×

Some stability questions concerning caustics for different propagation laws. (English) Zbl 0824.57022

The authors prove in detail that for any fixed embedding \(h:X\to Y\) and distance function \(d\) (induced either from a Riemannian metric, a Finsler metric, a positively homogeneous Hamiltonian on the dual tangent space, or the square of a topological distance function) there is an open and dense set of structures on \(Y\) for which \(d \cdot (h \times 1_ Y)\) is topologically stable (or, for \(C^ \infty\) data and \(\dim Y \leq 5\), \(C^ \infty\) stable) and so produces topologically stable focal and cut loci. In addition, a caustic is stable for perturbations of the initial wave front iff it is stable with respect to perturbations of the structure within one of the classes given above. They end with stating a stability theorem for the 4-vertex theorem: Since the 4-vertex theorem holds for closed curves on surfaces of constant negative curvature, its focal set has at least 4 cuspidal points for any small enough perturbation of the original metric.

MSC:

57R45 Singularities of differentiable mappings in differential topology
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
PDF BibTeX XML Cite
Full Text: EuDML