## 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

### Keywords:

cut locus; geodesic; cut points; a torus of revolution

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