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Homogeneous Finsler spaces and the flag-wise positively curved condition

  • Ming Xu und Shaoqiang Deng EMAIL logo
Veröffentlicht/Copyright: 7. August 2018

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 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.

MSC 2010: 22E46; 53C30

Communicated by Karl-Hermann Neeb


Award Identifier / Grant number: 11771331

Award Identifier / Grant number: 11671212

Award Identifier / Grant number: 51535008

Award Identifier / Grant number: 1182006

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

Acknowledgements

We would like to thank W. Ziller and L. Huang for helpful comments and discussions. This work is dedicated to the 80th birthday of our most respectable friend, teacher of mathematics and life, Professor Joseph A. Wolf.

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Received: 2018-05-25
Published Online: 2018-08-07
Published in Print: 2018-11-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 9.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2018-0130/pdf
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