Maria Aparecida Soares Ruas

The geometry of surfaces in 4-space from a contact viewpoint

We study the geometry of the surfaces embedded in ℝ4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ℝ4 has inflection points.

Inflection points and topology of surfaces in 4-space

We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincare-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.

Osculating Hyperplanes and Asymptotic Directions of Codimension Two Submanifolds of Euclidean Spaces

We define the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces and obtain necessary conditions, in terms of the existence of such directions, for the convexity and the sphericity of these submanifolds.

Total absolute horospherical curvature of submanifolds in hyperbolic space

We study the horospherical geometry of submanifolds in hyperbolicnspace. The main result is a formula for the total absolutenhorospherical curvature of

Topological triviality of families of functions on analytic varieties

M,

Generic sections of essentially isolated determinantal singularities

which implies, for the horosphericalngeometry, the analogues of classical inequalities of the EuclideannGeometry. We prove the horospherical Chern-Lashof inequality fornsurfaces in

Differential geometry from a singularity theory viewpoint

3

BILIPSCHITZ DETERMINACY OF QUASIHOMOGENEOUS GERMS

-space and the horospherical Fenchel andnFary-Milnors theorems.

Invariants of topological relative right equivalences

We present sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V . The main result is an infinitesimal criterion using the integral closure of a convenient ideal as the tangent space to a subset of the set of topologically trivial deformations of a given germ. Results of M. Saia [20] on the determination of the integral closure of an ideal in terms of its Newton Polyhedron are used to describe the topological triviality of Newton non degenerate families of map germs. Applications to the problem of equisingularity of families of sections of V are also discussed. March, 2001 ICMC-USP

Determinacy of determinantal varieties

We study the essentially isolated determinantal singularities (EIDS), defined by W. Ebeling and S. Gusein-Zade, as a generalization of isolated singularity. We prove in dimension