Poincaré–Hopf theorem
E156192
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Poincaré–Hopf theorem canonical | 3 |
| Hopf index theorem | 1 |
| Poincaré–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.
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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.
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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.
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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 \(\mathbb{R}^{2n}\)).
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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 |
hairy ball theorem on the 2-sphere
ⓘ
linked to:
Poincaré–Hopf theorem
|
| 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 ⓘ
linked to:
Gauss–Bonnet theorem (early form)
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 geometry
ⓘ
linked to:
Riemannian manifolds
algebraic topology ⓘ dynamical systems ⓘ foliation theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Poincaré–Hopf theorem Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Poincaré–Hopf theorem
linked to: Poincaré–Hopf theorem
linked to: Poincaré–Hopf theorem