A global version of a classical result of Joachimsthal

Guilfoyle, B.; Klingenberg, W.

Abstract

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.

Citation

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

• Journal Article Type: Article
• Acceptance Date: Dec 5, 2018
• Publication Date: 2019
• Deposit Date: Dec 6, 2018
• Publicly Available Date: Dec 7, 2018
• Journal: Houston journal of mathematics
• Print ISSN: 0362-1588
• Publisher: University of Houston
• Peer Reviewed: Peer Reviewed
• Volume: 45
• Issue: 2
• Pages: 455-467
• Public URL: https://durham-repository.worktribe.com/output/1312009
• Publisher URL: https://www.math.uh.edu/~hjm/Vol45-2.html