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

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 $(G,F)$.