Poincare-Hopf index theorem

This was used to derive Eq. (15). A special case of equation (56) was previously used to classify the ways to build a box The Poincare index theorem was extended by Hopf to vector fields on arbitrary manifolds. For vector fields with m isolated hyperbolic critical points, the Poincare-Hopf index theorem is " ... 

Method 4. Index theory approach.. This method is based on the Poincare-Hopf index theorem found in differential topology, see, e.g., Gillemin and Pollack (1974). Similarly to the univalence mapping approach, it requires a certain sign from the Hessian, but this requirement need hold only at the equilibrium point. 

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