<sub>1</sub> &ge 0} The Homeomorphism Must Send The Boundary Point To A Point With X<sub>1</sub> | 0 |
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| A '''Finsler Manifold''' Allows The Definition Of Distance, But Not Of Angle It Is An Analytic Manifold In Which Each | "http://wwwinformationdelightinfo/information/entry/tangent_space" class="copylinks">Tangent Space is equipped with a Norm , ·, in a manner which varies smoothly from point to point This norm can be extended to a Metric , defining the length of a curve but it cannot in general be used to define an inner product |
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