Editing Hodge star operator (section)

== On manifolds ==
For an ''n''-dimensional oriented [[pseudo-Riemannian manifold]] ''M'', we apply the construction above to each [[cotangent space]] <math>\text{T}^*_p M</math> and its exterior powers <math display="inline">\bigwedge^k\text{T}^*_p M</math>, and hence to the differential ''k''[[differential form|-forms]] <math display="inline">\zeta\in\Omega^k(M) = \Gamma\left(\bigwedge^k\text{T}^*\!M\right)</math>, the [[Sheaf (mathematics)|global sections]] of the [[Vector bundle|bundle]] <math display="inline">\bigwedge^k \mathrm{T}^*\! M\to M</math>. The Riemannian metric induces a scalar product on <math display="inline">\bigwedge^k \text{T}^*_p M</math> at each point <math>p\in M</math>. We define the '''Hodge dual''' of a ''k''[[differential form|-form]] <math> \zeta </math>, defining <math>{\star} \zeta</math> as the unique (''n'' – ''k'')-form satisfying
<math display="block">\eta\wedge {\star} \zeta \ =\ \langle \eta, \zeta \rangle \, \omega </math>
for every ''k''-form <math> \eta </math>, where <math>\langle\eta,\zeta\rangle</math> is a real-valued function on <math>M</math>, and the [[Volume form#Riemannian volume form|volume form]] <math> \omega </math> is induced by the pseudo-Riemannian metric. Integrating this equation over <math>M</math>, the right side becomes the <math>L^2</math> ([[Sobolev space|square-integrable]]) [[Exterior algebra#Inner product|scalar product on ''k''-forms]], and we obtain:
<math display="block">\int_M \eta\wedge {\star} \zeta
\ =\  \int_M  \langle\eta,\zeta\rangle\ \omega.</math>

More generally, if <math>M</math> is non-orientable, one can define the Hodge star of a ''k''-form as a (''n'' – ''k'')-[[pseudotensor|pseudo differential form]]; that is, a differential form with values in the [[Canonical bundle|canonical line bundle]].

=== Computation in index notation ===
We compute in terms of [[tensor index notation]] with respect to a (not necessarily orthonormal) basis <math display="inline">\left\{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right\}</math> in a tangent space <math>V = T_p M</math> and its dual basis <math>\{dx_1,\ldots,dx_n\}</math> in <math>V^* = T^*_p M</math>, having the metric matrix <math display="inline">(g_{ij}) = \left(\left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right\rangle\right)</math> and its inverse matrix <math>(g^{ij}) = (\langle dx^i, dx^j\rangle)</math>. The Hodge dual of a decomposable ''k''-form is:
<math display="block">
{\star}\left(dx^{i_1} \wedge \dots \wedge dx^{i_k}\right) 
\ =\  
\frac{\sqrt{\left|\det [g_{ij}]\right|}}{(n-k)!} g^{i_1 j_1} \cdots g^{i_k j_k} \varepsilon_{j_1 \dots j_n} dx^{j_{k+1}} \wedge \dots \wedge dx^{j_n}.
</math>

Here <math>\varepsilon_{j_1 \dots j_n}</math> is the [[Levi-Civita symbol]] with <math>\varepsilon_{1 \dots n} = 1</math>, and we [[Einstein notation|implicitly take the sum]] over all values of the repeated indices <math> j_1,\ldots,j_n</math>. The factorial <math>(n-k)!</math> accounts for double counting, and is not present if the summation indices are restricted so that <math>j_{k+1} < \dots < j_n</math>. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to [[Pseudo-Riemannian manifold#Lorentzian manifold|Lorentzian manifold]]s.

An arbitrary differential form can be written as follows:
<math display="block">
\alpha \ =\  \frac{1}{k!}\alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k} 
\ =\ \sum_{i_1 < \dots < i_k} \alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k}.
</math>

The factorial <math>k!</math> is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component <math>\alpha_{i_1, \dots, i_k}</math> so that the Hodge dual of the form is given by
<math display="block">
{\star}\alpha = \frac{1}{(n-k)!}({\star} \alpha)_{i_{k+1}, \dots, i_n} dx^{i_{k+1}} \wedge \dots \wedge dx^{i_n}.
</math>

Using the above expression for the Hodge dual of <math>dx^{i_1} \wedge \dots \wedge dx^{i_k}</math>, we find:<ref>{{cite book| last=Frankel|first=T.| title=The Geometry of Physics| publisher=Cambridge University Press| year=2012| isbn=978-1-107-60260-1| edition=3rd}}</ref>
<math display="block">
({\star} \alpha)_{j_{k+1}, \dots, j_n} = \frac{\sqrt{\left|\det [g_{ab}]\right|}}{k!} \alpha_{i_1, \dots, i_k}\,g^{i_1 j_1}\cdots g^{i_k j_k} \,\varepsilon_{j_1, \dots, j_n}\, .
</math>

Although one can apply this expression to any tensor <math>\alpha</math>, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor.  It is thus equivalent to antisymmetrization followed by applying the Hodge star.

The unit volume form <math display="inline">\omega = {\star} 1\in \bigwedge^n V^*</math> is given by:
<math display="block">\omega = \sqrt{ \left| \det [g_{ij}] \right| }\;dx^1 \wedge \cdots \wedge dx^n .</math>

=== Codifferential ===
<!-- This section is linked from [[Differential form]] -->
The most important application of the Hodge star on manifolds is to define the '''codifferential''' <math> \delta </math> on <math>k</math>-forms.  Let
<math display="block">\delta = (-1)^{n(k + 1) + 1} s\ {\star} d {\star} = (-1)^{k}\, {\star}^{-1} d {\star} </math>
where <math>d</math> is the [[exterior derivative]] or differential, and <math>s = 1</math> for Riemannian manifolds. Then
<math display="block">d:\Omega^k(M)\to \Omega^{k+1}(M)</math>
while
<math display="block">\delta:\Omega^k(M)\to \Omega^{k-1}(M).</math>

The codifferential is not an [[antiderivation]] on the exterior algebra, in contrast to the exterior derivative.

The codifferential is the [[Transpose of a linear map|adjoint]] of the exterior derivative with respect to the square-integrable scalar product:
<math display="block"> \langle\!\langle\eta,\delta \zeta\rangle\!\rangle \ =\ \langle\!\langle d\eta,\zeta\rangle\!\rangle, </math>
where <math> \zeta </math> is a <math>k</math>-form and <math> \eta </math> a <math>(k\!-\!1)</math>-form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved from Stokes' theorem for smooth forms:
<math display="block">
0 \ =\ \int_M d (\eta \wedge {\star} \zeta)  
\ =\ 
\int_M \left(d \eta \wedge {\star} \zeta + (-1)^{k-1}\eta \wedge {\star} \,{\star}^{-1} d\, {\star} \zeta\right) 
\ =\ 
\langle\!\langle d\eta,\zeta\rangle\!\rangle - \langle\!\langle\eta,\delta\zeta\rangle\!\rangle,
</math>
provided <math>M</math> has empty boundary, or <math> \eta </math> or <math>{\star}\zeta</math> has zero boundary values. (The proper definition of the above requires specifying a [[topological vector space]] that is closed and complete on the space of smooth forms. The [[Sobolev space]] is conventionally used; it allows the convergent sequence of forms <math>\zeta_i \to \zeta</math> (as <math>i \to \infty</math>) to be interchanged with the combined differential and integral operations, so that <math>\langle\!\langle\eta,\delta \zeta_i\rangle\!\rangle \to \langle\!\langle\eta,\delta \zeta\rangle\!\rangle</math> and likewise for sequences converging to <math>\eta</math>.)

Since the differential satisfies <math>d^2 = 0</math>, the codifferential has the corresponding property
<math display="block">\delta^2 = (-1)^n s^2 {\star} d {\star} {\star} d {\star} = (-1)^{nk+k+1} s^3 {\star} d^2 {\star} = 0. </math>

The [[Laplace–Beltrami operator|Laplace–deRham]] operator is given by
<math display="block">\Delta = (\delta + d)^2 = \delta d + d\delta</math>
and lies at the heart of [[Hodge theory]]. It is symmetric:
<math display="block">\langle\!\langle\Delta \zeta,\eta\rangle\!\rangle = \langle\!\langle\zeta,\Delta \eta\rangle\!\rangle</math>
and non-negative:
<math display="block">\langle\!\langle\Delta\eta,\eta\rangle\!\rangle \ge 0.</math>

The Hodge star sends [[harmonic form]]s to harmonic forms. As a consequence of [[Hodge theory]], the [[de Rham cohomology]] is naturally isomorphic to the space of harmonic {{mvar|k}}-forms, and so the Hodge star induces an isomorphism of cohomology groups
<math display="block">{\star} : H^k_\Delta (M) \to H^{n-k}_\Delta(M),</math>
which in turn gives canonical identifications via [[Poincaré duality]] of {{math|''H<sup> k</sup>''(''M'')}} with its [[dual space]].

In coordinates, with notation as above, the codifferential of the form <math>\alpha</math> may be written as
<math display="block">\delta \alpha=\  -\frac{1}{k!}g^{ml}\left(\frac{\partial}{\partial x_l} \alpha_{m,i_1, \dots, i_{k-1}} - \Gamma^j_{ml} \alpha_{j,i_1, \dots, i_{k-1}} \right) dx^{i_1} \wedge \dots \wedge dx^{i_{k-1}},</math>
where here <math>\Gamma^{j}_{ml}</math> denotes the [[Christoffel symbols]] of  <math display="inline">\left\{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right\}</math>.

==== Poincare lemma for codifferential ====
In analogy to the [[Poincare lemma]] for [[exterior derivative]], one can define its version for codifferential, which reads<ref name=":0">{{Cite journal |last=Kycia |first=Radosław Antoni |date=2022-07-29 |title=The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics |url=https://doi.org/10.1007/s00025-022-01646-z |journal=Results in Mathematics |language=en |volume=77 |issue=5 |pages=182 |doi=10.1007/s00025-022-01646-z |issn=1420-9012|arxiv=2009.08542 |s2cid=221802588 }}</ref>
: ''If'' <math>\delta\omega=0</math> ''for'' <math>\omega \in \Lambda^{k}(U)</math>'', where '' <math>U</math> ''is a [[star domain]] on a manifold, then there is'' <math>\alpha \in \Lambda^{k+1}(U)</math> ''such that'' <math>\omega=\delta\alpha</math>''.''

A practical way of finding <math>\alpha</math> is to use cohomotopy operator <math>h</math>, that is a local inverse of <math>\delta</math>. One has to define a [[homotopy operator]]<ref name=":0" />
: <math>H\beta = \int_{0}^{1} \mathcal{K}\lrcorner\beta|_{F(t,x)}t^{k}dt,</math> 
where <math>F(t,x)=x_{0}+t(x-x_{0})</math> is the linear homotopy between its center <math>x_{0}\in U</math> and a point <math>x \in U</math>, and the (Euler) vector <math>\mathcal{K}=\sum_{i=1}^{n}(x-x_{0})^{i}\partial_{x^{i}}</math> for <math>n=\dim(U)</math> is inserted into the form <math>\beta \in \Lambda^{*}(U)</math>. We can then define cohomotopy operator as<ref name=":0" />
: <math>h:\Lambda(U)\rightarrow \Lambda(U), \quad h:=\eta {\star}^{-1}H\star</math>,
where <math>\eta \beta = (-1)^{k}\beta</math> for <math>\beta \in \Lambda^{k}(U)</math>.

The cohomotopy operator fulfills (co)homotopy invariance formula<ref name=":0" /> 
: <math>\delta h + h\delta = I - S_{x_{0}} ,</math>
where <math>S_{x_{0}}={\star}^{-1}s_{x_{0}}^{*}{\star}</math>, and <math>s_{x_{0}}^{*}</math> is the [[Pullback (differential geometry)|pullback]] along the constant map <math>s_{x_{0}}:x \rightarrow x_{0}</math>.

Therefore, if we want to solve the equation <math>\delta \omega =0</math>, applying cohomotopy invariance formula we get
: <math> \omega= \delta h\omega + S_{x_{0}}\omega,</math> where <math>h\omega\in \Lambda^{k+1}(U)</math> is a differential form we are looking for, and "constant of integration" <math>S_{x_{0}}\omega</math> vanishes unless <math>\omega</math> is a top form.

Cohomotopy operator fulfills the following properties:<ref name=":0" /> <math>h^{2}=0, \quad \delta h \delta =\delta, \quad h\delta h =h</math>. They make it possible to use it to define<ref name=":0" /> ''anticoexact'' forms on <math>U</math> by <math>\mathcal{Y}(U)=\{ \omega\in\Lambda(U)| \omega = h\delta \omega \}</math>, which together with [[Exact form|exact forms]] <math>\mathcal{C}(U) =\{ \omega\in\Lambda(U)|\omega = \delta h\omega \}</math> make a [[direct sum]] decomposition<ref name=":0" /> 
: <math>\Lambda(U)=\mathcal{C}(U)\oplus \mathcal{Y}(U)</math>.

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the [[Projector operator|projector operators]] on the summands fulfills [[Idempotent (ring theory)|idempotence]] formulas:<ref name=":0" /> <math>(h\delta)^{2}=h\delta, \quad (\delta h)^{2}=\delta h</math>.

These results are extension of similar results for exterior derivative.<ref>{{Cite book |last=Edelen |first=Dominic G. B. |url=https://www.worldcat.org/oclc/56347718 |title=Applied exterior calculus |date=2005 |isbn=978-0-486-43871-9 |edition=Revised |location=Mineola, N.Y. |oclc=56347718}}</ref>