Publication details for Wilhelm KlingenbergGuilfoyle, B. & Klingenberg, W. (2019). A global version of a classical result of Joachimsthal. Houston Journal of Mathematics 45(2): 455-467.
- Publication type: Journal Article
- ISSN/ISBN: 0362-1588
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
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.