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).