### Abstract

Original language | English |
---|---|

Pages (from-to) | 433-443 |

Journal | The Quarterly Journal of Mathematics |

Volume | 49 |

DOIs | |

Publication status | Published - 1998 |

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*The Quarterly Journal of Mathematics*,

*49*, 433-443. https://doi.org/10.1093/qmathj/49.4.433

}

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

**Surfaces in ℝ⁴ and duality.** / Bruce, J.W.; Noguiera, A.C.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Surfaces in ℝ⁴ and duality

AU - Bruce, J.W.

AU - Noguiera, A.C.

PY - 1998

Y1 - 1998

N2 - 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]

AB - 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]

U2 - 10.1093/qmathj/49.4.433

DO - 10.1093/qmathj/49.4.433

M3 - Article

VL - 49

SP - 433

EP - 443

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

ER -