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Colóquio MAP 01/11/2013 - Qualitative Aspects of the Differential Equations of Principal Curvature Lines on Hypersurfaces of Euclidean 4-space and their Partially Umbilic Singularities

After a discussion of some historical landmarks for the study of the differential equations as in the title, going back to Euler, Monge and Darboux, for the case of surfaces in Euclidean 3-space, the results of Gutierrez and Sotomayor (1982-3) on surfaces with Structurally Stable. Configurations of principal curvature lines and umbilic singularities and their Bifurcations will be reviewed and briefly compared with Peixoto’s Theorem (1962) for Structurally Stable Differential Equations (Vector Fields) and their Bifurcations (Sotomayor, 1974) on compact surfaces. An extension of the results for surfaces to hypersurfaces in Euclidean 4-space will be presented in the form of an improved version of the Genericity Theorem (of Kupka – Smale type) due to R. Garcia (1992). New elements of this improvement include the stratified structure of the partially umbilic singularities and the analysis of the heteroclinic partially umbilic connections and attached separatrix surfaces. Open problems and, old and present day, situations where the analysis of Principal Structures, as those in the lecture, for modeling and applications will be mentioned. The mathematical ingredients of this lecture can be regarded as belonging to the elusive boundary between Geometry, Analysis and Dynamical Systems. Work in collaboration with R. Garcia (UFG) and D. Lopes (UFS).