A Global Version of a Classical Result of Joachimsthal

TL;DR

This paper extends Joachimsthal's classical result to a global setting, showing that two convex surfaces intersecting at a constant angle along a non-umbilic curve have principal foliations that are either both orientable or both non-orientable, using a novel geometric-topological approach.

Contribution

It provides a global analogue of Joachimsthal's classical result, characterizing the orientability of principal foliations along intersection curves of convex surfaces.

Findings

Principal foliations along the intersection are either both orientable or both non-orientable.

The intersection of surfaces can be characterized as a Lagrangian surface in the space of oriented lines.

The method links topology, geometry, and the structure of the space of lines in Euclidean space.

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 a global analogue of this result, as follows. 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 surface and a hypersurface in the space of oriented lines. The surface is Lagrangian, while the hypersurface is null, with respect to the canonical neutral Kaehler structure. We establish a relationship between the principal directions of the two surfaces along the intersection curve in Euclidean space, which yields the result. This method of proof is motivated by topology and, in particular, the slice problem for curves in the boundary of a 4-manifold.