Geodesic vectors of square metrics on 5- dimensional generalized symmetric spaces

Abstract

In this paper, we consider the $(\alpha, \beta)$-metric $F=\frac{(\alpha + \beta)^2}{\alpha}$ along with the function $\phi$ with the definition of $\phi(s)=1+2s+s^2$, which is known as a square metric, on 5-dimensional generalized symmetric spaces. Then we investigate and study the homogeneous geodesics of 5-dimensional generalized symmetric spaces equipped with a left invariant square metric. We also obtain and categorize the homogeneous geodesics of these spaces in some special cases, which are of type (2), (3) and (7). Also we show that for a 5-dimensional generalized symmetric space of type (2) and (7) equipped with a left invariant square metric, the geodesic vectors of $(M,F)$ are the same as the geodesic vectors of $(M, \tilde{a})$ and vice versa.

Keywords

• (α
• β)- metric
• geodesic vectors
• homogeneous geodesics
• homogeneous spaces
• square metric
• symmetric spaces