Realization of Bipartite Weighted Graphs by Stable Gauss Maps
Keywords:
Stable Gauss maps, Bipartite weighted graphs, Singularities, Parabolic curves, Euclidean spaceAbstract
The singularities of a stable Gauss map of a closed orientable surface immersed generically in three-dimensional Euclidean space, according to H. Whitney’s Theorem, are of the fold and cusp types. The singular set of a stable Gauss map of a surface, which consists of curves of fold points containing isolated cusp points, is the parabolic set on the surface. Each parabolic curve of the singular set separates a hyperbolic region from an elliptic region of the surface. In this work, we will explore how weighted graphs can be associated with stable Gauss maps and present a general result that determines necessary and sufficient conditions for a weighted graph to be associated with a stable Gauss map.
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References
T. Banchoff, T. Gaffney, C. Mccrory, and D. Dreibelbis. Cusps of Gauss Mappings. London: Pitman Books Limited, 1982. ISBN: 9780273085362. URL: https://www.emis.de/monographs/CGM/index.html.
M. P. do Carmo. Geometria diferencial de curvas e superfícies. 5ª ed. Textos Universitários. Rio de Janeiro: SBM, 2012. ISBN: 9788585818265.
C. M. de Jesus, S. M. Moraes, and M. C. Fuster. “Stable Gauss maps from a global viewpoint”. In: Bulletin of the Brazilian Mathematical Society 42 (2012), pp. 87–103. DOI: 10.1007/s00574-011-0005-8.
C. M. de Jesus and E. Sanabria-Codesal. “Realization of graphs by fold Gauss maps”. In: Topology and its Applications 234 (2018). DOI: 10.1016/j.topol.2017.11.016.
C. P. Mesquita. Grafos e suas aplicações no Ensino da Matemática. 2021.
I. V. Souza. “Singularidades de Aplicações de Gauss Estáveis”. Master dissertation. UFV, 2012.
H. Whitney. “On singularities of mappings of euclidean spaces i: Mappings of the plane into the plane”. In: Annals of Mathematics 62 (1955), pp. 374–410. DOI: 10.2307/1970070.
