Singularities of affine equidistants: extrinsic geometry of surfaces in 4-space

Abstract

For a generic embedding of a smooth closed surface $M$ into ${\mathbb{R}}^4$, the subset of ${\mathbb{R}}^4$ which is the affine $\lambda$-equidistant of $M$ appears as the discriminant set of a stable mapping $M\times M\to {\mathbb{R}}^4$, hence their stable singularities are $A_k,k=2,3,4,$ and $C_{2,2}^{\pm }$. In this paper, we characterize these stable singularities of $\lambda$-equidistants in terms of the bi-local extrinsic geometry of the surface, leading to a geometrical study of the set of weakly parallel points on $M$.

22 pages, 5 figures

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