Geodesic
Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 5, pp. 117-147.

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Closed locally minimal networks can be viewed as “branching” closed geodesics. We study such networks on the surfaces of convex polyhedra and discuss the problem of describing the set of all convex polyhedra that have such networks. A closed locally minimal network on a convex polyhedron is an embedding of a graph provided that all edges are geodesic arcs and at each vertex exactly three adges meet at angles of $120^{\circ}$. In this paper, we do not deal with closed (periodic) geodesics. Among other results, we prove that the natural condition on the curvatures of a polyhedron that is necessary for the polyhedron to have a closed locally minimal network on its surface is not sufficient. We also prove a new stronger necessary condition. We describe all possible combinatorial structures and edge lengths of closed locally minimal networks on convex polyhedra. We prove that almost all convex polyhedra with vertex curvatures divisible by $\frac{\pi}{3}$ have closed locally minimal networks.
Keywords: locally minimal network, geodesic net, convex polyhedron. 
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N. P. Strelkova. Closed Locally Minimal Networks on the Surfaces of Convex Polyhedra. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 5, pp. 117-147. http://geodesic.mathdoc.fr/item/MAIS_2013_20_5_a7/

[1] Strelkova N. P., “Ustoychivost lokalno minimalnykh setey”, Trudy seminara po vektornomu i tenzornomu analizu s ikh prilozheniyami k geometrii, mekhanike i fizike, 29, 2013, 148–170 (in Russian)

[2] A. O. Ivanov, A. A. Tuzhilin, Branching solutions to one-dimensional variational problems, World Scientific, Singapore, 2000 | MR

[3] W. P. Thurston, “Shapes of polyhedra and triangulations of the sphere”, The Epstein birthday schrift, Geom. Topol. Monogr., 1, 1998, 511–549 | MR | Zbl

[4] I. V. Ptitsyna, “Classification of closed minimal networks on tetrahedra”, Sci. Sb. Math., 82:1 (1995), 101–116 | DOI | MR | Zbl

[5] N. P. Strelkova, “Closed locally minimal nets on tetrahedra”, Sb. Math., 202:1 (2011), 135–153 | DOI | DOI | MR | Zbl

[6] V. Yu. Protasov, “Closed geodesics on the surface of a simplex”, Sbornik: Mathematics, 198:2 (2007), 243–260 | DOI | DOI | MR | Zbl

[7] A. D. Aleksandrov, Convex Polyhedra, Springer-Verlag, Berlin, 2005 | MR | MR

[8] N. P. Strelkova, “Realization of plane graphs as closed locally minimal nets on convex polyhedra”, Doklady Mathematics, 82:3 (2010), 939–941 | DOI | MR | Zbl

[9] B. Aronov, J. O'Rourke, “Nonoverlap of the star unfolding”, Discrete and Computational Geometry, 8:1 (1992), 219–250 | DOI | MR | Zbl

[10] Z. A. Melzak, “On the problem of Steiner”, Canad. Math. Bulletin, 4 (1961), 143–148 | DOI | MR | Zbl

[11] A. O. Ivanov, A. A. Tuzhilin, “The geometry of minimal networks with a given topology and a fixed boundary”, Izvestiya: Mathematics, 61:6 (1997), 1231–1263 | DOI | DOI | MR | Zbl