Chen Bang-yen | |
---|---|
陳邦彦 | |
Born | Toucheng, Yilan, Taiwan | October 3, 1943
Nationality | Taiwanese, American |
Alma mater | Tamkang University (BS) National Tsing Hua University (MS) University of Notre Dame (PhD) |
Known for | "Chen inequalities", "Chen invariants (or δ-invariants)", "Chen's conjectures", "Chen surface", "Chen–Ricci inequality", "Chen submanifold", "Chen equality", "Submanifolds of finite type", "Slant submanifolds", "ideal immersion", "(M+,M-)-theory for compact symmetric spaces & 2-numbers of Riemannian manifolds (joint with Tadashi Nagano)". |
Scientific career | |
Fields | Differential geometry, Riemannian Geometry, Geometry and topology |
Institutions | Michigan State University |
Thesis | On the G-total curvature and topology of immersed manifolds (1970) |
Doctoral advisor | Tadashi Nagano |
Doctoral students | Bogdan Suceavă |
Website | www |
Chen Bang-yen is a Taiwanese-American 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.
Chen Bang-yen (陳邦彦) is a Taiwanese-American mathematician. He received his B.S. from Tamkang University in 1965 and his M.S. 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. [1] [2]
Chen Bang-yen taught at Tamkang University between 1965 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 Distinguished Professor in 1990. After 2012 he became University Distinguished Professor Emeritus. [3] [4] [5]
Chen Bang-yen is the author of over 570 works including 12 books, mainly in differential geometry and related subjects. He also co-edited four books, three of them were published by Springer Nature and one of them by American Mathematical Society. [6] [7] His works have been cited over 37,000 times. [8] He was named as one of the top 15 famous Taiwanese scientists by SCI Journal. [9]
On October 20–21, 2018, at the 1143rd Meeting of the American Mathematical Society held at Ann Arbor, Michigan, one of the Special Sessions was dedicated to Chen Bang-yen's 75th birthday. [10] [11] The volume 756 in the Contemporary Mathematics series, published by the American Mathematical Society, is dedicated to Chen Bang-yen, and it includes many contributions presented in the Ann Arbor event. [12] The volume is edited by Joeri Van der Veken, Alfonso Carriazo, Ivko Dimitrić, Yun Myung Oh, Bogdan Suceavă, and Luc Vrancken.
Given an almost Hermitian manifold, a totally real submanifold is one for which the tangent space is orthogonal to its image under the almost complex structure. From the algebraic structure of the Gauss equation and the Simons formula, Chen and Koichi Ogiue derived a number of information on submanifolds of complex space forms which are totally real and minimal. By using Shiing-Shen Chern, Manfredo do Carmo, and Shoshichi Kobayashi's estimate of the algebraic terms in the Simons formula, Chen and Ogiue showed that closed submanifolds which are totally real and minimal must be totally geodesic if the second fundamental form is sufficiently small. [13] By using the Codazzi equation and isothermal coordinates, they also obtained rigidity results on two-dimensional closed submanifolds of complex space forms which are totally real.
In 1993, Chen studied submanifolds of space forms, showing that the intrinsic sectional curvature at any point is bounded below in terms of the intrinsic scalar curvature, the length of the mean curvature vector, and the curvature of the space form. In particular, as a consequence of the Gauss equation, given a minimal submanifold of Euclidean space, every sectional curvature at a point is greater than or equal to one-half of the scalar curvature at that point. Interestingly, the submanifolds for which the inequality is an equality can be characterized as certain products of minimal surfaces of low dimension with Euclidean spaces.
Chen introduced and systematically studied the notion of a finite type submanifold of Euclidean space, which is a submanifold for which the position vector is a finite linear combination of eigenfunctions of the Laplace-Beltrami operator. He also introduced and studied a generalization of the class of totally real submanifolds and of complex submanifolds; a slant submanifold of an almost Hermitian manifold is a submanifold for which there is a number k such that the image under the almost complex structure of an arbitrary submanifold tangent vector has an angle of k with the submanifold's tangent space.
In Riemannian geometry, Chen and Kentaro Yano initiated the study of spaces of quasi-constant curvature. Chen also introduced the δ-invariants (also called Chen invariants), which are certain kinds of partial traces of the sectional curvature; they can be viewed as an interpolation between sectional curvature and scalar curvature. Due to the Gauss equation, the δ-invariants of a Riemannian submanifold can be controlled by the length of the mean curvature vector and the size of the sectional curvature of the ambient manifold. Submanifolds of space forms which satisfy the equality case of this inequality are known as ideal immersions; such submanifolds are critical points of a certain restriction of the Willmore energy.
In the theory of symmetric spaces, Chen and Tadashi Nagano created the (M+,M-)-theory (also known as Chen-Nagano theory) for compact symmetric spaces with several nice applications. [14] [15] [16] One of advantages of their theory is that it is very useful for applying inductive arguments on polars or meridians. [17] [18]
Major articles
Surveys
Books
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Shing-Tung Yau 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.
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.
In mathematics, the Chern theorem states that the Euler–Poincaré characteristic of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial of its curvature form.
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
Mikhael Leonidovich Gromov is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University.
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case. Jeff Cheeger and Detlef Gromoll proved the theorem in 1972 by generalizing a 1969 result of Gromoll and Wolfgang Meyer. The related soul conjecture, formulated by Cheeger and Gromoll at that time, was proved twenty years later by Grigori Perelman.
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.
Jeff Cheeger is an American mathematician and Silver Professor at the Courant Institute of Mathematical Sciences of New York University. His main interest is differential geometry and its connections with topology and analysis.
The Geometry Festival is an annual mathematics conference held in the United States.
In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
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.
Kentaro Yano was a mathematician working on differential geometry who introduced the Bochner–Yano theorem.
In mathematics, the Lichnerowicz conjecture is a generalization of a conjecture introduced by Lichnerowicz. Lichnerowicz's original conjecture was that locally harmonic 4-manifolds are locally symmetric, and was proved by Walker (1949). The Lichnerowicz conjecture usually refers to the generalization that locally harmonic manifolds are flat or rank-1 locally symmetric. It has been proven true for compact manifolds with fundamental groups that are finite groups but counterexamples exist in seven or more dimensions in the non-compact case
Gerhard Huisken is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.
Bogdan Suceavă is a Romanian-American mathematician and writer, working since 2002 as professor of mathematics at California State University Fullerton. He is also a honorary research professor with the STAR-UBB Institute, Babeș-Bolyai University, Cluj-Napoca, Romania.
In the mathematical field of differential geometry, a biharmonic map is a map between Riemannian or pseudo-Riemannian manifolds which satisfies a certain fourth-order partial differential equation. A biharmonic submanifold refers to an embedding or immersion into a Riemannian or pseudo-Riemannian manifold which is a biharmonic map when the domain is equipped with its induced metric. The problem of understanding biharmonic maps was posed by James Eells and Luc Lemaire in 1983. The study of harmonic maps, of which the study of biharmonic maps is an outgrowth, had been an active field of study for the previous twenty years. A simple case of biharmonic maps is given by biharmonic functions.
David Allen Hoffman is an American mathematician whose research concerns differential geometry. He is an adjunct professor at Stanford University. In 1985, together with William Meeks, he proved that Costa's surface was embedded. He is a fellow of the American Mathematical Society since 2018, for "contributions to differential geometry, particularly minimal surface theory, and for pioneering the use of computer graphics as an aid to research." He was awarded the Chauvenet Prize in 1990 for his expository article "The Computer-Aided Discovery of New Embedded Minimal Surfaces". He obtained his Ph.D. from Stanford University in 1971 under the supervision of Robert Osserman.