# nLab Hodge star operator

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

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

## Definition

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

### Hodge inner product

The metric $g$ naturally induces a nondegenerate symmetric bilinear form

$(-\mid-) : \Omega^k(X) \otimes \Omega^k(X) \to \Omega^0(X) \,.$

If $X$ is compact then the integral of this against the volume form $vol_g$ exists. This is the Hodge inner product

$\langle - , - \rangle : \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}$
$\langle \alpha, \beta \rangle := \int_X (\alpha\mid \beta) vol \,.$

### Hodge star operator

The Hodge star operator is the unique linear function

${\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)$

defined by the identity

$\alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \bigwedge^k X \,,$

where $vol_g \in \Omega^n X$ is the volume form induced by $g$.

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

$\langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.$

## Properties

### Component formulas

If $e_1,\dots,e_n$ is a local basis on $X$ and $e^1,\dots,e^n$ is the dual basis, so that $\alpha = \frac{1}{k!} \alpha_{i_1,\dots,i_k} e^{i_1} \wedge \cdots \wedge e^{i_k}$, then

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

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

### Basic properties (Basis-independent formulas)

Let $(X,g)$ be a Riemannian manifold of dimension $n$ and let $\omega,\lambda \in \Omega^k(X)$. Then

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

• $\langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle$;

• $\star 1 = vol$.

### On a Kähler manifold

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

$\star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,.$

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

## Generalisations

The metric $g$ is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If $X$ is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes $k$-forms to $(n-k)$-vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes $k$-forms to $(n-k)$-pseudoforms. Finally, if $X$ is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on $X$), then the Hodge operator takes $k$-forms to $(n-k)$-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)

Revised on May 8, 2015 10:59:19 by Urs Schreiber (195.113.30.252)