# 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 $\left(X,g\right)$, the Hodge star operator “completes” a $k$-differential form to the volume form of $\left(X,g\right)$.

## Definition

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

### Hodge inner product

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

$\left(-\mid -\right):{\Omega }^{k}\left(X\right)\otimes {\Omega }^{k}\left(X\right)\to {\Omega }^{0}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$(-\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 ${\mathrm{vol}}_{g}$ exists. This is the Hodge inner product

$⟨-,-⟩:{\Omega }^{k}\left(X\right)\otimes {\Omega }^{k}\left(X\right)\to ℝ$\langle - , - \rangle : \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}
$⟨\alpha ,\beta ⟩:={\int }_{X}\left(\alpha \mid \beta \right)\mathrm{vol}\phantom{\rule{thinmathspace}{0ex}}.$\langle \alpha, \beta \rangle := \int_X (\alpha\mid \beta) vol \,.

### Hodge star operator

The Hodge star operator is the unique linear function

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

defined by the identity

$\alpha \wedge \star \beta =\left(\alpha \mid \beta \right){\mathrm{vol}}_{g},\phantom{\rule{2em}{0ex}}\forall \alpha ,\beta \in \stackrel{k}{\bigwedge }X\phantom{\rule{thinmathspace}{0ex}},$\alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \bigwedge^k X \,,

where ${\mathrm{vol}}_{g}\in {\Omega }^{n}X$ is the volume form induced by $g$.

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

$⟨\alpha ,\beta ⟩={\int }_{X}\alpha \wedge \star \beta \phantom{\rule{thinmathspace}{0ex}}.$\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

$\star \alpha =\frac{1}{k!\left(n-k\right)!}{ϵ}_{{i}_{1},\dots ,{i}_{n}}\sqrt{\mid \mathrm{det}\left(g\right)\mid }{\alpha }_{{j}_{1},\dots ,{j}_{k}}{g}^{{i}_{1},{j}_{1}}\cdots {g}^{{i}_{k},{j}_{k}}{e}^{{i}_{k+1}}\wedge \cdots \wedge {e}^{{i}_{n}},$\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 ${ϵ}_{{i}_{1},\dots ,{i}_{n}}$ is the sign of the permutation $\left(1,2,\dots ,n\right)↦\left({i}_{1},{i}_{2},\dots ,{i}_{n}\right)$ and $\mathrm{det}\left(g\right)$ is the determinant of $g$ in the local basis.

### Basis-independent formulas

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

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

• $⟨\star \omega ,\star \lambda ⟩=⟨\omega \mid \lambda ⟩$;

• $\star 1=\mathrm{vol}$.

## 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 $\left(n-k\right)$-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 $\left(n-k\right)$-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 $\left(n-k\right)$-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)

Revised on January 4, 2013 04:08:23 by Urs Schreiber (89.204.135.106)