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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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1435-5337
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Volume 29, Issue 5

# 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 $\left(G/H,F\right)$. 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. $\mathrm{SU}\left(4\right)/\mathrm{SU}{\left(2\right)}_{\left(1,2\right)}{\mathrm{S}}_{\left(1,1,1,-3\right)}^{1}$, $\mathrm{Sp}\left(2\right)/{\mathrm{S}}_{\left(1,1\right)}^{1}$, $\mathrm{Sp}\left(2\right)/{\mathrm{S}}_{\left(1,3\right)}^{1}$, $\mathrm{Sp}\left(3\right)/\mathrm{Sp}{\left(1\right)}_{\left(3\right)}{\mathrm{S}}_{\left(1,1,0\right)}^{1}$, and ${G}_{2}/\mathrm{SU}\left(2\right)$ with $\mathrm{SU}\left(2\right)$ the normal subgroup of $\mathrm{SO}\left(4\right)$ corresponding to the long root. Applying this classification to homogeneous positively curved reversible $\left(\alpha ,\beta \right)$ metrics, the number of exceptional candidates can be reduced to only two, i.e. $\mathrm{Sp}\left(2\right)/{\mathrm{S}}_{\left(1,1\right)}^{1}$ and $\mathrm{Sp}\left(3\right)/\mathrm{Sp}{\left(1\right)}_{\left(3\right)}{\mathrm{S}}_{\left(1,1,0\right)}^{1}$.

MSC 2010: 22E46; 53C30

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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,

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