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Berger Sphere
In the mathematical field of Riemannian geometry, the Berger spheres form a special class of examples of Riemannian manifolds diffeomorphic to the 3-sphere. They are named for Marcel Berger who introduced them in 1962. Geometry of the Berger spheres The Lie group is diffeomorphic to the 3-sphere. Its Lie algebra is a three-dimensional real vector space spanned by : u_1 = \begin 0 & i \\ i & 0 \end, \quad u_2 = \begin 0 & -1 \\ 1 & 0 \end, \quad u_3 = \begin i & 0 \\ 0 & -i \end~, which are complex multiples of the Pauli matrices. It is direct to check that the commutators are given by and and . Any positive-definite inner product on the Lie algebra determines a left-invariant Riemannian metric on the Lie group. A Berger sphere is a metric so obtained by making the inner product on the Lie algebra have matrix :\begint&0&0\\ 0&1&0\\ 0&0&1\end relative to the basis . Here is a positive number to be freely chosen; each choice produces a different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smoothly from point to point). This gives, in particular, local notions of angle, arc length, length of curves, surface area and volume. From those, some other global quantities can be derived by integral, integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in Three-dimensional space, R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini–Study Metric
In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study. A Hermitian form in (the vector space) C''n''+1 defines a unitary subgroup U(''n''+1) in GL(''n''+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CP''n'' is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2''n''+1)-sphere. In algebraic geometry, one uses a normalization making CP''n'' a Hodge manifold. Construction The Fubini–Study metric arises naturally in the quotient space construction of complex projective space. Specifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Katz
Mikhail "Mischa" Gershevich Katz (, ; born 1958)Curriculum vitae retrieved 2011-05-23. is an Israeli and professor of mathematics at . His main interests are , , |
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by '' Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chelsea Publishing Company
The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ..., founded in 1944 by Aaron Galuten while he was still a graduate student at Columbia. Its initial focus was to republish important European works that were unavailable in the United States because of wartime restrictions, such as Hausdorff's Mengenlehre, or because the works were out of print. This soon expanded to include translations of such works into English, as well as original works by American authors. As of 1985, the company's catalog included more than 200 titles. After Galuten's death in 1994, the company was acquired in 1997 by the AMS, which continues to publish a portion of the company's original catalog under the AMS Chelse ... [...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 internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Submersion
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let (''M'', ''g'') and (''N'', ''h'') be two Riemannian manifolds and f:M\to N a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution \mathrm(df)^ is a sub-bundle of the tangent bundle of TM which depends both on the projection f and on the metric g. The expression \mathrm(df)^ denotes the subbundle of TM that is the orthogonal complement of \mathrm(df_x) \sub T_M at each point x of ''M''. Then, ''f'' is called a Riemannian submersion if and only if, for all x\in M, the vector space isomorphism (df)_x : \mathrm(df_x)^ \rightarrow T_N is an isometry, or in other words it carries each vector to one of the same length. Examples An example of a Riemannian submersion arises when a Lie group G acts isometrically, freely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space. Definition Let X be a topological space, and let \sim be an equivalence relation on X. The quotient set Y = X/ is the set of equivalence classes of elements of X. The e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a pseudo-Riemannian manifold that preserves the metric tensor. Killing vector fields are the infinitesimal generators of isometries; that is, flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the ''Killing vector'' will not distort distances on the object. Definition Specifically, a vector field X is a Killing vector field if the Lie derivative with respect to X of the metric tensor g vanishes: : \mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is : g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 for all vectors Y and . In local coordinates, this amounts to the Killing equation : \nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,. This condition is expressed in covarian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collapsing Manifold
In Riemannian geometry, a collapsing or collapsed manifold is an ''n''-dimensional manifold ''M'' that admits a sequence of Riemannian metrics ''g''''i'', such that as ''i'' goes to infinity the manifold is close to a ''k''-dimensional space, where ''k'' < ''n'', in the Gromov–Hausdorff distance sense. Generally there are some restrictions on the s of (''M'', ''g''''i''). The simplest example is a flat manifold, whose metric can be rescaled by 1/''i'', so that the manifold is close to a point, but its curvature remains 0 for all ''i''. Examples Generally speaking ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
