Hodge star operator



Riemannian geometry

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Given a finite dimensional (pseudo)-Riemannian manifold (X,g)(X,g), the Hodge star operator “completes” a kk-differential form to the volume form of (X,g)(X,g).


Let (X,g)(X,g) be an oriented nn-dimensional smooth manifold XX endowed with a (pseudo)-Riemannian metric gg. For 0kn0 \leq k \leq n, write Ω k(X)\Omega^k(X) for the vector space of kk-forms on XX.

Hodge inner product

The metric gg naturally induces a nondegenerate symmetric bilinear form

():Ω k(X)Ω k(X)Ω 0(X). (-\mid-) \;\colon\; \Omega^k(X) \otimes \Omega^k(X) \to \Omega^0(X) \,.

If XX is compact then the integral of this against the volume form vol gvol_g exists. This is the Hodge inner product

,:Ω k(X)Ω k(X) \langle - , - \rangle \;\colon\; \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}
α,β:= X(αβ)vol. \langle \alpha, \beta \rangle := \int_X (\alpha\mid \beta) vol \,.

Hodge star operator

The Hodge star operator is the unique linear function

:Ω k(X)Ω nk(X) {\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)

defined by the identity

αβ=(αβ)vol g,α,β kX, \alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \bigwedge^k X \,,

where vol gΩ nXvol_g \in \Omega^n X is the volume form induced by gg.

Therefore in terms of the Hodge operator the Hodge inner product reads

α,β= Xαβ. \langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.


Component formulas

If e 1,,e ne_1,\dots,e_n is a local basis on XX and e 1,,e ne^1,\dots,e^n is the dual basis, so that α=1k!α i 1,,i ke i 1e i k\alpha = \frac{1}{k!} \alpha_{i_1,\dots,i_k} e^{i_1} \wedge \cdots \wedge e^{i_k}, then

α=1k!(nk)!ϵ i 1,,i n|det(g)|α j 1,,j kg i 1,j 1g i k,j ke i k+1e i n, \star \alpha = \frac{1}{k!(n-k)!} \epsilon_{i_1,\dots,i_n} \sqrt{|det(g)|} \alpha_{j_1,\dots,j_k} g^{i_1,j_1} \cdots g^{i_k,j_k} e^{i_{k+1}} \wedge \cdots \wedge e^{i_n},

where ϵ i 1,,i n\epsilon_{i_1,\dots,i_n} is the sign of the permutation (1,2,,n)(i 1,i 2,,i n)(1,2,\dots,n)\mapsto (i_1,i_2,\dots,i_n) and det(g)det(g) is the determinant of gg in the local basis.

Basic properties (Basis-independent formulas)

Let (X,g)(X,g) be a Riemannian manifold of dimension nn and let ω,λΩ k(X)\omega,\lambda \in \Omega^k(X). Then the following holds:

(1)(ω)=(1) k(n+1)ω=(1) k(nk)ω \star(\star\omega) = (-1)^{k(n+1)} \omega = (-1)^{k(n-k)} \omega
(2)ω,λ=ω|λ \langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle
(3)1=vol \star 1 = vol

(volume form)

On a Kähler manifold

On a Kähler manifold Σ\Sigma of dimension dim (Σ)=ndim_{\mathbb{C}}(\Sigma) = n the Hodge star operator acts on the Dolbeault complex as

:Ω p,q(X)Ω nq,np(X). \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.

(notice the exchange of the role of pp and qq). See e.g. (Biquerd-Höring 08, p. 79). See also at Serre duality.


The metric gg is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If XX is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes kk-forms to (nk)(n-k)-vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes kk-forms to (nk)(n-k)-pseudoforms. Finally, if XX is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on XX), then the Hodge operator takes kk-forms to (nk)(n-k)-pseudovector fields. (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)


Some useful basic formulas are listed in

  • Hodge theory on Riemannian manifolds , lecture notes (pdf)

Discussion in complex geometry includes

  • O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)

With an eye towards application in supergravity and string theory:

Last revised on June 25, 2019 at 04:43:57. See the history of this page for a list of all contributions to it.