Lenses on very curved zones of a singular foliation of C2

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 .

Entropy, transverse entropy and partitions of unity

AbstractThe topological entropy of a transformation is expressed in terms of partitions of unity. The transverse entropy of a flow tangential to a foliation is defined and expresed in a similar way. The geometric entropy of a foliation of a Riemannian manifold is compared with the transverse entropy of its geodesic flow.

Espace de Minkowski-Lorentz et des sphères : un état de l'art

International audience; Dans cet article, nous faisons une présentation de l'espace de Minkowski-Lorentz généralisant, à Ê 5 l'espace utilisé dans la théorie de la relativité. Cet espace de dimension 5 contient un paraboloïde de dimension 3 et isométrique à l'espace affine euclidien usuel E 3 , l'ensembles des sphères et plans orientés de E 3 regroupés sur une pseudo-sphère unité de dimension 4. Une premier avantage de cet espace est l'écriture intuitive d'une sphère qui est caractérisée par un point, un vecteur normal en ce point et une courbure. Un deuxième avantage est la manipulation de surfaces canal qui sont représentées par des courbes. Un troisième avantage concernant la simplificat…

Foliations of $\mathbb{S}^3$ by Cyclides

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…

THE ZONE MODULUS OF A LINK

In this paper, we construct a conformally invariant functional for two-component links called the zone modulus of the link. Its main property is to give a sufficient condition for a link to be split. The zone modulus is a positive number, and its lower bound is 1. To construct a link with modulus arbitrarily close to 1, it is sufficient to consider two small disjoint spheres each one far from the other and then to construct a link by taking a circle enclosed in each sphere. Such a link is a split link. The situation is different when the link is non-split: we will prove that the modulus of a non-split link is greater than [Formula: see text]. This value of the modulus is realized by a spec…

The geometry of canal surfaces and the length of curves in de Sitter space

Abstract We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.

Holomorphic Maps and Pencils of Circles

(2008). Holomorphic Maps and Pencils of Circles. The American Mathematical Monthly: Vol. 115, No. 8, pp. 690-700.

Blending canal surfaces along given circles using Dupin cyclides

We study blends between canal surfaces using Dupin cyclides via the space of spheres. We have already studied the particular case where it is possible to blend two canal surfaces using one piece of Dupin cyclide bounded by two characteristic circles, but this is not possible in the general case. That is why we solve this problem using two pieces of different cyclides, which is always possible. To get this conclusion and give the algorithms allowing to obtain such a result, we study, at first, the blend between two circles by a piece of cyclide. We impose to the cyclide to be tangent to a given sphere containing one of the circles. We give the existence condition on the previous circles to h…

Application of spaces of subspheres to conformal invariants of curves and canal surfaces

Habitat geometry of benthic substrata: effects on arrival and settlement of mobile epifauna

Abstract The effect of substratum complexity on the early stages of colonization by mobile epifauna was assessed through a comparative study based on the architecture of artificial substrata. We conducted field observations over 4 years, on six types of small plastic substrata placed in the low intertidal zone of an exposed rocky shore, for varied immersion periods (1, 2, 4 and 12 wk). The use of artificial substrata allowed us to manipulate independently structural and spatial features of the habitat, such as total area, amount of folds, intercepting area, total volume, and interstitial volume. The invertebrate fauna colonizing over 300 sample units was recorded, and their densities compar…

On bounds for total absolute curvature of surfaces in hyperbolic 3-space

Abstract We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space). To cite this article: R. Langevin, G. Solanes, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Osculating spheres to a family of curves.

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.

The non-degenerate Dupin cyclides in the space of spheres using Geometric Algebra

International audience; Dupin cyclides are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century and \textcolor{black}{were} introduced in CAD by R. Martin in 1982. A Dupin cyclide can be defined, in two different ways, as the envelope of a one-parameter family of oriented spheres. So, it is very interesting to model the Dupin cyclides in the space of spheres, space wherein each family of spheres can be seen as a conic curve. In this paper, we model the non-degenerate Dupin cyclides and the space of spheres using Conformal Geometric Algebra. This new approach permits us to benefit from the advantages of the use of Geometric Alge…

Blending Planes and Canal Surfaces Using Dupin Cyclides

We develop two different new algorithms of G1-blending between planes and canal surfaces using Dupin cyclides. It is a generalization of existing algorithms that blend revolution surfaces and planes using a plane called construction plane. Spatial constraints were necessary to do that. Our work consist in building three spheres to determine the Dupin cyclide of the blending. The first algorithm is based on one of the definitions of Dupin cyclides taking into account three spheres of the same family enveloping the cyclide. The second one uses only geometric properties of Dupin cyclide. The blending is fixed by a circle of curvature onto the canal surface. Thanks to this one, we can determine…

Differential Geometry of Curves and Surfaces

The goal of this article is to present the relation between some differential formulas, like the Gauss integral for a link, or the integral of the Gaussian curvature on a surface, and topological invariants like the linking number or the Euler characteristic.

Iterative construction of Dupin cyclides characteristic circles using non-stationary Iterated Function Systems (IFS)

International audience; A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by J.C. Fiorot. As we su…

Iterative constructions of central conic arcs using non-stationary IFS

Several methods of subdivision exist to build parabola arcs or circle arcs in the usual Euclidean affine plane. Using a compass and a ruler, it is possible to construct, from three weighted points, circles arcs in the affine space without projective considerations. This construction is based on rational quadratic Bézier curve properties. However, when the conic is an ellipse or a hyperbola, the weight computation is relatively hard. As the equation of a conic is $\qaff(x,y)=1$, where $\qaff$ is a quadratic form, one can use the pseudo-metric associed to $\qaff$ in the affine plane and then, the conic geometry is also handled as an Euclidean circle. At each step of the iterative algorithm, t…

Espace de Minkowski-Lorentz et des sphères : un état de l’art

International audience; Dans cet article, nous faisons une présentation de l'espace de Minkowski-Lorentz généralisant, a E 5 l'espace utilise dans la théorie de la relativité. Cet espace de dimension 5 contient un paraboloïde de dimension 3 et isométrique a l'espace affine euclidien usuel E 3 , l'ensemble des sphères et plans orientes de E 3 regroupes sur une pseudo-sphère unité de dimension 4. Une premier avantage de cet espace est l'écriture intuitive d'une sphère qui est caractérisée par un point, un vecteur normal en ce point et une courbure. Un deuxième avantage est la manipulation de surfaces canal qui sont représentées par des courbes. Un troisième avantage concernant la simplificati…

Canal foliations of S 3

The goal of the article is to classify foliations of S3 by regular canal surfaces, that is envelopes of one-parameter families of spheres which are immersed surfaces. We will add some extra information when the leaves are “surfaces of revolution” in a conformal sense.

Fenchel type theorems for submanifolds of S n

Darboux curves on surfaces I

International audience; In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable w…

Integral geometry from Buffon to geometers of today

La géométrie intégrale, aussi appelée théorie des probabilités géométriques, a accompagné pendant plus de deux siècles le développement des probabilités, de la théorie de la mesure et de la géométrie. Elle commence pour nous en 1777, date de la publication du « traité d'arithmétique morale » de Buffon. Ce n'est que presque un siècle plus tard que Crofton explicitera ce que veut dire mettre une mesure sur un ensemble continu comme l'ensemble des droites. Le sens de la formule de Cauchy-Crofton « la longueur d'une courbe plane est proportionnelle à la mesure pondérée de l'ensemble des droites qui la coupent », est maintenant clair. Au début du vingtième siècle, la géométrie intégrale commence…

Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension

AbstractWe construct an example of transitive Anosov flow on a compact 3-manifold, which admits a transversal torus and is not the suspension of an Anosov diffeomorphism.

Conformal invariance of the writhe of a knot

We give a new proof of an old theorem by Banchoff and White 1975 that claims that the writhe of a knot is conformally invariant.

Foliations making a constant angle with principal directions on ellipsoids

Lenses on very curved zones of a singular line field of ${\mathbb C}^2$ or of a singular plane field of ${\mathbb C}^3$

We renormalize, using suitable lenses, small domains of a singular holomorphic line field of ${\mathbb C}^2$ or plane field of ${\mathbb C}^3$ where the curvature of a plane-field is concentrated. At a proper scale the field is almost invariant by translations. When the field is integrable, the leaves are locally almost translates of a surface that we will call {\it profile}. When the singular rays of the tangent cone (a generalization to a plane-field of the tangent cone of a singular surface is defined) are isolated, we obtain more precise results. We also generalize a result of Merle (\cite{Me}) concerning the contact order of generic polar curves with the singular level $f=0$ when $\ome…

Espace de Minkowski-Lorentz et des sphères : un état de l'art

Dans cet article, nous faisons une présentation de l'espace de Minkowski-Lorentz généralisant, à Ê 5 l'espace utilisé dans la théorie de la relativité. Cet espace de dimension 5 contient un paraboloïde de dimension 3 et isométrique à l'espace affine euclidien usuel E 3 , l'ensembles des sphères et plans orientés de E 3 regroupés sur une pseudo-sphère unité de dimension 4. Une premier avantage de cet espace est l'écriture intuitive d'une sphère qui est caractérisée par un point, un vecteur normal en ce point et une courbure. Un deuxième avantage est la manipulation de surfaces canal qui sont représentées par des courbes. Un troisième avantage concernant la simplification des calculs quadrati…

Topological canal foliations

Regular canal surfaces of $\mathbb{R}^3$ or $\mathbb{S}^3$ admit foliations by circles: the characteristic circles of the envelope. In order to build a foliation of $\mathbb{S}^3$ with leaves being canal surfaces, one has to relax the condition “canal” a little (“weak canal condition”) in order to accept isolated umbilics. Here, we define a topological condition which generalizes this “weak canal” condition imposed on leaves, and classify the foliations of compact orientable 3-manifolds we can obtain this way.

Gluing Dupin cyclides along circles, finding a cyclide given three contact conditions.

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.