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Constant Curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature. Classification The Riemannian manifolds of constant curvature can be classified into the following three cases: * elliptic geometry – constant positive sectional curvature * Euclidean geometry – constant vanishing sectional curvature * hyperbolic geometry – constant negative sectional curvature. Properties * Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel \nabla \mathrm=0. * Every space of constant curvature is locally maximally symmetric, i.e. it has \frac n (n+1) number of local isometries, where ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing–Hopf Theorem
In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space form Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...s. The Killing–Hopf theorem was proved by and . References * * {{DEFAULTSORT:Killing-Hopf theorem Riemannian geometry Theorems in Riemannian geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry Of Surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: ''extrinsically'', relating to their embedding in Euclidean space and ''intrinsically'', reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The 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 n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frederick S
Frederick may refer to: People * Frederick (given name), the name Nobility Anhalt-Harzgerode * Frederick, Prince of Anhalt-Harzgerode (1613–1670) Austria * Frederick I, Duke of Austria (Babenberg), Duke of Austria from 1195 to 1198 * Frederick II, Duke of Austria (1219–1246), last Duke of Austria from the Babenberg dynasty * Frederick the Fair (Frederick I of Austria (Habsburg), 1286–1330), Duke of Austria and King of the Romans Baden * Frederick I, Grand Duke of Baden (1826–1907), Grand Duke of Baden * Frederick II, Grand Duke of Baden (1857–1928), Grand Duke of Baden Bohemia * Frederick, Duke of Bohemia (died 1189), Duke of Olomouc and Bohemia Britain * Frederick, Prince of Wales (1707–1751), eldest son of King George II of Great Britain Brandenburg/Prussia * Frederick I, Elector of Brandenburg (1371–1440), also known as Frederick VI, Burgrave of Nuremberg * Frederick II, Elector of Brandenburg (1413–1470), Margrave of Brandenburg * Frederick William, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moritz Epple
Moritz Epple (born 7 May 1960, in Stuttgart) is a German mathematician and historian of science. Biography Epple studied mathematics, philosophy and physics in Copenhagen, London, and at the University of Tübingen, where he received in 1987 his bachelor's degree (''Diplom'') in physics and in 1991 his Ph.D. (''Promotion)'' in mathematical physics. He then became an assistant in the history of mathematics and natural sciences at the University of Mainz, where he received in 1998 his '' Habilitation''. From 2001 to 2003 he was the head of the department of history of the natural sciences and technology at the University of Stuttgart. Since 2003 he has been a professor at the Goethe University of Frankfurt and head of the working group for the modern history of science at the historic seminary there. He was a visiting professor at several academic institutions including the Dibner Institute for the History of Science and Technology of Massachusetts Institute of Technology (MIT) an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis. In relativistic physics, the ''v'' represents the time or virtual dimension, and the ''p'' for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers implying ''r''= 0, or as an explicit list of signs of eigenvalues such as or for the signatures and , respectively. The signature is said to be indefinite or mixed if both ''v'' and ''p'' are nonzero, and degenerate if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Form
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the ''Timaeus'' of Plato, or Socrates in his reflections on what the Greeks called ''khôra'' (i.e. "space"), or in the ''Physics'' of Aristotle (Book IV, Delta) in the definition of ''topos'' (i.e. place), or in the later "geometrical conception of place" as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesically Complete
In mathematics, a complete manifold (or geodesically complete manifold) is a (pseudo-) Riemannian manifold for which, starting at any point , you can follow a "straight" line indefinitely along any direction. More formally, the exponential map at point , is defined on , the entire tangent space at . Equivalently, consider a maximal geodesic \ell\colon I\to M. Here I is an open interval of \mathbb, and, because geodesics are parameterized with "constant speed", it is uniquely defined up to transversality. Because I is maximal, \ell maps the ends of I to points of , and the length of I measures the distance between those points. A manifold is geodesically complete if for any such geodesic \ell, we have that I=(-\infty,\infty). Examples and non-examples Euclidean space \mathbb^n, the spheres \mathbb^n, and the tori \mathbb^n (with their natural Riemannian metrics) are all complete manifolds. All compact Riemannian manifolds and all homogeneous manifolds are geodesically com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of \mathbb R^n with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperbolic plane which are the other symmetric spaces of negative curvature. Hyperbolic space serves as the prototype of a Gromov hyperbolic space which is a far-reachin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Manifold
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by . Examples The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into \mathbb^3). Dimension 1 Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either \mathbb or S^1, it is straightforward to see that every connected one-dimensional Riemannian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
