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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 bili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and 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 manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...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 exterio ... [...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 1939 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cedric Villani
Cedric () is a masculine given name invented by Walter Scott in the 1819 novel ''Ivanhoe''.Sir Walter Scott, Graham Tulloch (ed.), ''Ivanhoe'', vol. 8 of The Edinburgh Edition of the Waverley Novels, Edinburgh University Press, 1998, , "explanatory notes", p. 511. The invented name is based on ''Cerdic'', the name of a 6th-century Anglo-Saxon king (itself from Brittonic ''Coroticus''). 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 ... [...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 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 Center 611 "Singular Phenomena in Mathematical Model ... [...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 adds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and 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 can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...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, then ... [...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. The Hopf-Rinow theorem therefore implies that M must be compact, as a closed (and hence compact) ball of radius \pi/\sqrt 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 nonc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Space Form
Space is the boundless Three-dimensional space, three-dimensional extent in which Physical body, objects and events have relative position (geometry), position and direction (geometry), direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional Continuum (theory), continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature, essence and the mode of existence of space date back to antiquity; namely, to treatises like the ''Timaeus (dialogue), Timaeus'' of Plato, or Socrates in his reflections on what the Greeks called ''khôra'' (i.e. "space"), or in the ''Physics (Aristotle), P ... [...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. *Chaplygin inequality *Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations. *Sturm comparison theorem *Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation. * Hille-Wintner comparison theorem Riemannian geometry In Riemannian geometry, it is a traditional name for a number of theorems that compare vario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigory Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. He is widely regarded as one of the greatest living mathematicians. 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 open problem in mathematics for the past century. The full details of Perelman's work were fil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
