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Calibrated Geometry
In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (''M'',''g'') of dimension ''n'' equipped with a differential ''p''-form ''φ'' (for some 0 ≤ ''p'' ≤ ''n'') which is a calibration, meaning that: * ''φ'' is closed: d''φ'' = 0, where d is the exterior derivative * for any ''x'' ∈ ''M'' and any oriented ''p''-dimensional subspace ''ξ'' of T''x''''M'', ''φ'', ''ξ'' = ''λ'' vol''ξ'' with ''λ'' ≤ 1. Here vol''ξ'' is the volume form of ''ξ'' with respect to ''g''. Set ''G''''x''(''φ'') = . (In order for the theory to be nontrivial, we need ''G''''x''(''φ'') to be nonempty.) Let ''G''(''φ'') be the union of ''G''''x''(''φ'') for ''x'' in ''M''. The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds w ... [...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] |
Stokes' Theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan(jp)(1979/01) [] (Written in Japanese) after Lord Kelvin and Sir George Stokes, 1st Baronet, George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical Stokes' theorem can be stated in one sentence: The line integral of a vector field over a loop is equal to the ''flux of its curl'' through the enclosed surface. Stokes' theorem is a special case of the generalized Stokes' theorem. In particular, a vector field on can be considered as a 1-form in which case its curl is its exterior derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi–Yau Manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after who first conjectured that such surfaces might exist, and who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. Definitions The motivational definition given by Shing-Tung Yau is of a compact Kähl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wirtinger Inequality (2-forms)
: ''For other inequalities named after Wirtinger, see Wirtinger's inequality.'' In mathematics, the Wirtinger inequality for 2-forms, named after Wilhelm Wirtinger, states that on a Kähler manifold , the exterior th power of the symplectic form (Kähler form) , when evaluated on a simple (decomposable) -vector of unit volume, is bounded above by . That is, : (\underbrace_)(v_1,\ldots,v_) \leq k ! for any orthonormal vectors . In other words, is a calibration on . An important corollary of the further characterization of equality is that every complex submanifold of a Kähler manifold is volume minimizing in its homology class. See also *2-form *Gromov's inequality for complex projective space * Systolic geometry Notes References *{{cite book, last = Federer, first = Herbert, author-link1=Herbert Federer, title = Geometric measure theory, place= Berlin–Heidelberg–New York, publisher = Springer-Verlag Springer Science+Business Media, commonly known as Springer, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Submanifold
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] |
Kähler Form
Kähler may refer to: ;People * Alexander Kähler (born 1960), German television journalist * Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and archaeologist *Luise Kähler (1869–1955), German trade union leader and politician *Martin Kähler (1835–1912), German theologian *Otto Kähler (1894–1967), German admiral *Wilhelmine Kähler (1864–1941), German politician ;Other * Kähler Keramik, a Danish ceramics manufacturer *Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ..., an important geometric complex manifold See also * Kahler (other) {{disambiguation, surname Occupational surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics. Definitions Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology (mathematics)
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion-Kähler Manifold
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1) for some n\geq 2. Here Sp(''n'') is the sub-group of SO(4n) consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n \times n matrix, while the group Sp(1) = S^3 of unit-length quaternions instead acts on quaternionic n-space ^n = ^ by right scalar multiplication. The Lie group Sp(n)\cdot Sp(1) \subset SO(4n) generated by combining these actions is then abstractly isomorphic to p(n) \times Sp(1) _2. Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by also requiring that the scalar curvature be non-zero— as is automatically true if the holonomy group equals the entire group Sp(''n'')·Sp(1). Early history Marcel Berger's 1955 paper on the classifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
