A Global Version of a Classical Result of Joachimsthal

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 total number of umbilic points, counted with index, on the discs enclosed by the intersection curve on the two surfaces are equal. 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.