Unduloids and their closed geodesics. (English) Zbl 1051.53005

Mladenov, Ivaïlo; Oprea, John

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 206-234 (2003).

The authors give parametrizations for the unduloids in terms of Jacobi elliptic functions and use Clairaut’s theorem to obtain explicit (in terms of an integral) parametrizations for geodesics on unduloids. Numerical integration and an intermediate value argument is then used to determine closed geodesics. The authors provide Maple code and surface graphics to illustrate their work.
For the entire collection see [Zbl 1008.00022].

Reviewer: Udo Hertrich-Jeromin (Bath)

MSC:

53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C22	Geodesics in global differential geometry

Keywords:

closed geodesic; Delaunay surface; unduloid; Jacobi elliptic function; Clairaut’s theorem

Software:

Maple