Please wait a minute...
Shopping Cart Email List Chinese
Advanced Search Fig/Tab
Frontiers of Mathematics in China

ISSN 1673-3452

ISSN 1673-3576(Online)

CN 11-5739/O1

Postal Subscription Code 80-964

2018 Impact Factor: 0.565

Featured Articles  |  Most Downloaded  |  Most Reading
Online Submission Subscribe to Our RSS Content Feed
Front Math Chin    2011, Vol. 6 Issue (2) : 309-323    https://doi.org/10.1007/s11464-011-0099-8
RESEARCH ARTICLE
A new characterization of Finsler metrics with constant flag curvature 1
Xiaohuan MO()
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China
 Download: PDF(177 KB)   HTML
 Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract

The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.
Keywords Finsler metric      constant flag curvature      Reeb vector field     
Corresponding Author(s): MO Xiaohuan,Email:moxh@pku.edu.cn   
Issue Date: 01 April 2011
 Cite this article:   
Xiaohuan MO. A new characterization of Finsler metrics with constant flag curvature 1[J]. Front Math Chin, 2011, 6(2): 309-323.
 URL:  
https://academic.hep.com.cn/fmc/EN/10.1007/s11464-011-0099-8
https://academic.hep.com.cn/fmc/EN/Y2011/V6/I2/309
1 Bao D, Chern S S, Shen Z. On the Gauss-Bonnet integrand for 4-dimensional Landsberg spaces. In: Bao D, Chern S S, Shen Z, eds. Finsler Geometry, Joint summer Research Conference on Finsler Geometry, July 16–20, 1995, Seattle, Washington. Contemp Math, 196 . Providence: Amer Math Soc, 1996, 15-25
2 Bao D, Chern S-S, Shen Z. An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, 200 . New York: Springer-Verlag, 2000
3 Bao D, Robles C, Shen Z. Zermelo navigation on Riemannian manifolds. J Differential Geom , 2004, 66: 377-435
4 Bao D, Shen Z. On the volume of unit tangent spheres in a Finsler manifold. Results Math , 1994, 26: 1-17
5 Bao D, Shen Z. Finsler metrics of constant positive curvature on the Lie group S3. J London Math Soc , 2002, 66(2): 453-467
doi: 10.1112/S0024610702003344
6 Bejancu A, Farran H R. A geometric characterization of Finsler manifolds of constant curvature K = 1. Inter J Math and Math Sci , 2000, 23: 399-407
doi: 10.1155/S0161171200002179
7 Bryant R L. Finsler structures on the 2-sphere satisfying K = 1. In: Bao D, Chern S S, Shen Z, eds. Finsler Geometry, Joint summer Research Conference on Finsler Geometry, July 16–20, 1995, Seattle, Washington. Contemp Math, 196 . Providence: Amer Math Soc, 1996, 27-41
8 Bryant R L. Projectively flat Finsler 2-spheres of constant curvature. Selecta Math (NS) , 1997, 3: 161-203
doi: 10.1007/s000290050009
9 Bryant R L. Some remarks on Finsler manifolds with constant flag curvature. Special Issue for S. S. Chern. Houston J Math , 2002, 28: 221-262
10 Chern S S. Riemannian geometry as a special case of Finsler geometry. Contem Math , 1996, 196: 51-58
11 Chern S S, Carmo M do, Kobayashi S. Minimal submanifolds of a sphere with second fundamental form of constant length. In: Browder F E, ed. Functional Analysis and Related Fields . Berlin: Springer-Verlag, 1970, 59-75
12 Chern S S, Shen Z. Riemann-Finsler Geometry. Nankai Tracts in Mathematics, 6 . Hackensack: World Scientific Publishing Co Pte Ltd, 2005
13 Li A, Li J. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch Math (Basel) , 1992, 58: 582-594
14 Mo X. Characterization and structure of Finsler spaces with constant flag curvature. Sci China, Ser A , 1998, 41: 910-917
doi: 10.1007/BF02879999
15 Mo X. Flag curvature tensor on a closed Finsler surface. Result in Math , 1999, 36: 149-159
16 Mo X. On the flag curvature of a Finsler space with constant S-curvature. Houston Journal of Mathematics , 2005, 31: 131-144
17 Mo X. An Introduction to Finsler Geometry. Singapore: World Scientific Press, 2006
doi: 10.1142/9789812773715
18 Mo X, Yu C. On the Ricci curvature of a Randers metrics of isotropic S-curvature. Acta Mathematica Sinica (NS) , 2008, 24: 991-996
19 Peng C, Terng C. The scalar curvature of minimal hypersurfaces in spheres. Math Ann , 1983, 266: 105-113
doi: 10.1007/BF01458707
20 Shen Y. On intrinsic rigidity for minimal submanifolds in a sphere. Sci China, Ser A , 1989, 32: 769-781
21 Shen Z. Projectively flat Finsler metrics of constant flag curvature. Trans Amer Math Soc , 2003, 355: 1713-1728
doi: 10.1090/S0002-9947-02-03216-6
22 Simons J. Minimal varieties in Riemannian manifolds. Ann Math , 1968, 83: 62-105
doi: 10.2307/1970556
23 Yano K. On harmonic and Killing vector fields. Ann Math , 1952, 55: 38-45
doi: 10.2307/1969418
24 Yano K. Integral Formulas in Riemannian Geometry. Pure and Applied Mathematics, No 1 . New York: Marcel Dekker, Inc, 1970
[1] Hongchuan XIA,Chunping ZHONG. A class of metrics and foliations on tangent bundle of Finsler manifolds[J]. Front. Math. China, 2017, 12(2): 417-439.
[2] RADEMACHER Hans-Bert. The second closed geodesic on a complex projective plane[J]. Front. Math. China, 2008, 3(2): 253-258.
[3] GUO Enli, MO Xiaohuan. Riemann-Finsler geometry[J]. Front. Math. China, 2006, 1(4): 485-498.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed