Lecture 51 Poincaré Hopf theorem

We extend the definition of the index of a vector field at a singular point to the case of isolated singularities. We present and show the Poincaré-Hopf theorem: the sum of the indexes of a vector field at its singular points is equal to the Euler characteristic (where the manifold is compact without boundary and the vector field has isolated singularities).

We see two alternative definitions of the Euler characteristic for the case of compact surfaces and their equivalence to our definition (the Euler characteristic is the self-intersection of the zero section of the tangent bundle). The first formula expresses the Euler characteristic in terms of the genus of the surface whereas the second one is the Euler's formula associated to a triangulation of the surface.

Finally, we show that a compact manifold without boundary has a vector field with no singular points if and only if the Euler characteristic is 0.

Lecture notes: