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A global version of a classical result of Joachimsthal.

Guilfoyle, B. and Klingenberg, W. (2019) 'A global version of a classical result of Joachimsthal.', Houston journal of mathematics., 45 (2). pp. 455-467.


A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove the following global analogue of this result. Suppose that two closed convex surfaces intersect with constant angle along a curve that is not umbilic in either surface. We prove that the principal foliations of the two surfaces along the curve are either both orientable, or both non-orientable. We prove this by characterizing the constant angle intersection of two surfaces in Euclidean 3-space as the intersection of a Lagrangian surface and a foliated hypersurface in the space of oriented lines, endowed with its canonical neutral Kähler structure. This establishes a relationship between the principal directions of the two surfaces along the intersection curve in Euclidean space. A winding number argument yields the result. The method of proof is motivated by topology and, in particular, the slice problem for curves in the boundary of a 4-manifold.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Date accepted:05 December 2018
Date deposited:07 December 2018
Date of first online publication:2019
Date first made open access:07 December 2018

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