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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Hodge star operator

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Summary

The metric- and orientation-dependent isomorphism between complementary-degree differential forms.

Record metadata

Canonical name: Hodge star operator
Concept ID: hodge_star_operator
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Differential geometry
Domains: Differential geometry
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Geometric operator

Carrier(s)

oriented pseudo-Riemannian manifold $(M,g)$

Data

linear maps $*\colon\Omega^k(M)\to\Omega^{n-k}(M)$

Axioms / constraints

defined by $\alpha\wedge *\beta=\langle\alpha,\beta\rangle_g\,\mathrm{vol}_g$

Canonically induces

codifferential
Hodge Laplacian

Notes

The square of $*$ depends on degree, dimension, and metric signature.

Concept sources

Wikipedia — Hodge star operator — Hodge star operator · encyclopedic reference · source ID wp-hodge_star_operator
nLab — Hodge star operator — Hodge star operator · mathematical reference · source ID nlab-hodge_star_operator
Cambridge — Electromagnetism — Electromagnetism · University of Cambridge lecture notes · source ID tong-em
Cambridge — General Relativity — General Relativity · University of Cambridge lecture notes · source ID tong-gr

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

metric + orientation(construction junction)Hodge star operator

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: hodge_from_forms
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: outputs the Hodge star

Authored explanation

A pseudo-Riemannian metric and an orientation on the same smooth manifold jointly define the Hodge star on differential forms.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Wikipedia — Differential form — Differential form · encyclopedic reference · source ID wp-differential_form
nLab — differential form — differential form · mathematical reference · source ID nlab-differential_form
Wikipedia — Hodge star operator — Hodge star operator · encyclopedic reference · source ID wp-hodge_star_operator
nLab — Hodge star operator — Hodge star operator · mathematical reference · source ID nlab-hodge_star_operator
Cambridge — Electromagnetism — Electromagnetism · University of Cambridge lecture notes · source ID tong-em
Cambridge — General Relativity — General Relativity · University of Cambridge lecture notes · source ID tong-gr

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.