Surfaces in ℝ⁴ and duality

Bruce, J.W. and Noguiera, A.C. (1998) Surfaces in ℝ⁴ and duality. The Quarterly Journal of Mathematics, 49. pp. 433-443. ISSN 0033-5606 DOI https://doi.org/10.1093/qmathj/49.4.433

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Abstract

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]

Item Type: Article
Subjects: Q Science > QA Mathematics
Date Deposited: 17 Jan 2011 13:39
URI: http://repository.edgehill.ac.uk/id/eprint/2233

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