Cut loci and conjugate loci on Liouville surfaces
In a previous paper by the same authors, cut loci and conjugate loci of general points on a two-dimensional elipsoid were studied. It was shown that the cut locus of a general point is a segment of the curvature line which passes through the antipodal point of the initial point and that the conjugate locus of a general point contains just four singularities, which are located on the curvature lines passing through the antipodal point of the initial point. In the present paper it is shown that certain Liouville surfaces have similar properties. For special compact Liouville surfaces, it is shown that the cut locus of a general point is a curve segment. For the noncompact case, it is shown that the possible shapes of cut loci are either the empty set, or a curve segment, or a disjoint union of two curve segments. For more special compact Liouville surfaces it is shown that the conjugate locus of a general point contains just four singular points located on the natural coordinate lines passing through the antipodal point of the original point. In the noncompact case, the conjugate locus of a point is not empty if and only if the cut locus is not empty and the singular points of the conjugate locus coincide with the end points of the cut locus.
scientific article; zbMATH DE number 5944332