Affine Lines in Spheres

Because of the hairy ball theorem, the only closed 2-manifold that supports a lattice in its tangent space is T . But, if singular points (i.e. points whose tangent space is not endowed with 2 distinct coordinate directions) are allowed, then it becomes possible to give the tangent space a lattice. Because the lattice is well defined everywhere around the points, the effect of moving around the singular points must map the lattice in the tangent space to itself, and as such, the holonomy of the points must be in SL(2,Z). This endows the manifold (minus the singular points) with an integral affine structure. This gives every point a sense of direction, but is not quite a metric (for that the holonomy of every point would also have to be in SO(2,R), giving each singularity a finite amount of Gaussian curvature). Thus, one can have paths in the manifold that are “straight lines” in some sense, which we refer to affine lines. The choices of the types of singular points yields varying affine structures. This paper investigates visualizing one particular type of construction of the the singular points for S given in the first two sections of “Integral Affine Structures on Spheres I” by Christian Haase and Ilia Zharkov [1] for the case where n = 2, as well as the affine lines in a particular example. This yields insight to the more general case.