Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare return map has eigenvalue $\leq 1$. What additional information do I need to infer that the eigenvalue is strictly less than 1? I know that in a neighborhood of the closed orbit, the Gaussian curvature is negative. How are such conclusions generally made? Thanks a lot!