The Poincare-Hopf Theorem relates the sum of the indices of the zeros of a vector field on a compact manifold to the manifold's Euler characteristic, showing this sum is the same for every such vector field with only finitely many, non-degenerate zeros. Named for Henri Poincare and Heinz Hopf, it is a foundational result of differential topology connecting local behavior of vector fields to a single global topological invariant.
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StatementRelates the zeros of a continuous vector field on a compact differential manifold to the Euler characteristic of that manifold. 1 Classification
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Source Poincare-Hopf theorem, Wikipedia
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1. Poincare-Hopf theorem, Wikipedia
Lead paragraph
relates zeros of continuous vector field on compact differential manifold to Euler characteristic of that manifold.
In Branch: Differential Topology, Lead sentence
or Hopf index theorem) is an important theorem in differential topology that relates zeros of continuous vector field on compact d
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