Hodge star operator

E876575

duality operator linear operator mathematical operator

The Hodge star operator is a linear map on differential forms in oriented Riemannian manifolds that sends a k-form to an (n−k)-form in a way that encodes the metric and orientation, enabling duality operations such as defining codifferentials and expressing vector calculus identities.

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Statements (47)

Predicate	Object
instanceOf	duality operator ⓘ linear operator ⓘ mathematical operator ⓘ
actsOn	differential forms ⓘ exterior algebra of the cotangent bundle ⓘ
appearsIn	Maxwell's equations in differential form notation ⓘ gauge theory ⓘ general relativity formulations using differential forms ⓘ
codomain	(n−k)-forms on an oriented Riemannian manifold ⓘ
context	oriented Riemannian manifolds ⓘ pseudo-Riemannian manifolds ⓘ
definedBy	requirement that α ∧ *β equals the inner product of α and β times the volume form ⓘ
dependsOn	dimension of the manifold ⓘ signature of the metric ⓘ
domain	k-forms on an oriented Riemannian manifold ⓘ
encodes	metric information ⓘ orientation information ⓘ
field	Riemannian geometry NERFINISHED ⓘ differential geometry ⓘ geometric analysis ⓘ
generalizationOf	cross product in three dimensions via forms ⓘ orthogonal complement operation in inner product spaces ⓘ
importantIn	Yang–Mills theory NERFINISHED ⓘ elliptic partial differential equations on manifolds ⓘ the study of harmonic forms ⓘ
is	linear over the complex numbers ⓘ linear over the real numbers ⓘ
namedAfter	W. V. D. Hodge NERFINISHED ⓘ
notation	* ⓘ
property	is an isometry up to sign with respect to the induced inner product on forms ⓘ maps volume form to 1 and 1 to volume form up to sign ⓘ
relatedTo	Hodge theory NERFINISHED ⓘ Laplace–de Rham operator NERFINISHED ⓘ Poincaré duality NERFINISHED ⓘ codifferential ⓘ exterior derivative ⓘ
requiresStructure	Riemannian metric NERFINISHED ⓘ orientation ⓘ
satisfies	* * α = (−1)^{k(n−k)} α on k-forms in n dimensions ⓘ is an involution up to sign ⓘ is an isomorphism between k-forms and (n−k)-forms ⓘ
usedFor	defining inner products on differential forms ⓘ defining self-dual and anti-self-dual forms ⓘ defining the Laplace–de Rham operator ⓘ defining the codifferential ⓘ expressing vector calculus identities ⓘ formulating Hodge decomposition ⓘ

Referenced by (1)

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

Levi-Civita symbol → relatedConcept → Hodge star operator ⓘ