### 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]

Original language | English |
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Pages (from-to) | 433-443 |

Journal | The Quarterly Journal of Mathematics |

Volume | 49 |

DOIs | |

Publication status | Published - 1998 |

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## Cite this

Bruce, J. W., & Noguiera, A. C. (1998). Surfaces in ℝ⁴ and duality.

*The Quarterly Journal of Mathematics*,*49*, 433-443. https://doi.org/10.1093/qmathj/49.4.433