On a class of locally projectively flat Finsler metrics

Document Type: Research Paper

Authors

1 Key Laboratory of Pure and Applied Mathematics‎, ‎School of Mathematical Sciences‎, ‎Peking University‎, ‎Beijing 100871‎, ‎China.

2 College of Mathematics and Information Science‎, ‎Henan Normal University‎, ‎Xinxiang‎, ‎453007‎, ‎China.

Abstract

‎In this paper we study Finsler metrics with orthogonal invariance‎. ‎We‎ ‎find a partial differential equation equivalent to these metrics being locally projectively flat‎. ‎Some applications are given‎. ‎In particular‎, ‎we give an explicit construction of a new locally projectively flat Finsler metric of vanishing flag curvature which differs from the Finsler metric given by Berwald in 1929.

Keywords

Main Subjects


D. Bao and Z.M. Shen, Finsler metrics of constant positive curvature on the Lie group S3, J. Lond. Math. Soc. (2) 66 (2002), no. 2, 453--467.

S.S. Chern and Z.M. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics 6, World Scientific Publ. Hackensack, 2005.

J. Douglas, The general geometry of paths, Ann. of Math. (2) 29 (1927/28), no. 1-4, 143--168.

D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 37 (2001), no. 4, 407--436.

L.B. Huang and X.H. Mo, On spherically symmetric Finsler metrics of scalar curvature, J. Geom. Phys. 62 (2012), no. 11, 2279--2287.

L.B. Huang and X.H. Mo, Projectively at Finsler metrics with orthogonal invariance, Ann. Polon. Math. 107 (2013), no. 3, 259--270.

B.L. Li, On the classification of projectively at Finsler metrics with constant ag curvature, Adv. Math. 257 (2014) 266--284.

B.L. Li and Z.M. Shen, On a class of projectively at Finsler metrics with constant ag curvature, Internat. J. Math. 18 (2007), no. 7, 749--760.

H.F. Liu and X.H. Mo, Examples of Finsler metrics with special curvature properties,Math. Nachr. 288 (2015), no. 13, 1527--1537.

M. Matsumoto, Projective changes of Finsler metrics and projectively at Finsler spaces, Tensor (N. S.) 34 (1980), no. 3, 303--315.

X.H. Mo, Z.M. Shen and C.H. Yang, Some constructions of projectively at Finsler metrics, Sci. China Ser. A 49 (2006), no. 5, 703--714.

X.H. Mo, N.M. Solorzano and K. Tenenblat, On spherically symmetric Finsler metrics with vanishing Douglas curvature, Differential Geom. Appl. 31 (2013), no. 6, 746--758.

X.H. Mo and C.T. Yu, On some explicit constructions of Finsler metrics with scalar ag curvature, Canad. J. Math. 62 (2010), no. 6, 1325--1339.

X.H. Mo, L.F. Zhou and H.M. Zhu, On a class of Finsler metrics of constant curvature, Houston J. Math. to appear.

X.H. Mo and H.M. Zhu, On a class of projectively at Finsler metrics of negative constant ag curvature, Internat. J. Math. 23 (2012), no. 8, Article ID 1250084, 14 pages.

E.S. Sevim, Z.M. Shen and S.Ulgen, Spherically symmetric Finsler metrics with constant Ricci and ag curvature, Publ. Math. Debrecen 87 (2015), no. 3-4, 463--472.

Z.M. Shen, Projectively at Randers metrics with constant ag curvature, Math. Ann. 325 (2003), no. 1, 19--30.

Z.M. Shen, On projectively at (α; β)-metrics, Canad. Math. Bull. 52 (2009), no. 1, 132--144.

L.F. Zhou, Projective spherically symmetric Finsler metrics with constant ag curvature in Rn, Geom. Dedicata 158 (2012) 353--364.

H.M. Zhu, A class of Finsler metrics of scalar curvature, Differential Geom. Appl. 40 (2015) 321--331.