Farid Tari

Dupin indicatrices and families of curve congruences

We study a number of natural families of binary differential equations (BDEs) on a smooth surface M in R-3. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDEs, another between the characteristic and principal BDEs. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets ( given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDEs on such a surface M, and show how the pencil of BDEs joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDEs are intimately related.

Bifurcations of implicit differential equations

In this paper we give a local classi¯cation of the integral curves of implicit di®erential equations F (x; y; p) = 0; where F is a smooth function and p = dy=dx, at points where Fp = 0, Fpp =6 0 and where the discriminant f(x; y) : F = Fp = 0g has a Morse singularity. We also produce models for generic bifurcations of such equations and apply the results to the di®erential geometry of smooth surfaces. This completes the local classi¯cation of generic one-parameter families of binary di®erential equations (BDEs)

Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes

We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces

Duality and implicit differential equations

M

FOLDING MAPS ON SPACELIKE AND TIMELIKE SURFACES AND DUALITY

in

Projections of surfaces in R4 to R3 and the geometry of their singular images

H^3_+(-1)

Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics

. We study the generic singularities of the projections of

Differential geometry from a singularity theory viewpoint

M

Extrema of principal curvature and symmetry

to horospheres and planes. We give geometric characterisations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koendrink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.

Real and Complex Singularities: Projections of timelike surfaces in the de Sitter space

We prove some duality results concerning various types of implicit differential equations where F is a smooth function. We show, for instance, that the well folded singularities are self-dual. The results are used to deduce some geometric properties of surfaces in 3-space. So, for example, there are three flecnodal curves at an elliptic flat umbilic and one at a hyperbolic flat umbilic. These curves are tangent to the separatrices of the binary differential equation determining the asymptotic lines.