The index of isolated umbilics on surfaces of non-positive curvature (Q5963022)

Summary: It is shown that if a \(C^2\) surface \(M \subset \mathbb R^3\) has negative curvature on the complement of a point \(q \in M\), then the \(\mathbb Z/2\)-valued Poincaré-Hopf index at \(q\) of either distribution of principal directions on \(M-\{ q \}\) is non-positive. Conversely, any non-positive half-integer arises in this fashion. The proof of the index estimate is based on geometric-topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincaré-Bendixson theory.