LEFT INVARIANT $(\alpha,\beta)$-METRICS ON 4-DIMENSIONAL LIE GROUPS

Mona Atashafrouz, Behzad Najafi, Laurian-Ioan Piscoran

DOI Number
https://doi.org/10.22190/FUMI2003727A
First page
727
Last page
740

Abstract


Let $G$ be a 4-dimensional Lie group with an invariant para-hypercomplex structure and let $F= \beta+ a\alpha+\beta^2/{\alpha}$ be a left invariant $(\alpha,\beta)$-metric, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on $G$, and $a$ is a real number. We prove that the flag curvature of $F$ with parallel 1-form $\beta$ is non-positive, except in Case 2, in which $F$ admits both negative and positive flag curvature. Then, we determine all geodesic vectors of $(G,F)$.  

Keywords

para-hypercomplex structure; $(\alpha,\beta)$-metric; Riemannian metric; flag curvature.

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DOI: https://doi.org/10.22190/FUMI2003727A

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