Poincaré–Hopf theorem

For a smooth vector field on a compact manifold with isolated zeros, the sum of the zeros’ indices equals the manifold’s Euler characteristic.

Vector field: A choice of tangent vector at each point of a manifold, varying smoothly when the field is smooth. The theorem begins with a smooth field whose isolated zeros contribute the indices.
Degree of a map: An integer measuring how many times an oriented space is covered by a continuous map, counted with signs. A zero’s index can be computed as the degree of the field’s normalized direction on a surrounding sphere.
Hairy ball theorem: The result that every continuous tangent vector field on an even-dimensional sphere has a zero. The sphere’s nonzero Euler characteristic forces a zero by Poincaré–Hopf.
Henri Poincaré: A French mathematician whose work helped establish topology and dynamical systems as mathematical fields. Poincaré developed early forms of the relation between vector-field indices and topology.
Index of a vector field: An integer assigned to an isolated zero of a vector field, measuring the field’s local winding around it. These local integers are the terms summed in the theorem.
Local degree: The degree associated with a map near an isolated preimage of a point. The theorem’s index at each zero is a local degree.
Poincaré–Hopf theorem on the sphere: The application of the index theorem to a sphere, where the indices of isolated zeros sum to two. It gives the simplest nonzero Euler-characteristic example.
Heinz Hopf: A German-Swiss mathematician known for foundational contributions to topology and differential geometry. Hopf established the theorem in its general manifold form.
Euler characteristic: A topological invariant equal to the alternating sum of cell counts in a finite cell decomposition. The sum of zero indices is constrained to equal this global invariant.
Winding number: An integer counting the signed turns of a closed curve around a point in the plane. In two dimensions, a zero’s index is the winding number of the field around it.

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