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Filling Radius
In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form. The filling radius of a simple loop ''C'' in the plane is defined as the largest radius, ''R'' > 0, of a circle that fits inside ''C'': :\mathrm(C\subset \mathbb^2) = R. Dual definition via neighborhoods There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the \varepsilon-neighborhoods of the loop ''C'', denoted :U_\varepsilon C \subset \mathbb^2. As \varepsilon>0 increases, the \varepsilon-neighborhood U_\varepsilon C swallows up more and more of the interior of the loop. The ''last' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in 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 dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction As with all projective spaces, RP''n'' is formed by taking the quotient of under the equivalence relation for all real numbers . For all ''x'' in one can always find a ''λ'' such that ''λx'' has norm 1. There are precisely two such ''λ'' differing by sign. Thus RP''n'' can also be formed by identifying antipodal points of the unit ''n''-sphere, ''S''''n'', in R''n''+1. One can further restrict to the upper hemisphere of ''S''''n'' and merely identify antipodal points on the bounding equator. This shows that RP''n'' is also equivalent to the closed ''n''-dimensional disk, ''D''''n'', with antipodal points on the boundary, , identified. Low-dimensional examples * RP1 is called the real projective line, which is topologically eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Katz
Mikhail "Mischa" Gershevich Katz (born 1958, in Chișinău)Curriculum vitae retrieved 2011-05-23. is an Israeli , a professor of mathematics at . His main interests are , and |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Filling Area Conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definitions and statement of the conjecture Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to ''along'' . For example, on a closed curve C of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from . A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve . The filling is said isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary. In other words, to fill a curve isometrically is to fill ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injectivity Radius
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provide more detailed expositions of the definitions given below. * Connection * Curvature * Metric space * Riemannian manifold See also: * Glossary of general topology * Glossary of differential geometry and topology * List of differential geometry topics Unless stated otherwise, letters ''X'', ''Y'', ''Z'' below denote metric spaces, ''M'', ''N'' denote Riemannian manifolds, , ''xy'', or , xy, _X denotes the distance between points ''x'' and ''y'' in ''X''. Italic ''word'' denotes a self-reference to this glossary. ''A caveat'': many terms in Riemannian and metric geometry, such as ''convex function'', ''convex set'' and others, do not have exactly the same meaning as in general mathematical usage. __NOTOC__ A Alexandrov space a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Manifold
In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov. Definition A closed manifold ''M'' is called essential if its fundamental class 'M''defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding Eilenberg–MacLane space ''K''(, 1), via the natural homomorphism :H_n(M)\to H_n(K(\pi,1)), where ''n'' is the dimension of ''M''. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise. Examples *All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere ''S2''. *Real projective space ''RPn'' is essential since the inclusion *:\mathbb^n \to \mathbb^\infty :is injective in homology, where ::\mathbb^\infty = K(\mathbb_2, 1) :is the Eilenberg–MacLane space of the finite cyclic group of order 2. *All comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski Embedding
In mathematics, the Kuratowski embedding allows one to view any metric space as a subset of some Banach space. It is named after Kazimierz Kuratowski. The statement obviously holds for the empty space. If (''X'',''d'') is a metric space, ''x''0 is a point in ''X'', and ''Cb''(''X'') denotes the Banach space of all bounded continuous real-valued functions on ''X'' with the supremum norm, then the map :\Phi : X \rarr C_b(X) defined by :\Phi(x)(y) = d(x,y)-d(x_0,y) \quad\mbox\quad x,y\in X is an isometry. The above construction can be seen as embedding a pointed metric space into a Banach space. The Kuratowski–Wojdysławski theorem states that every bounded metric space ''X'' is isometric to a closed subset of a convex subset of some Banach space. (N.B. the image of this embedding is closed in the convex subset, not necessarily in the Banach space.) Here we use the isometry :\Psi : X \rarr C_b(X) defined by :\Psi(x)(y) = d(x,y) \quad\mbox\quad x,y\in X The convex set m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, lengt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Class
In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.In past years mathematics.... Definition Closed, orientable When ''M'' is a connected orientable closed manifold of dimension ''n'', the top homology group is infinite cyclic: H_n(M,\mathbf) \cong \mathbf, and an orientation is a choice of generator, a choice of isomorphism \mathbf \to H_n(M,\mathbf). The generator is called the fundamental class. If ''M'' is disconnected (but still orientable), a fundamental class is the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component). In relation with de Rham cohomology it represents ''integration over M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
