Zhongmin Shen
Professor
- Office:
- LD 224A
- Phone:
- (317) 278-1065
- Email:
- zshen@iu.edu
- Website:
- https://math.indianapolis.iu.edu/~zshen/
- Research Areas:
- Modern Analysis/Geometry
Professor
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.
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.