Ana Claudia Nabarro

PROJECTIONS OF HYPERSURFACES IN R-4 WITH BOUNDARY TO PLANES

We study orthogonal projections of generic embedded hypersurfaces in R 4 with boundary to 2-spaces. Therefore we classify simple map germs from R 3 to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.

Families of surfaces and conjugate curve congruences

Given a smooth and oriented surface M in the Euclidean space R 3 , the conjugate curve congruence Cis a family of pairs of foliations on M that links the lines of curvature and the asymptotic curves of M. This family is first introduced in (21) and is studied in (8, 13). When the surface M = M0 is deformed in a 1-parameter family of surfaces Mt, we obtain a 2-parameter family of conjugate curve congruence C�,t. We study in this paper the generic local singularities in C�0,0 and the way they bifurcate in the family C�,t, with (�,t) close to (�0,0).

Families of curve congruences on Lorentzian surfaces and pencils of quadratic forms

We dene and study in this paper families of conjugate and reected curve congruences associated to a self-adjoint operator A on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the A-asymptotic and the A-principal curves, and the pencil joining the A-characteristic and the A-principal curves respectively. The binary dierential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE and of its A-conjugate equation, and on BDEs with orthogonal roots.

Second order geometry of spacelike surfaces in de Sitter 5-space

The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space S5 1 by using the action of GL(2,R) X SO(1,2) on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the LMN-ellipse. This ellipse gives information on the lightlike binormal directions and consequently about their associated asymptotic directions.