6533b862fe1ef96bd12c636c
RESEARCH PRODUCT
On stability of generic subriemannian caustic in the three-space
Andrei A. AgrachevVladimir M. ZakalyukinGrégoire CharlotJean-paul A. Gauthiersubject
Surface (mathematics)SingularityGeodesicDifferential geometrySettore MAT/05 - Analisi MatematicaMathematical analysisGravitational singularityGeneral MedicineCaustic (optics)Space (mathematics)Projection (linear algebra)Mathematicsdescription
Abstract The singularities of exponential mappings in subriemannian geometry are interesting objects, that are already non-trivial at the local level, contrarily to their Riemannian analogs. The simplest case is the three-dimensional contact case. Here we show that the corresponding generic caustics have moduli at the origin, and the first module that occurs has a simple geometric interpretation. On the contrary, we prove a stability result of the “big wave front”, that is, of the graph of the multivalued arclength function, reparametrized in a certain way. This object is a three-dimensional surface, which has also the natural structure of a wave front. The projection on the three-dimensional space of the singular set of this “big wave front” is nothing but the caustic.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2000-03-01 | Comptes Rendus de l'Académie des Sciences - Series I - Mathematics |