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Pu's Inequality
In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it. Statement A student of Charles Loewner, Pu proved in his 1950 thesis that every Riemannian surface M homeomorphic to the real projective plane satisfies the inequality : \operatorname(M) \geq \frac \operatorname(M)^2 , where \operatorname(M) is the systole of M . The equality is attained precisely when the metric has constant Gaussian curvature. In other words, if all noncontractible loops in M have length at least L , then \operatorname(M) \geq \frac L^2, and the equality holds if and only if M is obtained from a Euclidean sphere of radius r=L/\pi by identifying each point with its antipodal. Pu's paper also stated for the first time Loewner's inequality, a similar result for Riemannian metrics on the torus. Proof Pu's original proof relies on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner%27s Roman Surface , the professional wrestling "tag team" of real-life brothers Rick and Scott Steiner
{{disambiguation, geo ...
Steiner may refer to: Felix Steiner, German Waffen SS-commander Surname *Steiner (surname) Other uses *Steiner, Michigan, a village in the United States * Steiner, Mississippi * Steiner Studios, film and television production studio in New York City * Steiner's theorem, used to determine the mass moment of inertia around an axis. Also known as parallel axis theorem See also * Poncelet–Steiner theorem *Steiner point (other) * Steiner surface *Steiner system, a type of block design *Steiner tree *Waldorf education, also called Steiner education *The Steiner Brothers The Steiner Brothers are an American professional wrestling tag team consisting of brothers Robert "Rick Steiner" Rechsteiner and Scott "Scott Steiner" Rechsteiner. The brothers wrestled as amateurs at the University of Michigan. The team ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Inequalities
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries wi ... [...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 dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific J
The Pacific Ocean is the largest and deepest of Earth's five oceanic divisions. It extends from the Arctic Ocean in the north to the Southern Ocean (or, depending on definition, to Antarctica) in the south, and is bounded by the continents of Asia and Oceania in the west and the Americas in the east. At in area (as defined with a southern Antarctic border), this largest division of the World Ocean—and, in turn, the hydrosphere—covers about 46% of Earth's water surface and about 32% of its total surface area, larger than Earth's entire land area combined .Pacific Ocean . '' Britannica Concise.'' 2008: Encyclopædia Britannica, Inc. The centers of both the [...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 in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systoles Of Surfaces
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The ''systolic area'' of a metric is defined to be the ratio area/sys2. The ''systolic ratio'' SR is the reciprocal quantity sys2/area. See also Introduction to systolic geometry. Torus In 1949 Loewner proved his inequality for metrics on the torus T2, namely that the systolic ratio SR(T2) is bounded above by 2/\sqrt, with equality in the flat (constant curvature) case of the equilateral torus (see hexagonal lattice). Real projective plane A similar result is given by Pu's inequality for the real projective plane from 1952, due to Pao Ming Pu, with an upper bound of ''π''/2 for the systolic ratio SR(RP2), also attained in the constant curvature case. Klein bottl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Inequality For Complex Projective Space
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here \operatorname is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line \mathbb^1 \subset \mathbb^n in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms. Projective planes over division algebras \mathbb In the special case n=2, Gromov's inequality becomes \mathrm_2^2 \leq 2 \mathrm_4(\mathbb^2). This inequality can be thought of as an analog of Pu's inequality for the real projective plane \mathbb^2. In both cases, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Systolic Inequality For Essential Manifolds
In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;see it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane. Technically, let ''M'' be an essential Riemannian manifold of dimension ''n''; denote by sys''π''1(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form : \left(\operatorname_1(M)\right)^n \leq C_n \operatorname(M), where ''C''''n'' is a universal constant only depending on the dimension of ''M''. Essential manifolds A closed manifold is called ''essential'' if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Curve Theorem
In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. ''"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."'' (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetry
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n by its volume \operatorname(S), :\operatorname(S)\geq n \operatorname(S)^ \, \operatorname(B_1)^, where B_1\subset\R^n is a unit sphere. The equality holds only when S is a sphere in \R^n. On a plane, i.e. when n=2, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. ''Isoperimetric'' literally means "having the same perimeter". Specifically in \R ^2, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that : L^2 \ge 4\pi A, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose bo ... [...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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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