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Calabi–Eckmann Manifold
In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, homeomorphic to a product of two odd-dimensional spheres of dimension ≥ 3. The Calabi–Eckmann manifold is constructed as follows. Consider the space ^n\backslash \ \times ^m\backslash \, where m,n>1, equipped with an action of the group : : t\in , \ (x,y)\in ^n\backslash \ \times ^m\backslash \ \mid t(x,y)= (e^tx, e^y) where \alpha\in \backslash is a fixed complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for .... It is easy to check that this action is free and proper, and the corresponding orbit space ''M'' is homeomorphic to S^\times S^. Since ''M'' is a quot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Geometry
In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces such as complex manifolds and Complex algebraic variety, complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaf, coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenio Calabi
Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications. Early life and education Calabi was born in Milan, Italy on May 11, 1923, into a Jewish family. His sister was the journalist Tullia Zevi Calabi. In 1938, the family left Italy because of the racial laws, and in 1939 arrived in the United States. In the fall of 1939, aged only 16, Calabi enrolled at the Massachusetts Institute of Technology, studying chemical engineering. His studies were interrupted when he was drafted in the US military in 1943 and served during World War II. Upon his discharge in 1946, Calabi was able to finish his bachelor's degree under the G.I. Bill, and was a Putnam Fellow. He received a master's degree in mathematics from the University of Illinois Urbana-Champaign in 1947 and his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beno Eckmann
Beno Eckmann (31 March 1917 – 25 November 2008) was a Switzerland, Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry. Life Born to a Jewish family in Bern, Eckmann received his master's degree from ETH Zurich, Eidgenössische Technische Hochschule Zürich (ETH) in 1939. Later, he studied there under Heinz Hopf, obtaining his Ph.D. in 1941. His dissertation, on homotopy theory, was jointly supervised by Heinz Hopf and Ferdinand Gonseth. In 1942 he obtained a lecturer position at the University of Lausanne. He became an extraordinary professor there before, in 1948, taking a full professorship at ETH Zurich, where he remained until his 1984 retirement. He was also President of the Swiss Mathematical Society for 1961–1962, and the founding head of the Mathematics Research Institute at ETH Zurich from 1964 until his retirement. Recognition A colloquium in honor of Eckmann's 60th birthday was held in Zu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are Holomorphic function, 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) or an almost complex manifold, ''almost'' complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth manifold, smooth and complex manifolds have very different flavors: compact space, compact complex manifolds are much closer to algebraic variety, algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be Embedding, embedded as a smooth subma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and topological groups. More precisely, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), althou ... [...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] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space (where (x, g) is an element of P and h is the group element from G; the group action is conventionally a right action). # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unless it is the product space X \times G, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x \mapsto (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
