A generalized Clairaut’s theorem in Minkowski space. (English) Zbl 1328.53013

Summary: In Euclidean space, the geodesics on a surface of revolution can be characterized by means of Clairaut’s theorem, which essentially says that the geodesics are curves of fixed angular momentum. A similar result is known for three dimensional Minkowski space for timelike geodesics on surfaces of revolution about the time axis. Here, we extend this result to consider generalizations of surfaces of revolution to those surfaces generated by any one-parameter subgroup of the Lorentz group. We also observe that the geodesic flow in this case is easily seen to be a completely integrable system, and give the explicit formulae for the timelike geodesics.


53A35 Non-Euclidean differential geometry
53C22 Geodesics in global differential geometry
53B25 Local submanifolds