Abstract

The complex hyperbolic plane is a symmetric space of negative sectional curvature; hence, it has the structure of a 4-dimensional connected solvable real Lie group with a left-invariant metric. We consider all non-isometric left-invariant Riemannian metrics on this group, denoted by ${\mathcal {CH}}^2$, and search for real geodesics corresponding to them. Using Euler-Arnold equations, one can translate the second-order differential equations of the geodesics on the group into the first-order equations on its Lie algebra. In the Kähler case we solve these equations on the Lie algebra of ${\mathcal {CH}}^2$, i.e. we explicitly find curves on algebra corresponding to the geodesics of the standard Einstein metric. Numerical solutions are used to visualize geodesic lines and geodesic spheres of various left-invariant Riemannian metrics.

Keywords: Complex hyperbolic plane; left-invariant metric; Euler-Arnold equations; geodesic lines; geodesic spheres.

MSC: 53C22, 22E60

DOI: 10.57016/MV-MsnU3893

Pages:  105$-$117     

Volume  76 ,  Issue  1-2 ,  2024