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Gromov's Inequality (other)
The following pages deal with inequalities due to Mikhail Gromov: * Bishop–Gromov inequality * Gromov's inequality for complex projective space * 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 1 ... * Lévy–Gromov inequality {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Gromov (mathematician)
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; russian: link=no, Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Biography Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His Russian father Leonid Gromov and his Jewish mother Lea Rabinovitz were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years old, his mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bishop–Gromov Inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. Statement Let M be a complete ''n''-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound : \mathrm \geq (n-1) K for a constant K\in \R. Let M_K^n be the complete ''n''-dimensional simply connected space of constant sectional curvature K (and hence of constant Ricci curvature (n-1)K); thus M_K^n is the ''n''-sphere of radius 1/\sqrt if K>0, or ''n''-dimensional Euclidean space if K=0, or an appropriately rescaled version of ''n''-dimensional hyperbolic space if K<0. Denote by the ball of radius ''r'' around a point ''p'', defined with respect to the [...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] |