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Metaplectic Structure
In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry. Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for symplectic spin geometry. Formal definition A metaplectic structure on a symplectic manifold (M, \omega) is an equivariant lif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Symplectic Frame Bundle
In symplectic geometry, the symplectic frame bundle of a given symplectic manifold (M, \omega)\, is the canonical principal (n,)-subbundle \pi_\colon\to M\, of the tangent frame bundle \mathrm FM\, consisting of linear frames which are symplectic with respect to \omega\,. In other words, an element of the symplectic frame bundle is a linear frame u\in\mathrm_(M)\, at point p\in M\, , i.e. an ordered basis (_1,\dots,_n,_1,\dots,_n)\, of tangent vectors at p\, of the tangent vector space T_(M)\,, satisfying :\omega_(_j,_k)=\omega_(_j,_k)=0\, and \omega_(_j,_k)=\delta_\, for j,k=1,\dots,n\,. For p\in M\,, each fiber _p\, of the principal (n,)-bundle \pi_\colon\to M\, is the set of all symplectic bases of T_(M)\,. The symplectic frame bundle \pi_\colon\to M\,, a subbundle of the tangent frame bundle \mathrm FM\,, is an example of reductive G-structure on the manifold M\,. See also * Metaplectic group * Metaplectic structure * Symplectic basis * Symplectic structure * Symplectic geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Symplectic Spinor Bundle
In differential geometry, given a metaplectic structure \pi_\colon\to M\, on a 2n-dimensional symplectic manifold (M, \omega),\, the symplectic spinor bundle is the Hilbert space bundle \pi_\colon\to M\, associated to the metaplectic structure via the metaplectic representation. The metaplectic representation of the metaplectic group — the two-fold covering of the symplectic group — gives rise to an infinite rank vector bundle; this is the symplectic spinor construction due to Bertram Kostant. A section of the symplectic spinor bundle \, is called a symplectic spinor field. Formal definition Let (,F_) be a metaplectic structure on a symplectic manifold (M, \omega),\, that is, an equivariant lift of the symplectic frame bundle \pi_\colon\to M\, with respect to the double covering \rho\colon (n,)\to (n,).\, The symplectic spinor bundle \, is defined to be the Hilbert space bundle : =\times_L^2(^n)\, associated to the metaplectic structure via the metaplectic representation \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name " symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Symplectic Frame Bundle
In symplectic geometry, the symplectic frame bundle of a given symplectic manifold (M, \omega)\, is the canonical principal (n,)-subbundle \pi_\colon\to M\, of the tangent frame bundle \mathrm FM\, consisting of linear frames which are symplectic with respect to \omega\,. In other words, an element of the symplectic frame bundle is a linear frame u\in\mathrm_(M)\, at point p\in M\, , i.e. an ordered basis (_1,\dots,_n,_1,\dots,_n)\, of tangent vectors at p\, of the tangent vector space T_(M)\,, satisfying :\omega_(_j,_k)=\omega_(_j,_k)=0\, and \omega_(_j,_k)=\delta_\, for j,k=1,\dots,n\,. For p\in M\,, each fiber _p\, of the principal (n,)-bundle \pi_\colon\to M\, is the set of all symplectic bases of T_(M)\,. The symplectic frame bundle \pi_\colon\to M\,, a subbundle of the tangent frame bundle \mathrm FM\,, is an example of reductive G-structure on the manifold M\,. See also * Metaplectic group * Metaplectic structure * Symplectic basis * Symplectic structure * Symplectic geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Metaplectic Group
In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence. Definition The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2''n''(R) and called the metaplectic group. The metaplectic group Mp2(R) is ''not'' a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Cohomology Group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Spin Geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research. See also * Contact geometry * Symplectic topology * Spinor * Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ... * Spin manifold Books * * Differential topology Differential geometry {{physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Obstruction Theory
Obstruction may refer to: Places * Obstruction Island, in Washington state * Obstruction Islands, east of New Guinea Medicine * Obstructive jaundice * Obstructive sleep apnea * Airway obstruction, a respiratory problem ** Recurrent airway obstruction * Bowel obstruction, a blockage of the intestines. * Gastric outlet obstruction * Distal intestinal obstruction syndrome * Congenital lacrimal duct obstruction * Bladder outlet obstruction Politics and law * Obstruction of justice, the crime of interfering with law enforcement * Obstructionism, the practice of deliberately delaying or preventing a process or change, especially in politics * Emergency Workers (Obstruction) Act 2006 Science and mathematics * Obstruction set in forbidden graph characterizations, in the study of graph minors in graph theory * Obstruction theory, in mathematics * Propagation path obstruction ** Single Vegetative Obstruction Model Sports * Obstruction (baseball), when a fielder illegally hinders a baser ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Covering Map
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : \tilde X \to X is a covering, (\tilde X, p) is said to be a covering space or cover of X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |