  We show that Caratheodory's conjecture, on umbilical points of closed convex
surfaces, may be reformulated in terms of the existence of at least one umbilic
in the graphs of functions f: R^2-->R whose gradient decays uniformly faster
than 1/r. The divergence theorem then yields a pair of integral equations for
the normal curvatures of these graphs, which establish some weaker forms of the
conjecture. In particular, we show that there are uncountably many principal
lines in the graph of f whose projection into R^2 are parallel to any given
direction.
