Surfaces in ℝ⁴ and duality

IN this paper we study some aspects of the differential geometry of a surface M embedded in Euclidean (or indeed affine) 4-space R 4 • In particular, we shall consider the 3-parameter families of orthogonal projections of M to l-dimensional (resp. 3-dimensional) subspaces of R 4 . It is not hard to show that any projection n : R 4 R k (1 k 3) induces a map M Rk whose local type at any point depends only on the kernel of n. In the case k = 1 (resp. k = 3) these kernels are in 1-1 correspondence with points of the real projective 3-space P (resp. its dual P*) . Note that the above remark concerning the kernels show that there is nothing to be gained from considering more general families of projections. Each family of projections gives rise to a stratification of the space of such. In this paper we show that the dual of various 2-dimensional strata in P (for projections to lines) coincide with certain strata in P* (for projections to planes). This extends earlier results of the first author and Romero-Fuster in [2], and is used to correlate the geometry of the families. Of particular interest is the existence of flat umbilics on generic surfaces in R4 • The duality here gives rise to a blowing up of the set of critical values of a Gauss type map related to, but more general than, the blowing up of the focal set of an umbilic discussed in [3]. The duality result concerning one of the strata was found independently by Mochida in her thesis [9]. As a general background for results in singularity theory we recommend [11], and for those in generic geometry [1, 12]