DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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A smooth differential $k$ -form

Definition

Let $M$ be a smooth manifold and $k \geq 0$ . A smooth differential $k$ -form on $M$ is a smooth section of $⋀^{k} T^{*} M \to M$ . The space of such forms is denoted $Ω^{k} (M)$ , and $Ω^{0} (M) = C^{\infty} (M)$ .

Depends on

Used by

Smooth functions and tensor fields extend locally across the boundary Corollary
A graded derivation of the algebra of differential forms Definition
Bigraded complex forms and the Dolbeault operators Definition
Compact Kähler and rational Hodge polarization conventions Definition
Compact support of a differential form Definition
Connection one-form of an oriented orthonormal frame Definition
De rham cochain complex Definition
E-valued Dolbeault operators and their L2 structure Definition
Finite-dimensional vector-valued forms and their exterior derivative Definition
Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
Integration along the unit interval for a differential form Definition
Interior product of a form by a vector field Definition
The exterior derivative by the invariant vector-field formula Definition
The Lie derivative of a differential form Definition
The pullback of a differential form Definition
The wedge product of differential forms Definition
Flat torus harmonic forms are constant-coefficient forms Example
The canonical one-form on a cotangent bundle as a covariant tensor Example
Divisibility by a nowhere-vanishing one-form Lemma
Rational cohomology scalar comparison on compact smooth manifolds Lemma
Stokes formula for finite ordinary surface corners Lemma
The cut surface, primitives of closed forms, and their boundary jumps Lemma
The de rham mayer vietoris sequence is exact at the first two terms Lemma
The Euler characteristic of the structure sheaf is one minus the genus Lemma
Trace of a holomorphic differential along a nonconstant map to the sphere Lemma
Weak solutions of a degree-zero divisor and the logarithmic-derivative identity Lemma
Local coordinate expression for a differential form Proposition
Vector-bundle curvature is an endomorphism-valued two-form Proposition
Chern connection of a Hermitian holomorphic line bundle Theorem
Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
Orientability is equivalent to a nowhere-vanishing top form Theorem
The symplectic period formula for integrals of wedge products Theorem

Dependency tree · two levels

6 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)

A smooth differential kk-form