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Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace–Beltrami Operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami. For any twice-differentiable real-valued function ''f'' defined on Euclidean space R''n'', the Laplace operator (also known as the ''Laplacian'') takes ''f'' to the divergence of its gradient vector field, which is the sum of the ''n'' pure second derivatives of ''f'' with respect to each vector of an orthonormal basis for R''n''. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wasserstein Metric
In mathematics, the Wasserstein distance or Kantorovich– Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on ''M'', the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781. Because of this analogy, the metric is known in computer science as the earth mover's distance. The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after learning of it in the work of Leonid Vaseršteĭn on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by Leonid Kantorovich in ''The Mathematical Method of Production Planning and Organization'' (Russian original ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cedric Villani
Cedric () is a given name invented by Walter Scott in the 1819 novel ''Ivanhoe''. Etymology The invented name is based on '' Cerdic'', the name of a 6th-century Anglo-Saxon king (itself from Brittonic '' Coroticus''). Popularity The name was not popularly used until the children's book ''Little Lord Fauntleroy'' by Frances Hodgson Burnett was published in 1885 to 1886, the protagonist of which is called Cedric Errol. The book was highly successful, causing a fashion trend in children's formal dress in America and popularized the given name. People named Cedric born in the years following the novel's publication include British naval officer Cedric Holland (1889–1950), American war pilot Cedric Fauntleroy (1891–1973), Irish art director Austin Cedric Gibbons (1893–1960) and British actor Cedric Hardwicke (1893–1964). The name has ranked among the top 1,000 names for boys in the United States at different points since 1903. It ranked 958th on the popularity chart in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl-Theodor Sturm
Karl-Theodor "Theo" Sturm (born 7.November 1960) is a German mathematician working in stochastic analysis. Life and work After obtaining his Abitur from the Platen-Gymnasium Ansbach in 1980, Sturm began to study Mathematics and Physics at the University of Erlangen-Nuremberg where he graduated in 1986 with the Diploma in Mathematics and the State Examination in Mathematics and Physics. In 1989, he obtained his PhD (with a thesis on „Perturbation of Hunt processes by signed additive functionals“) under the supervision of Heinz Bauer and in 1993 he received his habilitation. Visiting and research positions led him to the universities of Stanford, Zurich, and Bonn as well as to the Max Planck Institute for Mathematics in the Sciences, MPI Leipzig. In 1994, he was awarded a Heisenberg fellowship of the DFG. Since 1997, he is professor of mathematics at the University of Bonn. From 2002 to 2012, he was vice spokesman and member of the executive board of the Collaborative Research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Lott (mathematician)
John William Lott (born January 12, 1959) is a professor of Mathematics at the University of California, Berkeley. He is known for contributions to differential geometry. Academic history Lott received his B.S. from the Massachusetts Institute of Technology in 1978 and M.A. degrees in mathematics and physics from University of California, Berkeley. In 1983, he received a Ph.D. in mathematics under the supervision of Isadore Singer. After postdoctoral positions at Harvard University and the Institut des Hautes Études Scientifiques, he joined the faculty at the University of Michigan. In 2009, he moved to University of California, Berkeley. Among his awards and honors: * Sloan Research Fellowship (1989-1991) * Alexander von Humboldt Fellowship (1991-1992) * U.S. National Academy of Sciences Award for Scientific Reviewing (with Bruce Kleiner) Mathematical contributions A 1985 article of Dominique Bakry and Michel Émery introduced a generalized Ricci curvature, in which one ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar Graustein Professor of Mathematics at Harvard, at which point he moved to Tsinghua. Yau was born in Shantou in 1949, moved to British Hong Kong at a young age, and then moved to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work are also seen in the mathematical and physical fields of convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bochner's Formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner. Formal statement If u \colon M \rightarrow \mathbb is a smooth function, then : \tfrac12 \Delta, \nabla u, ^2 = g(\nabla\Delta u,\nabla u) + , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) , where \nabla u is the gradient of u with respect to g, \nabla^2 u is the Hessian of u with respect to g and \mbox is the Ricci curvature tensor.. If u is harmonic (i.e., \Delta u = 0 , where \Delta=\Delta_g is the Laplacian with respect to the metric g ), Bochner's formula becomes : \tfrac12 \Delta, \nabla u, ^2 = , \nabla^2 u, ^2 + \mbox(\nabla u, \nabla u) . Bochner used this formula to prove the Bochner vanishing theorem. As a corollary, if (M, g) is a Riemannian manifold without boundary and u \colon M \rightarrow \mathbb is a smooth, compactly supported function, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myers's Theorem
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following: In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion. Corollaries The conclusion of the theorem says, in particular, that the diameter of (M, g) is finite. Therefore M must be compact, as a closed (and hence compact) ball of finite radius in any tangent space is carried onto all of M by the exponential map. As a very particular case, this shows that any complete and noncompact smooth Einstein manifold must h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Form
In mathematics, a space form is a complete Riemannian manifold ''M'' of constant sectional curvature ''K''. The three most fundamental examples are Euclidean ''n''-space, the ''n''-dimensional sphere, and hyperbolic space, although a space form need not be simply connected. Reduction to generalized crystallography The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an ''n''-dimensional space form M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean ''n''-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^. By rescaling the Riemannian metric on H^n, we may create a space M_K of constant curvature K for any K 0. Thus the universal cover of a space form M with constant curvature K is isometric to M_K. This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Theorem
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry. Differential equations In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. Differential (or integral) inequalities, derived from differential (respectively, integral) equations by replacing the equality sign with an inequality sign, form a broad class of such auxiliary relations. One instance of such theorem was used by Aronson and Weinberger to characterize solutions of Fisher's equation, a reaction-diffusion equation. Other examples of comparison theorems include: * Chaplygin's theorem * Grönwall's inequality, and its various generalizations, prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigori Perelman
Grigori Yakovlevich Perelman (, ; born 13June 1966) is a Russian mathematician and geometer who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman resigned from his research post in Steklov Institute of Mathematics and in 2006 stated that he had quit professional mathematics, owing to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg and has declined requests for interviews since 2006. In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
