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arXiv · 1401.0472

$(\alpha_1,\alpha_2)$-Spaces and Clifford-Wolf Homogeneity

Ming Xu·Shaoqiang Deng
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Abstract

In this paper, we introduce a new type of Finsler metrics, called $(\alpha_1,\alpha_2)$-metrics. We define the notion of the good datum of a homogeneous $(\alpha_1,\alpha_2)$-metric and use that to study the geometric properties. In particular, we give a formula of the S-curvature and deduce a condition for the S-curvature to be vanishing identically. Moreover, we consider the restrictive Clifford-Wolf homogeneity of left invariant $(\alpha_1,\alpha_2)$-metrics on compact connected simple Lie groups. We prove that, in some special cases, a restrictively Clifford-Wolf homogeneous $(\alpha_1,\alpha_2)$-metric must be Riemannian. An unexpected interesting observation contained in the proof reveals the fact that the S-curvature may play an important role in the study of Clifford-Wolf homogeneity in Finsler geometry.

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BibTeXRIS

Ming Xu, Shaoqiang Deng. 2014-01-02. $(\alpha_1,\alpha_2)$-Spaces and Clifford-Wolf Homogeneity. https://arxiv.org/abs/1401.0472

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