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The geometry of fronts. (English) Zbl 1177.53014

In this very interesting paper, the authors investigate the generic singularities of wave fronts from the viewpoint of differential geometry. They introduce the concept of singular curvature function on cuspidal edges of surfaces, and investigate its properties. They also generalize Gauss-Bonnet formulas for compact surfaces given by R. Langevin, G. Levitt and H. Rosenberg in [Can. J. Math. 47, No. 3, 544–572 (1995; Zbl 0837.57025)], and by M. Kossowski in [Ann. Global Anal. Geom. 21, No. 1, 19–29 (2002; Zbl 1006.53006)], to fronts which admit finitely many corank one “peak” singularities.
Their results on the behavior of Gaussian curvature \(K\) near singular points of surfaces show that \(K\) is generically unbounded near cuspidal edges and swallowtails, and on the special occasions that \(K\) is bounded, the shape of the singularities is very restricted. Moreover, they obtain a geometric formula for the Maslov index or zigzag number, which is a well known topological invariant for fronts. The paper also gives some results for the geometry of fronts that are hypersurfaces in \(n\)-space. In the final section, a discussion on the intrinsic formulation of singularities of wave fronts is presented.

MSC:

53A05 Surfaces in Euclidean and related spaces
57R45 Singularities of differentiable mappings in differential topology
58K05 Critical points of functions and mappings on manifolds
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