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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

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

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1435-5337
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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

## Abstract

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 ${S}^{3}×ℝ$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on ${S}^{2}×{S}^{3}$ and ${S}^{6}×{S}^{7}$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on ${S}^{3}×{S}^{3}$, shedding some light on the long standing general Hopf conjecture.

MSC 2010: 22E46; 53C30

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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,

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