Hodge decomposition

E553414

result in Hodge theory result in differential geometry theorem

Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.

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All labels observed (4)

Label	Occurrences
Hodge decomposition canonical	3
Helmholtz decomposition	1
Hodge–Kodaira decomposition	1
Stokes–Helmholtz decomposition	1

Statements (48)

Predicate	Object
instanceOf	result in Hodge theory ⓘ result in differential geometry ⓘ theorem ⓘ
appliesTo	Riemannian manifolds ⓘ compact Riemannian manifolds ⓘ differential forms ⓘ
assumption	Riemannian metric ⓘ compactness of manifold ⓘ ellipticity of Laplacian ⓘ
conclusion	each de Rham cohomology class has a unique harmonic representative ⓘ every differential form splits into exact, co-exact, and harmonic parts ⓘ harmonic forms represent de Rham cohomology classes ⓘ
context	Riemannian manifolds without boundary ⓘ elliptic partial differential equations ⓘ
domain	smooth differential forms ⓘ space of differential k-forms ⓘ
field	Hodge theory NERFINISHED ⓘ Riemannian geometry ⓘ differential geometry ⓘ global analysis ⓘ
generalizationOf	Helmholtz decomposition NERFINISHED ⓘ
hasVariant	Hodge decomposition with boundary conditions ⓘ L2 Hodge decomposition on non-compact manifolds ⓘ
implies	Hk(M,R) is isomorphic to space of harmonic k-forms ⓘ isomorphism between harmonic forms and de Rham cohomology ⓘ
namedAfter	W. V. D. Hodge NERFINISHED ⓘ
property	finite-dimensional space of harmonic forms on compact manifolds ⓘ orthogonal decomposition with respect to L2 inner product ⓘ uniqueness of decomposition ⓘ
relatedTo	Dolbeault cohomology NERFINISHED ⓘ Hodge numbers ⓘ Hodge theorem NERFINISHED ⓘ Hodge theory NERFINISHED ⓘ Kähler manifolds ⓘ de Rham theorem NERFINISHED ⓘ
usedIn	algebraic geometry ⓘ gauge theory ⓘ mathematical physics ⓘ topology ⓘ
usesConcept	Hodge star operator ⓘ Laplace–Beltrami operator NERFINISHED ⓘ co-exact forms ⓘ codifferential ⓘ de Rham cohomology NERFINISHED ⓘ elliptic operators ⓘ exact forms ⓘ exterior derivative ⓘ harmonic forms ⓘ

Referenced by (6)

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

Hodge theory → studies → Hodge decomposition ⓘ

Hodge Conjecture → concerns → Hodge decomposition ⓘ

Clebsch representation → relatedTo → Hodge decomposition ⓘ

this entity surface form: Helmholtz decomposition

William Vallance Douglas Hodge → notableWork → Hodge decomposition ⓘ

Kunihiko Kodaira → notableWork → Hodge decomposition ⓘ

this entity surface form: Hodge–Kodaira decomposition

George Gabriel → notableIdea → Hodge decomposition ⓘ

subject surface form: George Gabriel Stokes

this entity surface form: Stokes–Helmholtz decomposition