Lines of Curvature on Surfaces, Historical Comments and Recent Developments

Details Lines of Curvature on Surfaces, Historical Comments and Recent Developments Lines of Curvature on Surfaces, Historical Comments and Recent Developments

2007-12-10

arxiv.org/abs/0712.1585v1

Mathematics

Differential Geometry

40 pages, 25 figures

Scientific paper

This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincar\'e in 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections.

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