[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b150765

RIGIDITY THEOREMS OF SOME DUALLY FLAT FINSLER METRICS AND ITS APPLICATIONS

Shen, Bin (Department of Mathematics Southeast University)
Tian, Yanfang (Logistical Engineering University of PLA)

Publication Information

Bulletin of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1457-1469 More about this Journal

Abstract

In this paper, we study a class of Finsler metric. First, we find some rigidity results of the dually flat ( ${\alpha}$ , ${\beta}$ )-metric where the underline Riemannian metric ${\alpha}$ satisfies nonnegative curvature properties. We give a new geometric approach of the Monge- $Amp{\acute{e}}re$ type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.

Keywords

Finsler metric; ( ${\alpha}$ , ${\beta}$ )-metric; dually flat; Monge- $Amp{\acute{e}}re$ equation; Bernstein type theorem;

Citations & Related Records

Reference

1	L. Caffarelli and Y. Y Li, An extension to a theorem of Jorgens, Calabi, and Pogorelov, Comm. Pure Appl. Math. 56 (2003), no. 5, 549-583. DOI
2	E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jorgens, Michigan Math. J. 5 (1958), 105-126. DOI
3	A. Caminha, P. Souza, and F. Camargo, Complete foliations of space forms by hypersurfaces , Bull. Braz. Math. Soc. (N.S.) 41 (2010), no. 3, 339-353. DOI
4	X. Cheng and Z. Shen, Finsler Geometry-An approach via Ransders spaces, Springer, Heidelberg, Science Press, Beijing, 2012.
5	X. Cheng, Z. Shen, and Y. Zhou, On locally dually flat Randers metrics, Internat. J. Math. 21 (2010), 1531-1543. DOI
6	J. Jost and Y. L. Xin, Some aspects of the global geometry of entire space-like submanifolds, Dedicated to Shiing-Shen Chern on his 90th birthday, Results Math. 40 (2001), no. 1-4, 233-245. DOI
7	A. V. Pogorelov, Improper convex affine hyperspheres, Dokl. Akad. Nauk SSSR 202 (1972), 1008-1011.
8	Z. Shen, Riemannian-Finsler geometry with applications to information geometry, Chinese Ann. Math. Ser. B 27 (2006), no. 1, 73-94. DOI
9	B. Shen and Y. Tian, Dually Flat Kropina Metrics, Advances in Mathematics(China), to be appear.
10	Q. Xia, On locally dually flat ( ${\alpha},{\beta}$ )-metrics, Differential Geom. Appl. 29 (2011), no. 2, 233-243. DOI
11	C. Yu, On dually flat Randers metrics, Nonlinear Anal. 95 (2014), 146-155. DOI