Hodge star operator

Let V be a n-dimensional (n finite) vector spaceMathworldPlanetmath with inner product g. The Hodge star operator (denoted by ) is a linear operator mapping http://planetmath.org/node/3050p-forms on V to (n-p)-forms, i.e.,


In terms of a basis {e1,,en} for V and the corresponding dual basisMathworldPlanetmath {e1,,en} for V* (the star used to denote the dual space is not to be confused with the Hodge star!), with the inner product being expressed in terms of componentsPlanetmathPlanetmathPlanetmath as g=i,j=1ngijeiej, the -operator is defined as the linear operator that maps the basis elements of Ωp(V) as

(ei1eip) = |g|(n-p)!gi1l1giplpεl1lplp+1lnelp+1eln.

Here, |g|=detgij, and ε is the Levi-Civita permutation symbol

This operator may be defined in a coordinate-free manner by the condition


where the notation g(u,v) denotes the inner product on p-forms (in coordinates, g(u,v)=gi1j1gipjpui1ipvj1jp) and 𝐕𝐨𝐥(g) is the unit volume form associated to the metric. (in coordinates, 𝐕𝐨𝐥(g)=det(g)e1en)

Generally =(-1)p(n-p)id, where id is the identity operator in Ωp(V). In three dimensionsPlanetmathPlanetmath, =id for all p=0,,3. On 3 with Cartesian coordinatesMathworldPlanetmath, the metric tensorMathworldPlanetmath is g=dxdx+dydy+dzdz, and the Hodge star operator is


The Hodge star operationMathworldPlanetmath occurs most frequently in differential geometryMathworldPlanetmath in the case where Mn is a n-dimensional orientable manifold with a Riemannian (or pseudo-Riemannian) tensor g and V is a cotangent vector space of Mn. Also, one can extend this notion to antisymmetric tensor fields by computing Hodge star pointwise.

Title Hodge star operator
Canonical name HodgeStarOperator
Date of creation 2013-03-22 13:31:41
Last modified on 2013-03-22 13:31:41
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Definition
Classification msc 53B21
Synonym Hodge operator
Synonym star operator
Defines hodge star operator