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27 relations: American Mathematical Society, Area, Charles Loewner, Conformal map, Differential geometry, Filling area conjecture, Gaussian curvature, Gromov's inequality for complex projective space, Gromov's systolic inequality for essential manifolds, Haar measure, Introduction to systolic geometry, Isoperimetric inequality, Jordan curve theorem, Journal of Differential Geometry, Length, Loewner's torus inequality, Pacific Journal of Mathematics, Pao Ming Pu, Real projective plane, Riemannian circle, Riemannian manifold, Rotation, Sphere, Systoles of surfaces, Systolic geometry, Torus, Uniformization theorem.
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.
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Area
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.
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Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician.
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Conformal map
In mathematics, a conformal map is a function that preserves angles locally.
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Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
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Filling area conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the surfaces that fill a closed curve of given length without introducing shortcuts between its points.
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Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: For example, a sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere.
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Gromov's inequality for complex projective space
In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality \;\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.
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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.
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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.
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Introduction to systolic geometry
Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality.
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Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the surface area of a set and its volume.
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Jordan curve theorem
In topology, a Jordan curve, sometimes called a plane simple closed curve, is a non-self-intersecting continuous loop in the plane.
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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.
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Length
In geometric measurements, length is the most extended dimension of an object.
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Loewner's torus inequality
In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner.
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Pacific Journal of Mathematics
The Pacific Journal of Mathematics (ISSN 0030-8730) is a mathematics research journal supported by a number of American, Asian and Australian universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation.
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Pao Ming Pu
Pao Ming Pu (the form of his name he used in Western languages, although the Wade-Giles transliteration would be Pu Baoming;; August 1910 – February 22, 1988), was a mathematician born in Jintang County, Sichuan, China.
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Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.
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Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle equipped with its great-circle distance.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Rotation
A rotation is a circular movement of an object around a center (or point) of rotation.
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Sphere
A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").
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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).
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Systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.
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Redirects here:
Pu's inequality for real projective plane, Pu's inequality for the real projective plane, Pu's theorem.
References
[1] https://en.wikipedia.org/wiki/Pu's_inequality
