Hodge Laplacian

E577497

differential operator elliptic differential operator geometric operator

The Hodge Laplacian is a differential operator on differential forms of a Riemannian manifold that combines the exterior derivative and its adjoint to study harmonic forms and de Rham cohomology.

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Observed surface forms (2)

Surface form	Occurrences
Bochner Laplacian	1
Laplace–Beltrami operator	1

Statements (48)

Predicate	Object
instanceOf	differential operator ⓘ elliptic differential operator ⓘ geometric operator ⓘ
actsOn	differential forms ⓘ
alsoKnownAs	Hodge–de Rham Laplacian NERFINISHED ⓘ Laplace–de Rham operator NERFINISHED ⓘ
associatedWith	Hodge heat equation NERFINISHED ⓘ Hodge–de Rham complex NERFINISHED ⓘ
bundleVersionActsOn	exterior algebra bundle of cotangent bundle ⓘ
characterizes	harmonic forms ⓘ
codifferentialDefinedAs	δ = (−1)^{n(k+1)+1} * d * on k-forms in dimension n ⓘ
commutesWith	pullback by isometries ⓘ
definedOn	Riemannian manifold ⓘ
definition	Δ = d δ + δ d ⓘ
dependsOn	Hodge star operator NERFINISHED ⓘ Riemannian metric ⓘ
domain	space of smooth differential forms ⓘ
eigenformsCalled	Laplacian eigenforms ⓘ
field	Hodge theory NERFINISHED ⓘ Riemannian geometry NERFINISHED ⓘ differential geometry ⓘ global analysis ⓘ
generalizes	Laplace–Beltrami operator NERFINISHED ⓘ
historicallyNamedAfter	W. V. D. Hodge NERFINISHED ⓘ
isElliptic	true ⓘ
isInvariantUnder	Riemannian isometries ⓘ
isNonNegative	true ⓘ
isSelfAdjoint	true ⓘ
kernelConsistsOf	harmonic forms ⓘ
linearity	linear operator ⓘ
localExpressionDependsOn	Levi-Civita connection NERFINISHED ⓘ
order	2 ⓘ
property	kernel on k-forms is isomorphic to k-th de Rham cohomology group on compact manifolds ⓘ
reducesTo	Laplace–Beltrami operator on functions NERFINISHED ⓘ
relatedTheory	Hodge decomposition ⓘ Hodge theorem NERFINISHED ⓘ de Rham cohomology NERFINISHED ⓘ
requires	orientation to define codifferential via Hodge star ⓘ
spectrum	discrete on compact manifolds ⓘ
symbol	Δ ⓘ
type	second-order linear elliptic operator on vector bundles ⓘ
usedFor	Hodge decomposition of differential forms ⓘ index theory ⓘ spectral geometry ⓘ study of heat kernel on forms ⓘ study of topology via analysis ⓘ
usesOperator	codifferential ⓘ exterior derivative ⓘ

Referenced by (3)

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

Laplace operator → generalization → Hodge Laplacian ⓘ

Dirac operator → squareRelatesTo → Hodge Laplacian ⓘ

this entity surface form: Laplace–Beltrami operator

Dirac operator → squareRelatesTo → Hodge Laplacian ⓘ

this entity surface form: Bochner Laplacian