A statistical cohomogeneity one metric on the upper plane with constant negative curvature (Q2019268)

Summary: we analyze the geometrical structures of the statistical manifold \(S\) consisting of all the wrapped Cauchy distributions. We prove that \(S\) is a simply connected manifold with constant negative curvature \(K=-2\). However, it is not isometric to the hyperbolic space because \(S\) is noncomplete. In fact, \(S\) is approved to be a cohomogeneity one manifold. Finally, we use several tricks to get the geodesics and explore the divergence performance of them by investigating the Jacobi vector field.