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A global version of a classical result of Joachimsthal

Guilfoyle, B.; Klingenberg, W.

A global version of a classical result of Joachimsthal Thumbnail


Authors

B. Guilfoyle



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
Publisher URL https://www.math.uh.edu/~hjm/Vol45-2.html

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