Manifold - Wikipedia, the free encyclopedia Page 1 of 19 Manifold From Wikipedia, the free encyclopedia In mathematics (specifically in differential geometry and topology), a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one- dimensional manifolds, a plane and sphere (the surface of a ball) are two-dimensional manifolds, and so on into high-dimensional space. More formally, every point of an n-dimensional manifold has a neighborhood homeomorphic to an open subset of the n-dimensional space Rn. Although manifolds resemble Euclidean spaces near each point ("locally"), the global structure of a manifold may be more complicated. For example, The sphere (surface of a ball) is a two-dimensional any point on the usual two-dimensional surface of a manifold since it can be represented by a collection sphere is surrounded by a circular region that can be of two-dimensional maps. flattened to a circular region of the plane, as in a geographical map. However, the sphere differs from the plane "in the large": in the language of topology, they are not homeomorphic. The structure of a manifold is encoded by a collection of charts that form an atlas , in analogy with an atlas consisting of charts of the surface of the Earth. The concept of manifolds is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. For example, a manifold is typically endowed with a differentiable structure that allows one to do calculus and a Riemannian metric that allows one to measure distances and angles. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model space-time in general relativity. Contents 1 Motivational examples 1.1 Circle 1.2 Other curves 1.3 Enriched circle 2 History 2.1 Early development 2.2 Synthesis 2.3 Topology of manifolds: highlights 3 Mathematical definition 3.1 Broad definition 4 Charts, atlases, and transition maps 4.1 Charts http://en.wikipedia.org/wiki/Manifold 5/23/2011 Manifold - Wikipedia, the free encyclopedia Page 2 of 19 4.2 Atlases 4.3 Transition maps 4.4 Additional structure 5 Construction 5.1 Charts 5.1.1 Sphere with charts 5.2 Patchwork 5.2.1 Intrinsic and extrinsic view 5.2.2 n-Sphere as a patchwork 5.3 Identifying points of a manifold 5.4 Manifold with boundary 5.5 Gluing along boundaries 5.6 Cartesian products 6 Manifolds with additional structure 6.1 Topological manifolds 6.2 Differentiable manifolds 6.3 Riemannian manifolds 6.4 Finsler manifolds 6.5 Lie groups 6.6 Other types of manifolds 7 Classification and invariants 8 Examples of surfaces 8.1 Orientability 8.1.1 Möbius strip 8.1.2 Klein bottle 8.1.3 Real projective plane 8.2 Genus and the Euler characteristic 9 Maps of manifolds 9.1 Scalar-valued functions 10 Generalizations of manifolds 11 See also 11.1 By dimension 12 Notes 13 References 14 External links Motivational examples Circle Main article: Circle After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top half of the unit http://en.wikipedia.org/wiki/Manifold 5/23/2011 Manifold - Wikipedia, the free encyclopedia Page 3 of 19 circle, x2 + y2 = 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, projection onto the first coordinate is a continuous, and invertible, mapping from the upper semicircle to the open interval (−1,1): Such functions along with the open regions they map are called chart s. 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 Figure 1: The four charts each map quarter of the circle where both the x- and the y-coordinates are part of the circle to an open interval, positive. The two charts χtop and χright each map this part into the and together cover the whole circle. interval (0, 1). Thus a function T from (0, 1) to itself can be constructed, which first uses the inverse of the top chart to reach the circle and then follows the right 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 Figure 2: A circle manifold chart (x,y) and the fixed pivot point (−1, 0); t is the mirror image, with based on slope, covering all but one pivot point (+1, 0). The inverse mapping from s to ( x, y) is given point of the circle. by http://en.wikipedia.org/wiki/Manifold 5/23/2011 Manifold - Wikipedia, the free encyclopedia Page 4 of 19 It can easily be confirmed that x2 + y2 = 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 can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. Other curves Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. In this example we see that a manifold need not have any well-defined notion of distance, for there is no way to define the distance between points that don't lie in the same piece. Manifolds need not be closed; thus a line segment without its end points is a manifold. And they are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola (two open, infinite pieces) and the locus of points on a cubic curve y2 = x3−x (a closed loop piece and an open, infinite piece). However, we exclude examples like two touching circles that Four manifolds from algebraic share a point to form a figure-8; at the shared point we cannot curves: create a satisfactory chart. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a ■ circles, ■ parabola, ■ hyperbola, line (a + is not homeomorphic to a closed interval (line segment) ■ cubic. since deleting the center point from the + gives a space with four components (i.e., pieces) whereas deleting a point from a closed interval gives a space with at most two pieces; topological operations always preserve the number of pieces). 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 http://en.wikipedia.org/wiki/Manifold 5/23/2011 Manifold - Wikipedia, the free encyclopedia Page 5 of 19 example, the circle has a notion of distance between two points, the arc -length between the points; hence it is a Riemannian manifold . History For more details on this topic, see History of manifolds and varieties. The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Early development 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 manifolds 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.