Geodesics on an invariant surface

Abstract

We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces; the local description of the geodesics; the explicit description of geodesic curves on an invariant surface with constant Gauss curvature.