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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


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Volume 29, Issue 5

Issues

Reversible homogeneous Finsler metrics with positive flag curvature

Ming Xu / Wolfgang Ziller
Published Online: 2016-10-14 | DOI: https://doi.org/10.1515/forum-2016-0173

Abstract

In this work, we continue with the classification for positively curve homogeneous Finsler spaces (G/H,F). With the assumption that the homogeneous space G/H is odd dimensional and the positively curved metric F is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group H is regular in G. Applying the fixed point set technique and the homogeneous flag curvature formulas, we show that the classification of odd dimensional positively curved reversible homogeneous Finsler spaces coincides with that of L. Bérard Bergery in Riemannian geometry except for five additional possible candidates, i.e. SU(4)/SU(2)(1,2)S(1,1,1,-3)1, Sp(2)/S(1,1)1, Sp(2)/S(1,3)1, Sp(3)/Sp(1)(3)S(1,1,0)1, and G2/SU(2) with SU(2) the normal subgroup of SO(4) corresponding to the long root. Applying this classification to homogeneous positively curved reversible (α,β) metrics, the number of exceptional candidates can be reduced to only two, i.e. Sp(2)/S(1,1)1 and Sp(3)/Sp(1)(3)S(1,1,0)1.

Keywords: Homogeneous Finsler space; flag curvature; totally geodesic subspace

MSC 2010: 22E46; 53C30

References

  • [1]

    S. Aloff and N. Wallach, An infinite family of 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93–97. Google Scholar

  • [2]

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

  • [3]

    L. Bérard Bergery, Les variétés Riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive, J. Math. Pures Appl. (9) 55 (1976), 47–68. Google Scholar

  • [4]

    M. Berger, Les varietes Riemanniennes homogenes normales simplement connexes a courbure strictment positive, Ann. Sc. Norm. Super. Pisa Sci. Fis. Mat. III. Ser. 15 (1961), 191–240. Google Scholar

  • [5]

    S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012. Google Scholar

  • [6]

    S. Deng and M. Xu, Left invariant Clifford–Wolf homogeneous (α,β)-metrics on compact semisimple Lie groups, Transform. Groups 20 (2015), no. 2, 395–416. Google Scholar

  • [7]

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

  • [8]

    L. Kozma, Weinstein’s theorem for Finsler manifolds, Kyoto J. Math. 46 (2006), 377–382. Google Scholar

  • [9]

    N. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277–295. Google Scholar

  • [10]

    B. Wilking and W. Ziller, Revisiting homogeneous spaces with positive curvature, J. Reine Angew. Math. (2015), 10.1515/crelle-2015-0053. Google Scholar

  • [11]

    J. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Boston, 1984. Google Scholar

  • [12]

    M. Xu and S. Deng, Normal homogeneous Finsler spaces, preprint (2014), http://arxiv.org/abs/1411.3053; to appear in Transform. Groups.

  • [13]

    M. Xu and S. Deng, Towards the classication of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature, preprint (2015), http://arxiv.org/abs/1504.03018.

  • [14]

    M. Xu and S. Deng, Homogeneous Finsler spaces and the flag-wise positively curved condition, preprint (2016), http://arxiv.org/abs/1604.07695.

  • [15]

    M. Xu, S. Deng, L. Huang and Z. Hu, Even dimensional homogeneous Finsler spaces with positive flag curvature, preprint (2014), http://arxiv.org/abs/1407.3582; to appear in Indiana Univ. Math. J.

  • [16]

    M. Xu and J. Wolf, Sp(2)/U(1) and a positive curvature problem, Differential Geom. Appl. 42 (2015), 115–124. Google Scholar

  • [17]

    W. Ziller, Examples of Riemannian manifolds with nonnegative sectional curvature, Metric and Comparison Geometry, Surv. Differ. Geom. 11, International Pres, Somerville (2007), 63–102. Google Scholar

About the article


Received: 2016-08-12

Published Online: 2016-10-14

Published in Print: 2017-09-01


The first author was supported by the NSFC (no. 11271216), the Science and Technology Development Fund for Universities and Colleges in Tianjin (no. 20141005), and the Doctor Fund of Tianjin Normal University (no. 52XB1305). The second author was supported by a grant from the National Science Foundation.


Citation Information: Forum Mathematicum, Volume 29, Issue 5, Pages 1213–1226, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0173.

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