Hodge star operator


Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

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& Rh & \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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

  • (ω)=(1) k(n+1)ω=(1) k(nk)ω\star(\star\omega) = (-1)^{k(n+1)} \omega = (-1)^{k(n-k)} \omega;

  • ω,λ=ω|λ\langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle;

  • 1=vol\star 1 = vol.

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)

Revised on March 11, 2018 05:37:43 by Urs Schreiber (