Information AboutManifold |
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| 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. Lines are one-dimensional, 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 a pair of 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 a working knowledge of Calculus and Topology would be helpful. 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 , where the ''y''-coordinate is positive (the yellow part in ''Figure 1''). Any point in 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): : 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: : 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 and the Locus of points on the Cubic Curve ''y''2−''x''3+''x''=0. 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 generalises concepts such as Curves and Surfaces as well as ideas from Linear Algebra and Topology . Certain special classes of manifolds also have additional algebraic structure; they may behave like Group s, for instance. Prehistory Before the modern concept of a manifold there were several important results. 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 the Euler Characteristic . For a non-intersecting Graph (graph Theory) in Euclidean 2-dimensional space, with ''V'' vertices (or corners), ''E'' edges and ''F'' faces (counting the exterior) Euler showed that ''V''-''E''+''F''= 2. Thus 2 is called the Euler characteristic of Euclidean 2-dimensional space. The Euler characteristic of other 2-dimensional spaces is a useful Topological Invariant , which can be extended to higher dimensions using Betti Number s. 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 manifolds with constant negative and positive Curvature , respectively. Synthesis Bernhard Riemann was the first 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 ''discrete'' ''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. In the study of Complex Variables , the process of Analytic Continuation leads to the construction of manifolds. Abelian Varieties were already implicitly known in Riemann's time as Complex Manifold s. Lagrangian Mechanics and Hamiltonian Mechanics , when considered geometrically, are also naturally manifold theories. All these use the notion of several characteristic Axes or Dimension s (known as Generalized Coordinates in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth. Henri Poincaré studied three-dimensional manifolds and raised a question, today known as the Poincaré Conjecture . As of 2006, a consensus among experts is that recent work by Grigori Perelman may have answered this question, after nearly a century of effort by many mathematicians. Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. 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. 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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| A '''Finsler Manifold''' Allows The Definition Of Distance, But Not Of Angle It Is An Analytic Manifold In Which Each | "http://wwwinformationdelightinfo/encyclopedia/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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