Geodesics on an invariant surface

TL;DR

This paper investigates the properties of geodesics on invariant surfaces within three-dimensional Riemannian manifolds, providing characterizations, relations, and explicit descriptions especially for surfaces with constant Gauss curvature.

Contribution

It offers new characterizations of geodesic orbits, Clairaut's relation interpretations, and explicit descriptions of geodesics on invariant surfaces, including those with constant Gauss curvature.

Findings

Characterization of geodesic orbits

Clairaut's relation and geometric interpretation

Explicit description of geodesics on constant Gauss curvature surfaces

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.