0000000000034562
AUTHOR
Maria Aparecida Soares Ruas
The geometry of corank 1 surfaces in ℝ4
Abstract We study the geometry of surfaces in ℝ4 with corank 1 singularities. For such surfaces, the singularities are isolated and, at each point, we define the curvature parabola in the normal space. This curve codifies all the second-order information of the surface. Also, using this curve, we define asymptotic and binormal directions, the umbilic curvature and study the flat geometry of the surface. It is shown that we can associate to this singular surface a regular one in ℝ4 and relate their geometry.
Generalized distance-squared mappings of the plane into the plane
Abstract We define generalized distance-squared mappings, and we concentrate on the plane to plane case. We classify generalized distance-squared mappings of the plane into the plane in a recognizable way.
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.
ASYMPTOTIC CURVES ON SURFACES IN ℝ5
We study asymptotic curves on generically immersed surfaces in ℝ5. We characterize asymptotic directions via the contact of the surface with flat objects (k-planes, k = 1 - 4), give the equation of the asymptotic curves in terms of the coefficients of the second fundamental form and study their generic local configurations.
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.