Poincaré–Hopf theorem

E156192

mathematical theorem theorem in differential topology

The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.

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All labels observed (3)

Label	Occurrences
Poincaré–Hopf theorem canonical	2
Hopf index theorem	1
hairy ball theorem on the 2-sphere	1

How this entity was disambiguated

This entity first appeared as the object of triple T1358651 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Poincaré–Hopf theorem
Context triple: [Henri Poincaré, notableWork, Poincaré–Hopf theorem]

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
E. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Poincaré–Hopf theorem
Target entity description: The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.

A. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
B. Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
E. Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in differential topology ⓘ
appliesTo	compact differentiable manifold ⓘ continuous vector fields with isolated zeros ⓘ smooth manifold ⓘ smooth vector fields ⓘ
assumes	vector field with isolated zeros ⓘ
category	global differential geometry result ⓘ topology theorem ⓘ
conclusion	sum of indices of zeros equals Euler characteristic ⓘ
coreIdea	topological invariant equals sum of local differential invariants ⓘ
example	Poincaré–Hopf theorem self-linksurface differs ⓘ surface form: hairy ball theorem on the 2-sphere
field	differential geometry ⓘ differential topology ⓘ
generalizationOf	results on indices of planar vector fields ⓘ
hasConsequence	existence of nowhere-vanishing vector fields on tori ⓘ nonexistence of nowhere-vanishing tangent vector fields on even-dimensional spheres ⓘ
historicalPeriod	20th century mathematics ⓘ
holdsFor	compact manifolds with boundary under suitable conditions ⓘ compact oriented manifolds ⓘ
implies	existence of zeros of vector fields on manifolds with nonzero Euler characteristic ⓘ
invariantUnder	homotopy of vector fields avoiding creation or annihilation of zeros on the boundary ⓘ
namedAfter	Heinz Hopf ⓘ Henri Poincaré ⓘ
relatedTo	Brouwer fixed-point theorem ⓘ Gauss–Bonnet theorem (early form) ⓘ surface form: Gauss–Bonnet theorem Lefschetz fixed-point theorem ⓘ Morse theory ⓘ characteristic classes ⓘ degree of a map ⓘ tangent bundle ⓘ
relatesConcept	Euler characteristic ⓘ index of a vector field ⓘ isolated zero of a vector field ⓘ vector field ⓘ
requires	Euler characteristic of a topological space ⓘ notion of index of an isolated singularity of a vector field ⓘ
specialCaseOf	Atiyah–Singer index theorem ⓘ
statementForm	sum of indices of isolated zeros of a vector field on a compact manifold equals the Euler characteristic of the manifold ⓘ
topic	global analysis on manifolds ⓘ index theory ⓘ
usedFor	computing Euler characteristic via vector fields ⓘ obstructing existence of nonvanishing vector fields ⓘ
usedIn	Riemannian manifolds ⓘ surface form: Riemannian geometry algebraic topology ⓘ dynamical systems ⓘ foliation theory ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Henri Poincaré → notableWork → Poincaré–Hopf theorem ⓘ

Atiyah–Singer index theorem → generalizes → Poincaré–Hopf theorem ⓘ

this entity surface form: Hopf index theorem

Poincaré–Hopf theorem → example → Poincaré–Hopf theorem self-linksurface differs ⓘ

this entity surface form: hairy ball theorem on the 2-sphere

Lefschetz fixed-point theorem → relatedTo → Poincaré–Hopf theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (4)

How this entity was disambiguated

How these facts were elicited