Math 2.0

Manifolds and cell complexes Differential topology: smooth manifolds and maps Theorem

Poincaré–Hopf theorem

Builds on Smooth manifoldVector fieldEuler characteristicDegree of a continuous map

Statement

Let MM be a compact smooth manifold of dimension nn without boundary. Let vv be a continuous vector field on MM with only isolated zeros. For a zero xx of vv, choose a chart around xx and a small sphere SS around xx in it that encloses no other zero. The index ind⁡x(v)\operatorname{ind}_x(v) is the degree of the map S→Sn−1S \to S^{n-1}, u↦v(u)/∥v(u)∥u \mapsto v(u) / \norm{v(u)}. Then vv has finitely many zeros, and ∑v(x)=0ind⁡x(v)=χ(M),\sum_{v(x) = 0} \operatorname{ind}_x(v) = \chi(M), the Euler characteristic of MM.

Remarks

The index does not depend on the chart. One proof first shows that the sum of indices does not depend on vv. For this it compares vv with a field on a tubular neighbourhood of MM in ℝN\R^N and uses the degree of the Gauss map. Then one computes the sum for the gradient of a Morse function, whose zeros are the critical points. The index at a critical point of Morse index λ\lambda is (−1)λ(-1)^\lambda, and the sum of these signs is χ(M)\chi(M) by Morse theory. For a manifold with boundary the same formula holds if vv points outward along the boundary.

Example: on the sphere S2S^2, the gradient of the height function vanishes only at the two poles, a source and a sink, each of index +1+1, and 1+1=2=χ(S2)1 + 1 = 2 = \chi(S^2). On the torus, the field tangent to one family of circles has no zero, and χ(T2)=0\chi(T^2) = 0.

Compactness is needed: the constant field on ℝn\R^n has no zero, but χ(ℝn)=1\chi(\R^n) = 1.

Going deeper

For a student

The theorem links analysis, through vector fields, with topology, through the Euler characteristic. It gives the hairy ball theorem at once. It shows that a closed manifold has a nowhere vanishing vector field only if χ(M)=0\chi(M) = 0, and Hopf proved the converse for connected closed manifolds. It can be derived from the Lefschetz fixed-point theorem applied to the flow of vv. It is closely related to the Gauss–Bonnet theorem and to the Euler class of the tangent bundle.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Degree of a continuous map, concept, no evidence levelDegree of a continuou…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectEuler characteristic, concept, no evidence levelEuler characteristicconceptFunction, concept, no evidence levelFunctionconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptVector-valued function, concept, no evidence levelVector-valued functionconceptVector field, concept, no evidence levelVector fieldconceptPoincaré–Hopf theorem, statement, no evidence levelPoincaré–Hopf theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Degree of a continuous map concept
  2. Euclidean space object
  3. Euler characteristic concept
  4. Function concept
  5. Smooth manifold concept
  6. Vector-valued function concept
  7. Vector field concept
  8. Poincaré–Hopf theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincaré–Hopf_theorem
About this page
Tag4XTPY82PP4
URIm2://4XTPY82PP4
Kindstatement, theorem
WritingW0 stub
MSC57R25, 55M25
StatusThis copy of the site has no build data. Run m2 status 4XTPY82PP4.
AboutVector field, Euler characteristic
UsesSmooth manifold, Vector field, Euler characteristic, Degree of a continuous map
Same as
Changed2 November 2026

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