Euclidean transnormality

Talaat, Ramy Mohamed Kamel (1981) Euclidean transnormality. University of Southampton, Doctoral Thesis.

Abstract

The concept of transnormality generalises the idea of constant width for closed plane curves and closed surface in Euclidean space. The idea of constant width in this sense originated in the work of Euler on ovals, and was developed from the nineteenth century in two directions : towards the general theory of convexity on the one hand, and into differential geometry on the other. The latter line of development has grown over the past fifteen years into the theory of transnormality. This thesis makes a number of contributions to the Euclidean aspects of this theory. It is shown that the generating frame of any transnormal torus or Klein bottle in Euclidean four-dimensional space is a rectangle. The same conclusion holds for any compact orientable transnormal n-manifold of codimension two whose space of normal planes has Abelian fundamental group. A second group of theorems relates to the rectangular product construction, under which the family of transnormal manifolds, and its subfamilies of gradient manifolds and pernormal manifolds are closed. The behaviour of Euler planes of pernormal manifolds is explored in relation to the structure of the generating frame, with explicit calculations in low dimensions. The thesis concludes with a study of transnormal manifolds with axial or helical symmetry.

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