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Nearly Kähler Manifold
In mathematics, a nearly Kähler manifold is an almost Hermitian manifold M, with almost complex structure J, such that the (2,1)-tensor \nabla J is skew-symmetric. So, : (\nabla_X J)X =0 for every vector field X on M. In particular, a Kähler manifold is nearly Kähler. The converse is not true. For example, the nearly Kähler six-sphere S^6 is an example of a nearly Kähler manifold that is not Kähler. The familiar almost complex structure on the six-sphere is not induced by a complex atlas on S^6. Usually, non Kählerian nearly Kähler manifolds are called "strict nearly Kähler manifolds". Nearly Kähler manifolds, also known as almost Tachibana manifolds, were studied by Shun-ichi Tachibana in 1959 and then by Alfred Gray from 1970 on. For example, it was proved that any 6-dimensional strict nearly Kähler manifold is an Einstein manifold and has vanishing first Chern class (in particular, this implies spin). In the 1980s, strict nearly Kähler manifolds obtained a l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Hermitian Manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure. A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure ( U(n) structure) on the manifold. By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed (i.e., it is a symplectic form), we get an almost Kähler structure. If both the almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Complex Structure
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry. The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s. Formal definition Let ''M'' be a smooth manifold. An almost complex structure ''J'' on ''M'' is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field ''J'' of degree such that J^2=-1 when regarded as a vector bundle isomorphism J\colon TM\to TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold. If ''M'' admits an almost complex structure, it must be even-dimensional. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-symmetric Matrix
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because A^\textsf = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = -A . Properties Throughout, we assume that all matrix entries belong to a field \mathbb whose characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-symmetric matrices is skew-symmetric. * A scalar ... [...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 vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
North Holland
North Holland (, ) is a Provinces of the Netherlands, province of the Netherlands in the northwestern part of the country. It is located on the North Sea, north of South Holland and Utrecht (province), Utrecht, and west of Friesland and Flevoland. As of January 2023, it had a population of about 2,952,000 and a total area of , of which is water. From the 9th to the 16th century, the area was an integral part of the County of Holland. During this period West Friesland (region), West Friesland was incorporated. In the 17th and 18th centuries, the area was part of the province of Holland and commonly known as the Noorderkwartier (English: "Northern Quarter"). In 1840, the province of Holland was split into the two provinces of North Holland and South Holland. In 1855, the Haarlemmermeer was drained and turned into land. The provincial capital is Haarlem (pop. 161,265). The province's largest city and also the largest city in the Netherlands is the Dutch capital Amsterdam, with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, with more than 170 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology and Medicine. Company structure The company head office is in Singapore. The Chairman and Editor-in-Chief is Dr Phua Kok Khoo, while the Managing Director is Doreen Liu. The company was co-founded by them in 1981. Imperial College Press In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology Information technology (IT) is a set of related fields within information and communications technology (ICT), that encompass computer systems, software, programming languages, data processing, data and information processing, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Gray (mathematician)
Alfred Gray (October 22, 1939 – October 27, 1998) was an American mathematician whose main research interests were in differential geometry. He also made contributions in the fields of complex variables and differential equations. Short biography Alfred Gray was born in Dallas, Texas to Alfred James Gray & Eloise Evans and studied mathematics at the University of Kansas. He received a Ph.D. from the University of California, Los Angeles in 1964 and spent four years at University of California, Berkeley. From 1970–1998 he was a professor at the University of Maryland, College Park. He died in Bilbao, Spain of a heart attack while working with students in a computer lab at Colegio Mayor Miguel de Unamuno around 4 AM, on October 27, 1998. Mathematical contributions In the broad area of differential geometry, he made specific contributions in classifying various types of geometrical structures, such as (Kähler manifolds and almost Hermitian manifolds). Gray introduced the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds (including the four-dimensional Lorentzian manifolds usually studied in general relativity). Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons. If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that :\mathrm = kg for some constant k, where \operatorname denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds. The Einstein condition and Einstein's equation In loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Spinor
Killing, Killings, or The Killing may refer to: Types of killing *-cide, a suffix that refers to types of killing (see List of types of killing), such as: ** Homicide, one human killing another *** Murder, unlawful killing of another human without justification or valid excuse *Animal slaughter, the killing of animals * Assisted dying Arts, entertainment, and media Films * ''Killing'' (film), a 2018 Japanese film * ''The Killing'' (film), a 1956 film noir directed by Stanley Kubrick * '' Encounter: The Killing'', a 2002 Indian film by Ajay Phansekar Television * ''The Killing'' (Danish TV series), a police procedural drama first broadcast in 2007 * ''The Killing'' (American TV series), a crime drama based on the Danish television series, first broadcast in 2011 Literature * ''Killing'' (comics), Italian photo comic series about a vicious vigilante-criminal * ''Killing'', a series of historical nonfiction books by Bill O'Reilly and Martin Dugard * "Killings" (short story), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Kähler Manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure. A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure ( U(n) structure) on the manifold. By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed (i.e., it is a symplectic form), we get an almost Kähler structure. If both the almos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Torsion (mechanics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a Set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and Homotopy, homotopies. A property that is invariant under such deformations is a to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
