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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / 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

Online
ISSN
1435-5337
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Volume 27, Issue 5

# Homogeneous (α,β)-metrics of Douglas type

Huaifu Liu
/ Shaoqiang Deng
Published Online: 2014-02-19 | DOI: https://doi.org/10.1515/forum-2014-0001

## Abstract

In this paper, we consider two related problems concerning homogeneous (α,β)-metrics. In the first part we consider homogeneous (α,β)-spaces of Douglas type. We prove that a homogeneous (α,β)-metric is a Douglas metric if and only if either F is a Berwald metric or F is a Douglas metric of Randers type. In the second part, we prove that if F is a homogeneous (α,β)-metric which is neither a Riemannian metric nor a Minkowski metric, then F is locally projectively flat if and only if F is a locally projectively flat left invariant Randers metric on the hyperbolic space ${𝐇}^{n}$ as a solvable Lie group. We also give all the explicit forms of the metric F in the second case. This result also provides new examples of locally projectively flat metrics which have not been described before in the literature, presenting some new insight in the study of the Hilbert's fourth problem.

MSC: 22E46; 53C30

Revised: 2014-01-19

Published Online: 2014-02-19

Published in Print: 2015-09-01

Funding Source: NSFC

Award identifier / Grant number: 11271198

Funding Source: NSFC

Award identifier / Grant number: 11221091

Funding Source: SRFDP of China

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 3149–3165, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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