Degenerate Umbilic Points of Analytic Surfaces

Abstract

Umbilics are points of a surface embedded in three space where normal curvatures are independent of direction. The (in)famous Carath\'{e}odory Conjecture states that a compact simply connected embedded surface has at least two umbilic points. A counterexample to this conjecture would be a surface whose principal foliation has index two at a single umbilic. All (purported) proofs of the Carath\'{e}odory Conjecture are based on analyses of the index of an umbilic, concluding that it is at most one. This investigation gives a much simpler geometric argument that the index of an umbilic on an analytic surface cannot be an integer larger than one, providing new insight into the Carath\'{e}odory Conjecture. The results also establish lower bounds for the index of an umbilic based on its degeneracy.