  The Poincare-Hopf Theorem is one of the most used in other areas of science.
There are applications of the Poincare-Hopf Theorem in physics, chemistry,
biology and even in economics, psychology, etc ... The Poincare-Hopf Theorem
connects an invariant of combinatorial, the character of Euler-Poincare to an
invariant of differential geometry, index of vector fields. The results that
connect two very different areas of mathematics can be considered as the most
beautiful, useful and fruitful.
