GEODESICS OF RIEMANNIAN COMPLEX HYPERBOLIC PLANE

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 CH2, 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 CH2, 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.