# Bang-Yen Chen

Chen Bang-yen (traditional Chinese: 陳邦彦; born October 3, 1943) 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.

## Biography

Chen Bang-yen (陳邦彦) was born in Toucheng, Taiwan. 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 the University of Notre Dame in 1970 under the supervision of Tadashi Nagano.
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.
Chen Bang-yen is the author of over 580 works including 12 books, mainly in differential geometry and related subjects. He also co-edited four books, three of them were published by Springer and one of them by American Mathematical Society. His works have been cited over 37,000 times. In 2008, Bang-Yen Chen was presented with the first Geometry Prize from the Simon Stevin Institute for Geometry, Netherlands, for his seminal contributions to differential geometry. Chen was named as one of the top 15 famous Taiwanese scientists by SCI Journal in 2022.
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. 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. The volume is edited by Joeri Van der Veken, Alfonso Carriazo, Ivko Dimitrić, Yun Myung Oh, Bogdan Suceavă, and Luc Vrancken.

## Research contributions

In 1993, Bang-Yen 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.
In symmetric spaces, Bang-Yen Chen and Tadashi Nagano created the (M+,M-)-theory (also known as Chen-Nagano theory) for compact symmetric spaces with many nice applications. One of advantages of their theory is that it is very useful for applying inductive arguments on polars or meridians. In particularly, Chen and Nagano initiated the study of maximal antipodal set and 2-number (also known as Chen-Nagano invariant or Chen-Nagano number); as an application Chen and Nagano were able to completely determine 2-rank of all compact simple Lie groups and thus they settled a problem in group theory raised by Armand Borel and Jean-Pierre Serre.
In Riemannian geometry, Bang-Yen Chen invented the δ-invariants (also known as 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, allowing for more nuanced analysis of submanifolds. 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. Also in Riemannian geometry, Bang-Yen Chen and Kentaro Yano initiated the study of spaces of quasi-constant curvature.
In differential geometry, Bang-Yen Chen created the theory of slant 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 differential geometry, Bang-Yen Chen also initialed 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. As a by-product, Chen proposed his longstanding biharmonic conjecture in 1991 stating that any biharmonic submanifold in a Euclidean space must be a minimal submanifold.
Given a 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, Bang-Yen 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. 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.
Within the context of almost Hermitian manifolds, Bang-Yen Chen also initiated the study of CR-warped product submanifolds, providing a new method to investigate CR-submanifolds and their extensions by utilizing the concept of warped product.
In general relativity and gravitational theory, Bang-Yen Chen established a very simple and useful characterization of generalized Robertson-Walker spacetimes; namely, a Lorentzian manifold is a generalized Robertson-Walker spacetime if and only if it admits a timelike concircular vector field.

## Publications

Major articles
* Chen Bang-yen and Koichi Ogiue. On totally real submanifolds. Trans. Amer. Math. Soc. 193 (1974), 257–266. doi:10.1090/S0002-9947-1974-0346708-7
* Chen Bang-yen. Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60 (1993), no. 6, 568–578. doi:10.1007/BF01236084
Surveys
* Chen Bang-yen. Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), no. 2, 169–188.
* Chen Bang-yen. A report on submanifolds of finite type. Soochow J. Math. 22 (1996), no. 2, 117–337.
* Chen Bang-yen. Riemannian submanifolds. Handbook of Differential Geometry, Vol. I (2000), 187–418. North-Holland, Amsterdam. doi:10.1016/S1874-5741(00)80006-0 ; arXiv:1307.1875
Books
* Chen Bang-yen. Geometry of submanifolds. Pure and Applied Mathematics, No. 22. Marcel Dekker, Inc., New York, 1973. vii+298 pp.
* Chen Bang-yen. Geometry of submanifolds and its applications. Science University of Tokyo, Tokyo, 1981. iii+96 pp.
* Chen Bang-yen. Finite type submanifolds and generalizations. Università degli Studi di Roma "La Sapienza", Istituto Matematico "Guido Castelnuovo", Rome, 1985. iv+68 pp.
* Chen Bang-yen. A new approach to compact symmetric spaces and applications. A report on joint work with Professor T. Nagano. Katholieke Universiteit Leuven, Louvain, 1987. 83 pp.
* Chen Bang-yen. Geometry of slant submanifolds. Katholieke Universiteit Leuven, Louvain, 1990. 123 pp. arXiv:1307.1512
* Chen Bang-yen and Leopold Verstraelen. Laplace transformations of submanifolds. Centre for Pure and Applied Differential Geometry (PADGE), 1. Katholieke Universiteit Brussel, Group of Exact Sciences, Brussels; Katholieke Universiteit Leuven, Department of Mathematics, Leuven, 1995. x+126 pp.
* Chen Bang-yen. Pseudo-Riemannian geometry, δ-invariants and applications. With a foreword by Leopold Verstraelen. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. xxxii+477 pp. ISBN 978-981-4329-63-7, 981-4329-63-0. doi:10.1142/8003
* Chen Bang-yen. Total mean curvature and submanifolds of finite type. Second edition of the 1984 original. With a foreword by Leopold Verstraelen. Series in Pure Mathematics, 27. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. xviii+467 pp. ISBN 978-981-4616-69-0, 978-981-4616-68-3. doi:10.1142/9237
* Chen Bang-yen. Differential geometry of warped product manifolds and submanifolds. With a foreword by Leopold Verstraelen. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. xxx+486 pp. ISBN 978-981-3208-92-6
* Ye-Lin Ou and Chen Bang-yen. Biharmonic submanifolds and biharmonic maps in Riemannian geometry. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2020. xii+528 pp. ISBN 978-981-121-237-6