Rigidity theorems of some dually flat Finsler metrics and its applications
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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\'ere type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.
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\'ere type equation on $R^n$ by using those results. We also get the non-existence of the compact globally dually flat Riemannian manifold.
참고문헌 (11)
- L. Caffarelli, "An extension to a theorem of Jorgens, Calabi, and Pogorelov", Comm. Pure Appl. Math, 56(5), pp.549~583 (2003)
- E. Calabi, "Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jorgens", Michigan Math. J, 5, pp.105~126 (1958)
- A. Caminha, "Complete foliations of space forms by hypersurfaces", Bull. Braz. Math. Soc. (N.S.), 41(3), pp.339~353 (2010)
- X. Cheng, "Finsler Geometry–An approach via Ransders spaces" (2012), Springer, Science Press
- X. Cheng, "On locally dually flat Randers metrics", Internat. J. Math, 21, pp.1531~1543 (2010)
- J. Jost, "Some aspects of the global geometry of entire space-like submanifolds, Dedicated to Shiing-Shen Chern on his 90th birthday", Results Math, 40(1-4), pp.233~245 (2001)
- A. V. Pogorelov, "Improper convex affine hyperspheres", Dokl. Akad. Nauk SSSR, 202, pp.1008~1011 (1972)
- Z. Shen, "Riemannian-Finsler geometry with applications to information geometry", Chinese Ann. Math. Ser. B, 27(1), pp.73~94 (2006)
- B. Shen, "Dually Flat Kropina Metrics", Advances in Mathematics(China)
- Q. Xia, "On locally dually flat (α, β)-metrics", Differential Geom. Appl, 29(2), pp.233~243 (2011)
- C. Yu, "On dually flat Randers metrics", Nonlinear Anal, 95, pp.146~155 (2014)