Zhongmin Shen

Research

My primary research interests are in Riemannian Geometry and Finsler Geometry. Roughly speaking, Riemannian metrics are quadratic metrics while Finsler metrics are metrics without quadratic restriction. In Finsler Geometry, there are several geometric quantities. The Riemann curvature and its mean (the Ricci curvature) are the natural extensions of the Riemann curvature and the Ricci curvature in Riemannian geometry. Other quantities are Cartan torsion, S-curvature and Landsberg curvature, etc. They all vanish when the metric is Riemannian. These non-Riemannian quantities interact with the Riemann/Ricci curvature. I investigate Finsler metrics of constant/scalar flag curvature and of isotropic Ricci curvature (Einstein metrics). Recently, I pay more attention to Ricci-flat metrics which have applications to Finslerian extension of general relativity.

Education

• 1990 Ph.D.(Differential Geometry), State University of New York at Stony Brook
• 1986 M.Sc.(Differential Geometry), Academia Sinica, P.R. China
• 1983 B.Sc. (Mathematics), University of Science and Technology of China, P.R.China

Publications & Professional Activity

Z. Shen, On complete manifolds of nonpositive kth-Ricci curvature, Transactions of American Mathematical Society, 338 (1993), no.1, 289-310.

Z. Shen, Complete manifolds with nonnegative Ricci curvature and large volume growth, Inventiones mathematicae, 125 (1996), 393-404.

Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Advances in Mathematics 128 (1997), no. 2, 306-328.

Z. Shen and C. Sormani, The codimension one homology of a complete manifold with nonnegative Ricci curvature, American Journal of Mathematics, 123(2001), 515-524.

Z. Shen, On projectively related Einstein metrics in Riemann-Finsler geometry, Mathematische Annalen, 320 (2001), 625-647.

Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Transactions of the American Mathematical Society, 355(4)(2003), 1713-1728.

X. Chen, X. Mo and Z. Shen, On the flag curvature of Finsler metrics of scalar curvature, Journal of the London Mathematical Society, (2) 68 (2003), 762-780.

D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, Journal of Differential Geometry, 66(2004), 391-449.

X. Mo and Z. Shen, On negatively curved Finsler manifolds of scalar curvature, Canadian Mathematical Bulletin, 48(2005), 112-120.

Z. Shen, On projectively flat (α,β)-metrics, Canadian Mathematical Bulletin, 52(1)(2009), 132-144

Z. Shen, On a class of Landsberg metrics in Finsler Geometry, Canadian Journal of Mathematics, 61(6) (2009), 1357-1374.

X. Cheng and Z. Shen, Randers metrics of scalar flag curvature, Journal of Australian Mathematical Society, 87 (2009), 359-370.

Z.Shen and G. C. Yildirim, A characterization of Randers metrics of scalar flag curvature, Survey in Geometric Analysis and Relativity, ALM 23 (2012), 330-343.

Awards

• NSF grant (DMS 0810159, award for research project: $125377), Project Title: Finsler metrics of Scalar Flag Curvature, , 09/01/2008-08/31/2012
• NSF grant (DMS 0804228, award for conference: $20000), Project Title: International Conference on Riemann-Finsler Geometry, 06/01/2008-05/31/2009.
• NSF grant (DMS 0943046, awarded for conference: $10000), Project Title: Workshop on Riemannian and Non-Riemnnian Geometry, 08/15/2009-07/31/2010.
• Teaching Excellence Recognition Award awarded by IUPUI in 1998.
• Guest Professor of Mathematics, Chongqing Institute of Technology, P.R. China (2002-present).
• Guest Professor of Mathematics, University of Science and Technology of China, P.R. China (2008-present).
• Guanbiao Professor, Zhejiang University, P.R. China (May  2007-May 2010).