0000000000148035
AUTHOR
Jean-claude Sifre
showing 4 related works from this author
Lenses on very curved zones of a singular foliation of C2
2018
Abstract We renormalize, using suitable lenses, small domains of a singular holomorphic foliation of C 2 where the curvature is concentrated. At a proper scale, the leaves are almost translates of a graph that we will call profile. When the leaves of the foliations are levels f = λ , where f is a polynomial in 2 variables, this graph is polynomial. Finally we will indicate how our methods may be adapted to study levels of polynomials and 1-forms in C 3 .
Foliations of $\mathbb{S}^3$ by Cyclides
2018
Throughout the last 2–3 decades, there has been great interest in the extrinsic geometry of foliated Riemannian manifolds (see [2], [4] and [22]). ¶One approach is to build examples of foliations with reasonably simple singularities with leaves admitting some very restrictive geometric condition. For example (see [22], [23] and [17]), consider in particular foliations of $\mathbb{S}^{3}$ by totally geodesic or totally umbilical leaves with isolated singularities. ¶The article [14] provides families of foliations of $\mathbb{S}^{3}$ by Dupin cyclides with only one smooth curve of singularities. Quadrics and other families of cyclides like Darboux cyclides provide other examples. These foliat…
Osculating spheres to a family of curves.
2021
The authors study the extrinsic conformal geometry of space forms involving pencils of circles or spheres. They consider curves orthogonal to a foliation of an open set of a 3-sphere by spheres and prove that the osculating spheres to the curves at points of a leaf form a pencil. They first prove the analogous result in a lower-dimensional case, that is, foliations of the 2-dimensional sphere and their orthogonal foliations. The 3-dimensional result, that is, the result for a foliation of (an open subset of) the 3-dimensional sphere by 2-dimensional spheres, is obtained using the de Sitter space, which is a model for the set of oriented spheres of the 3-dimensional sphere.
Gluing Dupin cyclides along circles, finding a cyclide given three contact conditions.
2013
Dupin cyclides form a 9-dimensional set of surfaces which are, from the viewpoint of differential geometry, the simplest after planes and spheres. We prove here that, given three oriented contact conditions, there is in general no Dupin cyclide satisfying them, but if the contact conditions belongs to a codimension one subset, then there is a one-parameter family of solutions, which are all tangent along a curve determined by the three contact conditions.