Projections of surfaces with a connected fold curve

Abstract

Let S be a surface (compact, connected and without boundary) and ƒ: S → R 2 a generic smooth mapping. Suppose the apparent contour γ is irreducible (which means the fold curve of ƒ is connected). We give a criterion to decide if the number of singularities in the contour is the least possible or not, and we see that these minimal values for the cusps and double points of γ depend on the the Euler-Poincaré characteristic χ( S) of the surface. In the course of our proof, we illustrate the construction of such “minimal” projections for surfaces of arbitrary genus. All operations involved can be performed by using manifolds immersed in R 3.