Geodesic Bifurcation on Smooth Surfaces

Within Riemannian geometry the geodesic exponential map is an essential tool for various distance related investigations and computations. Several natural questions can be formulated in terms of its preimages, usually leading to quite challenging nonlinear problems. In this context we recently proposed an approach for computing multiple geodesics connecting two arbitrary points on two-dimensional surfaces in situations where an ambiguity of these connecting geodesics is indicated by the presence of so-called focal curves. The essence of the approach consists in exploiting the structure of the associated focal curve and using a suitable curve for a homotopy algorithm to collect the geodesic connections. In this follow-up discussion we extend those constructions to overcome a significant limitation inherent in the previous method, i.e. the necessity to construct homotopy curves artificially. We show that considering homotopy curves meeting a focal curve tangentially leads to a singularity that we investigate thoroughly. Solving this so–called geodesic bifurcation analytically and dealing with it numerically provides not only theoretical insights, but also allows geodesics to be used as homotopy curves. This yields a stable computational tool in the context of computing distances. This is applicable in common situations where there is a curvature induced non-injectivity of the exponential map. In particular we illustrate how applying geodesic bifurcation solves the distance problem on compact manifolds with a single closed focal curve. Furthermore the presented investigations provide natural initial values for computing cut loci and medial curves using the medial differential equation.