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Myers 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] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smoothly from point to point). This gives, in particular, local notions of angle, arc length, length of curves, surface area and volume. From those, some other global quantities can be derived by integral, integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in Three-dimensional space, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumner Byron Myers
Sumner Byron Myers (February 19, 1910 – October 8, 1955) was an American mathematician specializing in topology and differential geometry. He studied at Harvard University under H. C. Marston Morse, Tucker, A: Interview with Albert Tucker'', Princeton University, July 11, 1984. Last accessed January 1, 2010. where he graduated with a Ph.D. in 1932.Mathematics Genealogy Project: Sumner Byron Myers', no date. Last accessed December 5, 2005. Myers then pursued postdoctoral studies at Princeton University (1934–1936)Princeton University: Members of the School of Mathematics'', no date. Last accessed December 5, 2005. before becoming a professor for mathematics at the University of Michigan. He died unexpectedly from a heart attack during the 1955 Michigan–Army football game at Michigan Stadium. Sumner B. Myers Prize The Sumner B. Myers Prize was created in his honor for distinguished theses within the LSA Mathematics Department.University of Michigan: Sumner Myers Award', no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Ossian Bonnet
Pierre Ossian Bonnet (; 22 December 1819, Montpellier – 22 June 1892, Paris) was a French mathematician. He made some important contributions to the differential geometry of surfaces, including the Gauss–Bonnet theorem. Biography Early years Pierre Bonnet attended the Collège in Montpellier. In 1838 he entered the École Polytechnique in Paris. He also studied at the École Nationale des Ponts et Chaussées. Middle years In graduating he was offered a post as an engineer. After some thought Bonnet decided on a career in teaching and research in mathematics instead. Turning down the engineering post had not been an easy decision since Bonnet was not well off financially. He had to do private tutoring so that he could afford to accept a position at the Ecole Polytechnique in 1844. One year before this, in 1843, Bonnet had written a paper on the convergence of series with positive terms. Another paper on series in 1849 was to earn him an award from the Brussels Academy. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectional Curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface (topology), surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map (Riemannian geometry), exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, Riemann curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter Of A Set
In mathematics, the diameter of a set of points in a metric space is the largest distance between points in the set. As an important special case, the diameter of a metric space is the largest distance between any two points in the space. This generalizes the diameter of a circle, the largest distance between two points on the circle. This usage of diameter also occurs in medical terminology concerning a lesion or in geology concerning a rock. A bounded set is a set whose diameter is finite. Within a bounded set, all distances are at most the diameter. Formal definition The diameter of an object is the least upper bound (denoted "sup") of the set of all distances between pairs of points in the object. Explicitly, if S is a set of points with distances measured by a Metric (mathematics), metric \rho, the diameter is \operatorname(S) = \sup_ \rho(x, y). Of the empty set The diameter of the empty set is a matter of convention. It can be defined to be zero, -\infty, or undefined. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Manifold
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to Lorentzian manifolds (including the four-dimensional Lorentzian manifolds usually studied in general relativity). Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons. If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that :\mathrm = kg for some constant k, where \operatorname denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds. The Einstein condition and Einstein's equation In loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from University of California at Berkeley. Cheng then spent some years as a post-doctoral fellow and assistant professor at Princeton University and the State University of New York at Stony Brook. Then he became a full professor at University of California at Los Angeles. Cheng chaired the Mathematics departments of both the Chinese University of Hong Kong and the Hong Kong University of Science and Technology in the 1990s. In 2004, he became the Dean of Science at HKUST. In 2012, he became a fellow of the American Mathematical Society. He is well known for contributions to differential geometry and partial differential equations, including Cheng's eigenvalue comparison theorem, Cheng's maximal diameter theorem, and a number of works with Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift
''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It was initially edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Because Lichtenstein was Jewish, he was forced to step down as editor in 1933 under the Nazi rule of Germany; he fled to Poland and died soon after. The editorship was offered to Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ..., but he refused, Translated by Bärbel Deninger from the 1982 German original. and Konrad Knopp took it over. Other past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Hel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Inequalities
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. 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. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries. During t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
