|
Bang-Yen Chen
Chen Bang-yen is a Taiwanese mathematician who works mainly on differential geometry and related subjects. He was a University Distinguished Professor of Michigan State University from 1990 to 2012. After 2012 he became University Distinguished professor emeritus. Biography Chen Bang-yen (陳邦彦) is a Taiwanese-American mathematician. He received his B.S. from Tamkang University in 1965 and his M.Sc. from National Tsing Hua University in 1967. He obtained his Ph.D. degree from University of Notre Dame in 1970 under the supervision of Tadashi Nagano. Chen Bang-yen taught at Tamkang University between 1966 and 1968, and at the National Tsing Hua University in the academic year 1967–1968. After his doctoral years (1968-1970) at University of Notre Dame, he joined the faculty at Michigan State University as a research associate from 1970 to 1972, where he became associate professor in 1972, and full professor in 1976. He was presented with the title of University Distingu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toucheng, Yilan
Toucheng Township () is an urban township in Yilan County, Taiwan. The township includes Guishan Island and Guiluan Island in the Philippine Sea. The Senkaku Islands, known in Mandarin as the Diaoyu Islands, are claimed as part of the township. History Toucheng was formerly called ''Thau-ui'' (). Toucheng Township () was established on 9 September 1946. Toucheng Township was upgraded to an urban township () on 1 January 1948. Geography * Area: 100.89 km2 * Population: 29,890 people (2014) Administrative divisions Toucheng includes twenty-five urban villages: *Shicheng/Shihcheng (Shih-ch'eng-tzu, Sekijōshi; ), Dali (Ta-li-chien, Dairikan; , 大里簡), Guishan (Kuei-shan, Kīzan; ), Daxi (Ta-ch'i, Taikei; ), Gexing (Ho-hsing, Gōkō; ), Gengxin (), Waiao (Wai-ao, Gaiō; ), Gangkou (Chiang-k'ou, Kōkō; ), Wuying (), Dakeng (), Chengtung (), Chengbei (), Chengxi (), Chengnan (), Zhuan (), Xinjian (), Baya (), Fucheng (), Jinmian (Hsiao-chin-mien, Shō-kimmen; , 小金 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of t ... [...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 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 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, 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'' is the Riemann curvature tensor, defined here by the convention R ... [...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] |
Isothermal Coordinates
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form : g = \varphi (dx_1^2 + \cdots + dx_n^2), where \varphi is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.) Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally flat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simons' Formula
In the mathematical field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental equation in the study of minimal submanifolds. It was discovered by James Simons in 1968. It can be viewed as a formula for the Laplacian of the second fundamental form of a Riemannian submanifold A Riemannian submanifold ''N'' of a Riemannian manifold ''M'' is a submanifold of ''M'' equipped with the Riemannian metric inherited from ''M''. The image of an isometric immersion In mathematics, an embedding (or imbedding) is one instance of .... It is often quoted and used in the less precise form of a formula or inequality for the Laplacian of the length of the second fundamental form. In the case of a hypersurface of Euclidean space, the formula asserts that :\Delta h=\operatornameH+Hh^2-, h, ^2h, where, relative to a local choice of unit normal vector field, is the second fundamental form, is the mean c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shoshichi Kobayashi
was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras. Biography Kobayashi graduated from the University of Tokyo in 1953. In 1956, he earned a Ph.D. from the University of Washington under Carl B. Allendoerfer. His dissertation was ''Theory of Connections''. He then spent two years at the Institute for Advanced Study and two years at MIT. He joined the faculty of the University of California, Berkeley in 1962 as an assistant professor, was awarded tenure the following year, and was promoted to full professor in 1966. Kobayashi served as chairman of the Berkeley Mathematics Department for a three-year term from 1978 to 1981 and for the 1992 Fall semester. He chose early retirement under the VERIP plan in 1994. The two-volume book ''Foundations of Differential Geometry'', which he coau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manfredo Do Carmo
Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil. Education and career Do Carmo studied civil engineering at the University of Recife from 1947 to 1951. After working a few years as engineer, he accepted a teaching position at the newly created Institute of Physics and Mathematics at Recife. On suggestion of Elon Lima, in 1959 he went to Instituto Nacional de Matemática Pura e Aplicada to improve his background and in 1960 he moved to the USA to pursue a Ph.D. in mathematics at the University of California, Berkeley under the supervision of Shiing-Shen Chern. He defended his thesis, entitled "''The Cohomology Ring of Certain Kahlerian Manifolds''", in 1963. After working again at University of Recife and at the University of Brasilia, in 1966 he became professor at Instituto Nacional de Matemática Pura e Aplica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
