nLab
Hodge star operator

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          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).

          Definition

          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 \,.

          Properties

          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.

          Generalisations

          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.)

          References

          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)

          Last revised on March 11, 2018 at 05:37:43. See the history of this page for a list of all contributions to it.