Immersions with fractal set of points of zero Gauss-Kronecker curvature

Abstract.

We construct, for any “good” Cantor set F of S ^n-1, an immersion of the sphere S ⁿ with set of points of zero Gauss-Kronecker curvature equal to F × D ¹, where D ¹ is the 1-dimensional disk. In particular these examples show that the theorem of Matheus-Oliveira strictly extends two results by do Carmo-Elbert and Barbosa-Fukuoka-Mercuri.