Geometric invariants of cuspidal edges (Q2798994)

In this very well written paper the authors obtain differential geometric invariants for cuspidal edges which determine them up to order 3.NEWLINENEWLINECuspidal edges appear naturally in differential geometry, for example in focal sets or parallel surfaces of regular surfaces in \(\mathbb R^3\). They are also a generic singularity of wave-fronts.NEWLINENEWLINEThe authors obtain a West-type normal form for a generic cuspidal ege using diffeomorphisms in the source and only isometries in the target (in order to preserve the geometry).NEWLINENEWLINEThey introduce the singular and limiting normal curvatures, \(\kappa_s\) and \(\kappa_n\) resp., which were defined in [Ann. Math. (2) 169, No. 2, 491--529 (2009; Zbl 1177.53014)] by the second author et al. as limits of the geodesic and normal curvatures at the cuspidal edge. They prove that \(|\kappa_n|\) coincides with the umbilic curvature, \(\kappa_u\), defined by the first author and \textit{J. J. Nuño-Ballesteros} [Tohoku Math. J. (2) 67, No. 1, 105--124 (2015; Zbl 1320.58023)]. Furthermore, they show that the relation NEWLINE\[NEWLINE\kappa^2=\kappa_s^2+\kappa_n^2NEWLINE\]NEWLINE holds, where \(\kappa\) is the curvature of the space curve defined by the cuspidal edge, as an analogy to the case of regular surfaces with the geodesic and normal curvatures.NEWLINENEWLINEThey then define 3 more invariants, the cuspidal curvature, the cusp-directional torsion and the edge-inflectional curvature (\(\kappa_c\), \(\kappa_t\) and \(\kappa_i\) resp.). They give all of the invariants in terms of the coefficients of their normal form and prove that \(\tau\), \(\kappa_s\), \(\kappa_n\), \(\kappa_c\), \(\kappa_t\) and \(\kappa_i\) define the cuspidal edge up to order 3 terms, where \(\tau\) is the torsion of the cuspidal edge seen as a space curve.NEWLINENEWLINEThis paper has led to further research on cuspidal edges by many authors and motivated the study in a similar way of other singularities from the differential geometric point of view.