We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.
In this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.
The classical singularity theory deals with singularities of various mathematical objects: curves and surfaces, mappings, solutions of differential equations, etc. In particular, singularity theory treats the tasks of recognition, description and classification of singularities in each of these cases. In many applications of singularity theory it is important to sharpen its basic results, making them "quantitative", i.e. providing explicit and effectively computable estimates for all the important...
A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.
La caustique d?un point sur une variété riemannienne est l?ensemble des points d?intersection des géodésiques infiniment voisins partant de ce point. Jacobi a remarqué, en utilisant un raisonnement topologique, que la caustique d?un point sur une surface convexe fermée doit avoir des points de rebroussement. Il a aussi annoncé (sans démonstration) que le nombre de ces points est quatre pour les caustiques sur les surfaces d?ellipsoïdes (Jacobi, 1964). Dans cette note j?essaie d?inclure les théorèmes...