DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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A smooth vector field is a smooth section of the tangent bundle

Definition

Assume ${A C}_{ω}$ , so that $T M$ carries its canonical smooth structure. Let $M$ be a smooth manifold. A smooth vector field on $M$ is a smooth section

$X : M \to T M$

of the tangent-bundle projection $π : T M \to M$ . Thus $π \circ X = {id}_{M}$ .

Equivalently, for each $p \in M$ the value $X (p)$ is a tangent vector in $T_{p} M$ , and the dependence on $p$ is smooth with respect to the canonical smooth structure on $T M$ .

Depends on

Used by

An inward radial field violates the outward boundary formula Counterexample
Downward gradient-like vector fields for a Morse function Definition
F-related vector fields Definition
Fundamental vector fields for a left action Definition
Integral curves of a vector field Definition
Isolated zero and local index of a vector field Definition
Left- and right-invariant vector fields Definition
Nondegenerate zero of a vector field Definition
Principal connections and vertical curvature Definition
The action of a vector field on smooth functions Definition
The Bott partial connection on the normal bundle of a foliation Definition
The Riemannian gradient is the metric dual of the differential Definition
Time-dependent vector fields and their evolution operators Definition
Vector fields tangent to a distribution Definition
A nowhere-zero vector field on an odd sphere Example
Source, sink and saddle indices on a surface Example
The hairy-ball theorem for even spheres Example
The outward radial field on a disk Example
FALSE: every pointwise assignment of a tangent vector is a smooth vector field False statement
A vector field along an embedded submanifold extends to a neighbourhood and globally when the submanifold is closed Lemma
Isotopy extension for a compact source with boundary Lemma
A smooth distribution is exactly a locally framed constant-rank family of tangent spaces Proposition
A vector field tangent to an embedded submanifold restricts to a vector field on it Proposition
Smoothness of a vector field is equivalent to smooth coordinate components Proposition
Compactly supported smooth vector fields are complete Theorem
Converse Poincare-Hopf for nowhere-zero fields Theorem
Derivations of smooth functions are exactly smooth vector fields Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)