Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

See all formats and pricing
More options …
Volume 30, Issue 6


Homogeneous Finsler spaces and the flag-wise positively curved condition

Ming Xu / Shaoqiang Deng
Published Online: 2018-08-07 | DOI: https://doi.org/10.1515/forum-2018-0130


In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space S3× which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on S2×S3 and S6×S7 which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on S3×S3, shedding some light on the long standing general Hopf conjecture.

Keywords: Finsler metric; homogeneous Finsler space; flag curvature; flag-wise positively curved condition

MSC 2010: 22E46; 53C30


  • [1]

    S. Bácsó, X. Cheng and Z. Shen, Curvature properties of (α,β)-metrics, Finsler Geometry (Sapporo 2005), Adv. Stud. Pure Math. 48, Mathematical Society of Japan, Tokyo (2007), 73–110. Google Scholar

  • [2]

    D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000. Google Scholar

  • [3]

    D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004), no. 3, 377–435. CrossrefGoogle Scholar

  • [4]

    L. Berard-Bergery, Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive, J. Math. Pures Appl. (9) 55 (1976), no. 1, 47–67. Google Scholar

  • [5]

    S.-S. Chern and Z. Shen, Riemann–Finsler Geometry, Nankai Tracts Math. 6, World Scientific, Hackensack, 2005. Google Scholar

  • [6]

    S. Deng and Z. Hou, Invariant Finsler metrics on homogeneous manifolds, J. Phys. A 37 (2004), no. 34, 8245–8253. CrossrefGoogle Scholar

  • [7]

    S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240 (2013), 194–226. CrossrefWeb of ScienceGoogle Scholar

  • [8]

    Z. Hu and S. Deng, Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature, Math. Z. 270 (2012), no. 3–4, 989–1009. CrossrefWeb of ScienceGoogle Scholar

  • [9]

    L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differential Geom. Appl. 40 (2015), 187–208. Web of ScienceCrossrefGoogle Scholar

  • [10]

    L. Huang and X. Mo, On the flag curvature of a class of Finsler metrics produced by the navigation problem, Pacific J. Math. 277 (2015), no. 1, 149–168. Web of ScienceCrossrefGoogle Scholar

  • [11]

    X. Mo and L. Hang, On curvature decreasing property of a class of navigation problems, Publ. Math. Debrecen 71 (2007), no. 1–2, 141–163. Google Scholar

  • [12]

    G. Randers, On an asymmetrical metric in the fourspace of general relativity, Phys. Rev. (2) 59 (1941), 195–199. CrossrefGoogle Scholar

  • [13]

    M. Xu, Examples of flag-wise positively curved spaces, Differential Geom. Appl. 52 (2017), 42–50. Web of ScienceCrossrefGoogle Scholar

  • [14]

    M. Xu and S. Deng, Homogeneous (α,β)-spaces with positive flag curvature and vanishing S-curvature, Nonlinear Anal. 127 (2015), 45–54. Web of ScienceGoogle Scholar

  • [15]

    M. Xu and S. Deng, Normal homogeneous Finsler spaces, Transform. Groups 22 (2017), no. 4, 1143–1183. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    M. Xu and S. Deng, Towards the classification of odd-dimensional homogeneous reversible Finsler spaces with positive flag curvature, Ann. Mat. Pura Appl. (4) 196 (2017), no. 4, 1459–1488. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    M. Xu, S. Deng, L. Huang and Z. Hu, Even-dimensional homogeneous Finsler spaces with positive flag curvature, Indiana Univ. Math. J. 66 (2017), no. 3, 949–972. CrossrefGoogle Scholar

  • [18]

    M. Xu and J. A. Wolf, Killing vector fields of constant length on Riemannian normal homogeneous spaces, Transform. Groups 21 (2016), no. 3, 871–902. Web of ScienceCrossrefGoogle Scholar

  • [19]

    M. Xu and L. Zhang, δ-homogeneity in Finsler geometry and the positive curvature problem, Osaka J. Math. 55 (2018), no. 1, 177–194. Google Scholar

  • [20]

    M. Xu and W. Ziller, Reversible homogeneous Finsler metrics with positive flag curvature, Forum Math. 29 (2017), no. 5, 1213–1226. Web of ScienceGoogle Scholar

About the article

Received: 2018-05-25

Published Online: 2018-08-07

Published in Print: 2018-11-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11771331

Award identifier / Grant number: 11671212

Award identifier / Grant number: 51535008

Funding Source: Natural Science Foundation of Beijing Municipality

Award identifier / Grant number: 1182006

Supported by NSFC (no. 11771331, 11671212, 51535008), Beijing Natural Science Foundation (no. 1182006) and the Fundamental Research Funds for the Central Universities.

Citation Information: Forum Mathematicum, Volume 30, Issue 6, Pages 1521–1537, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2018-0130.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in