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The cut locus of a torus of revolution. (English) Zbl 1080.53005

Let \((M, g)\) be a complete Riemannian manifold and \(\gamma: [0, t_0]\to M\) a minimal geodesic segment emanating from a point \(p:=\gamma(0)\). The endpoint \(\gamma(t_0)\) of the geodesic segment is called a cut point of \(p\) along \(\gamma\) if any extended geodesic segment \(\widetilde\gamma:[0, t_1]\to M\) of \(\gamma\), where \(t_1>t_0\), is not a minimizing arc joining \(p\) to \(\widetilde\gamma(t_1)\) anymore. The cut locus \(C_p\) of the point \(p\) is defined by the set of the cut points along all geodesic segments emanating from \(p\). In the paper under review the authors determine the structure of the cut locus of a class of tori of revolution, which includes the standard tori in 3-dimensional Euclidean space.

MSC:

53A05 Surfaces in Euclidean and related spaces
53C22 Geodesics in global differential geometry

Software:

Loki
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