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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] |
Wirtinger's Inequality (other)
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Wirtinger's inequality is either of two inequalities named after Wilhelm Wirtinger: * Wirtinger's inequality for functions * 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelm Wirtinger
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory. Biography He was born at Ybbs on the Danube and studied at the University of Vienna, where he received his doctorate in 1887, and his habilitation in 1890. Wirtinger was greatly influenced by Felix Klein with whom he studied at the University of Berlin and the University of Göttingen. Honours In 1907 the Royal Society of London awarded him the Sylvester Medal, for his contributions to the general theory of functions. Work Research activity He worked in many areas of mathematics, publishing 71 works. His first significant work, published in 1896, was on theta functions. He proposed as a generalization of eigenvalues, the concept of the spectrum of an operator, in an 1897 paper; the concept was further extended by David Hilbert and now it forms the main object of investigation in the field of spectral ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Form
In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument separately; ; Alternating: holds for all ; and ; Non-degenerate: for all implies that . If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, ''ω'' can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If ''V'' is finite-dimensional, then its dimension must necessarily be even s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Inequality For Complex Projective Space
In Riemannian geometry, Gromov's optimal stable 2- systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here \operatorname is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line \mathbb^1 \subset \mathbb^n in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms. Projective planes over division algebras \mathbb In the special case n=2, Gromov's inequality becomes \mathrm_2^2 \leq 2 \mathrm_4(\mathbb^2). This inequality can be thought of as an analog of Pu's inequality for the real projective plane \mathbb^2. In both cases, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systolic Geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slower-paced Introduction to systolic geometry. The notion of systole The ''systole'' of a compact metric space ''X'' is a metric invariant of ''X'', defined to be the least length of a noncontractible loop in ''X'' (i.e. a loop that cannot be contracted to a point in the ambient space ''X''). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of ''X''. When ''X'' is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequalities
Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * Spatial inequality, the unequal distribution of income and resources across geographical regions * Income inequality metrics, used to measure income and economic inequality among participants in a particular economy * International inequality, economic differences between countries Healthcare * Health equity, the study of differences in the quality of health and healthcare across different populations Mathematics * Inequality (mathematics), a relation between two values when they are different Social sciences * Educational inequality, the unequal distribution of academic resources to socially excluded communities * Gender inequality, unequal treatment or perceptions of individuals due to their gender * Participation inequality, the phe ... [...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 st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
