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Dianalytic Manifold
In mathematics, dianalytic manifolds are possibly non-orientable generalizations of complex analytic manifolds. A dianalytic structure on a manifold is given by an atlas of charts such that the transition maps are either complex analytic maps or complex conjugates of complex analytic maps. Every dianalytic manifold is given by the quotient of an analytic manifold (possibly non-connected) by a fixed-point-free involution changing the complex structure to its complex conjugate structure. Dianalytic manifolds were introduced by , and dianalytic manifolds of 1 complex dimension are sometimes called Klein surface In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over ...s. References * Riemann surfaces {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analytic Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be embedded as a smooth submanifold of R2''n'', whereas it is "rare" for a complex manifold to have a holomorphic embedding into C''n''. Consider for example any compact connected complex manifold ''M'': any holomorphic function on it is const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles. Charts The definition of an atlas depends on the notion of a ''chart''. A chart for a topological space ''M'' (also called a coordinate chart, coordinate patch, coordinate map, or local frame) is a homeomorphism \varphi from an open subset ''U'' of ''M'' to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi). Formal definition of atlas An atlas for a topological space M is an indexed family \ of charts on M which covers M (that is, \bigcup_ U_ = M). If the codomain of each chart is the ''n''-dimensiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
