Poincaré-Hopf theorem
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf.
The Poincaré–Hopf theorem is often illustrated by the special case of the Hairy ball theorem, which simply states that there is no smooth vector field on a sphere having no sources or sinks.
Let M be a differentiable manifold, of dimension n, and v a vector field on M. Suppose that x is an isolated zero of v, and fix some local coordinates near x. Pick a closed ball D centered at x, so that x is the only zero of v in D. Then we define the index of v at x, indexx(v), to be the degree of the map u:∂D→Sn-1 from the boundary of D to the (n-1)-sphere given by u(z)=v(z)/| v(z) |.
Theorem. Let M be a compact differentiable manifold. Let v be a vector field on M with isolated zeroes. If M has boundary, then we insist that v be pointing in the outward normal direction along the boundary. Then we have the formula
where the sum of the indices is over all the isolated zeroes of v and is the Euler characteristic of M. A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0.
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