Inmathematics, theHodge star operator orHodge star is alinear map defined on theexterior algebra of afinite-dimensionalorientedvector space endowed with anondegeneratesymmetric bilinear form. Applying the operator to an element of the algebra produces theHodge dual of the element. This map was introduced byW. V. D. Hodge.

For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by theexterior product of two basis vectors, and its Hodge dual is thenormal vector given by theircross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to ann-dimensional vector space, the Hodge star is a one-to-one mapping ofk-vectors to(n – k)-vectors; the dimensions of these spaces are thebinomial coefficients${\displaystyle (\genfrac{}{}{0ex}{}{n}{k})=(\genfrac{}{}{0ex}{}{n}{n-k})}$.

Thenaturalness of the star operator means it can play a role in differential geometry when applied to the cotangentbundle of apseudo-Riemannian manifold, and hence todifferentialk-forms. This allows the definition of the codifferential as the Hodge adjoint of theexterior derivative, leading to theLaplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in whichdivergence of a vector field may be realized as the codifferential opposite to thegradient operator, and theLaplace operator on a function is the divergence of its gradient. An important application is theHodge decomposition of differential forms on aclosed Riemannian manifold.

LetV be ann-dimensionalorientedvector space with a nondegenerate symmetric bilinear form${\displaystyle ⟨\cdot ,\cdot ⟩}$, referred to here as a scalar product. (In more general contexts such as pseudo-Riemannian manifolds andMinkowski space, the bilinear form may not be positive-definite.) This induces ascalar product onk-vectors$\alpha ,\beta \in \stackrel{k}{\bigwedge }V$, for${\displaystyle 0\le k\le n}$, by defining it on simplek-vectors${\displaystyle \alpha ={\alpha }_1\wedge \cdots \wedge {\alpha }_k}$ and${\displaystyle \beta ={\beta }_1\wedge \cdots \wedge {\beta }_k}$ to equal theGram determinant^[1]: 14

${\displaystyle ⟨\alpha ,\beta ⟩=det\left({⟨{\alpha }_i,{\beta }_j⟩}_{i,j=1}^k\right)}$

extended to$\stackrel{k}{\bigwedge }V$ through linearity.

The unitn-vector${\displaystyle \omega \in {\bigwedge }^{n}V}$ is defined in terms of an orientedorthonormal basis${\displaystyle \{e_1,\dots ,e_n\}}$ ofV as:

${\displaystyle \omega :=e_1\wedge \cdots \wedge e_n.}$

(Note: In the general pseudo-Riemannian case, orthonormality means${\displaystyle ⟨e_i,e_j⟩\in \{{\delta }_{ij},-{\delta }_{ij}\}}$ for all pairs of basis vectors.)TheHodge star operator is a linear operator on theexterior algebra ofV, mappingk-vectors to (n –k)-vectors, for${\displaystyle 0\le k\le n}$. It has the following property, which defines it completely:^[1]: 15

${\displaystyle \alpha \wedge (\star \beta )=⟨\alpha ,\beta ⟩\omega }$ for allk-vectors${\displaystyle \alpha ,\beta \in {\bigwedge }^{k}V.}$

Dually, in the space${\displaystyle {\bigwedge }^n{V}^{\ast }}$ ofn-forms (alternatingn-multilinear functions on${\displaystyle V^n}$), the dual to${\displaystyle \omega }$ is thevolume form${\displaystyle det}$, the function whose value on${\displaystyle v_1\wedge \cdots \wedge v_n}$ is thedeterminant of the${\displaystyle n\times n}$ matrix assembled from the column vectors of${\displaystyle v_j}$ in${\displaystyle e_i}$-coordinates. Applying${\displaystyle det}$ to the above equation, we obtain the dual definition:

${\displaystyle det(\alpha \wedge \star \beta )=⟨\alpha ,\beta ⟩}$ for allk-vectors${\displaystyle \alpha ,\beta \in {\bigwedge }^{k}V.}$

Equivalently, taking${\displaystyle \alpha ={\alpha }_1\wedge \cdots \wedge {\alpha }_k}$,${\displaystyle \beta ={\beta }_1\wedge \cdots \wedge {\beta }_k}$, and${\displaystyle \star \beta ={\beta }_1^{\star }\wedge \cdots \wedge {\beta }_{n-k}^{\star }}$:

${\displaystyle det\left({\alpha }_1\wedge \cdots \wedge {\alpha }_k\wedge {\beta }_1^{\star }\wedge \cdots \wedge {\beta }_{n-k}^{\star }\right)=det\left(⟨{\alpha }_i,{\beta }_j⟩\right).}$

This means that, writing an orthonormal basis ofk-vectors as${\displaystyle e_I=e_{i_1}\wedge \cdots \wedge e_{i_k}}$ over all subsets of${\displaystyle [n]=\{1,\dots ,n\}}$, the Hodge dual is the (n – k)-vector corresponding to the complementary set:

${\displaystyle \star e_I=s\cdot t\cdot e_{\overline{I}},}$

where${\displaystyle s\in \{1,-1\}}$ is thesign of the permutation${\displaystyle i_1\cdots i_k{\overline{i}}_1\cdots {\overline{i}}_{n-k}}$and${\displaystyle t\in \{1,-1\}}$ is the product${\displaystyle ⟨e_{i_1},e_{i_1}⟩\cdots ⟨e_{i_k},e_{i_k}⟩}$. In the Riemannian case,${\displaystyle t=1}$.

Since Hodge star takes an orthonormal basis to an orthonormal basis, it is anisometry on the exterior algebra$\bigwedge V$.

The Hodge star is motivated by the correspondence between a subspaceW ofV and its orthogonal subspace (with respect to the scalar product), where each space is endowed with anorientation and a numerical scaling factor. Specifically, a non-zero decomposablek-vector${\displaystyle w_1\wedge \cdots \wedge w_k\in \stackrel{k}{\bigwedge }V}$ corresponds by thePlücker embedding to the subspace${\displaystyle W}$ with oriented basis${\displaystyle w_1,\dots ,w_k}$, endowed with a scaling factor equal to thek-dimensional volume of the parallelepiped spanned by this basis (equal to theGramian, the determinant of the matrix of scalar products${\displaystyle ⟨w_i,w_j⟩}$). The Hodge star acting on a decomposable vector can be written as a decomposable (n −k)-vector:

${\displaystyle \star (w_1\wedge \cdots \wedge w_k)=u_1\wedge \cdots \wedge u_{n-k},}$

where${\displaystyle u_1,\dots ,u_{n-k}}$ form an oriented basis of theorthogonal space${\displaystyle U=W^{\perp }}$. Furthermore, the (n −k)-volume of the${\displaystyle u_i}$-parallelepiped must equal thek-volume of the${\displaystyle w_i}$-parallelepiped, and${\displaystyle w_1,\dots ,w_k,u_1,\dots ,u_{n-k}}$ must form an oriented basis of${\displaystyle V}$.

A generalk-vector is a linear combination of decomposablek-vectors, and the definition of Hodge star is extended to generalk-vectors by defining it as being linear.

In two dimensions with the normalized Euclidean metric and orientation given by the ordering(x,y), the Hodge star onk-forms is given by

A common example of the Hodge star operator is the casen = 3, when it can be taken as the correspondence between vectors and bivectors. Specifically, forEuclideanR³ with the basis${\displaystyle dx,dy,dz}$ ofone-forms often used invector calculus, one finds that

The Hodge star relates the exterior and cross product in three dimensions:^[2]$${\displaystyle \star (\mathbf{u}\wedge \mathbf{v})=\mathbf{u}\times \mathbf{v}\star (\mathbf{u}\times \mathbf{v})=\mathbf{u}\wedge \mathbf{v}.}$$ Applied to three dimensions, the Hodge star provides anisomorphism betweenaxial vectors andbivectors, so each axial vectora is associated with a bivectorA and vice versa, that is:^[2]${\displaystyle \mathbf{A}=\star \mathbf{a},\mathbf{a}=\star \mathbf{A}}$.

The Hodge star can also be interpreted as a form of the geometric correspondence between anaxis of rotation and aninfinitesimal rotation (see also:3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space${\displaystyle V}$ gives anisomorphism${\displaystyle V\cong V^{\ast }}$ identifying${\displaystyle V}$ with itsdual space, and the vector space${\displaystyle L(V,V)}$ is naturally isomorphic to thetensor product${\displaystyle V^{\ast }\otimes V\cong V\otimes V}$. Thus for${\displaystyle V={\mathbb{R}}^3}$, the star mapping$\star :V\to \stackrel{2}{\bigwedge }V\subset V\otimes V$ takes each vector${\displaystyle \mathbf{v}}$ to a bivector${\displaystyle \star \mathbf{v}\in V\otimes V}$, which corresponds to a linear operator${\displaystyle L_{\mathbf{v}}:V\to V}$. Specifically,${\displaystyle L_{\mathbf{v}}}$ is askew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis${\displaystyle \mathbb{v}}$ are given by thematrix exponential${\displaystyle \exp (t{L}_{\mathbf{v}})}$. With respect to the basis${\displaystyle dx,dy,dz}$ of${\displaystyle {\mathbb{R}}^3}$, the tensor${\displaystyle dx\otimes dy}$ corresponds to a coordinate matrix with 1 in the${\displaystyle dx}$ row and${\displaystyle dy}$ column, etc., and the wedge${\displaystyle dx\wedge dy=dx\otimes dy-dy\otimes dx}$ is the skew-symmetric matrix, etc. That is, we may interpret the star operator as: Under this correspondence, cross product of vectors corresponds to the commutatorLie bracket of linear operators:${\displaystyle L_{\mathbf{u}\times \mathbf{v}}=L_{\mathbf{v}}{L}_{\mathbf{u}}-L_{\mathbf{u}}{L}_{\mathbf{v}}=-\left[L_{\mathbf{u}},L_{\mathbf{v}}\right]}$.

In case${\displaystyle n=4}$, the Hodge star acts as anendomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since4 − 2 = 2). If the signature of themetric tensor is all positive, i.e. on aRiemannian manifold, then the Hodge star is aninvolution. If the signature is mixed, i.e.,pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see§ Duality below. This particular endomorphism property of 2-forms in four dimensions makesself-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues${\displaystyle \pm 1}$ (or${\displaystyle \pm i}$, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where${\displaystyle n=4}$ with metric signature(− + + +) and coordinates${\displaystyle (t,x,y,z)}$. Thevolume form is oriented as${\displaystyle {\epsilon }_{0123}=1}$. Forone-forms,while for2-forms,

These are summarized in the index notation as

Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature,${\displaystyle {\star }^2=1}$ for odd-rank forms and${\displaystyle {\star }^2=-1}$ for even-rank forms. An easy rule to remember for these Hodge operations is that given a form${\displaystyle \alpha }$, its Hodge dual${\displaystyle \star \alpha }$ may be obtained by writing the components not involved in${\displaystyle \alpha }$ in an order such that${\displaystyle \alpha \wedge (\star \alpha )=dt\wedge dx\wedge dy\wedge dz}$.^{[verification needed]} An extra minus sign will enter only if${\displaystyle \alpha }$ contains${\displaystyle dt}$. (For(+ − − −), one puts in a minus sign only if${\displaystyle \alpha }$ involves an odd number of the space-associated forms${\displaystyle dx}$,${\displaystyle dy}$ and${\displaystyle dz}$.)

Note that the combinations$${\displaystyle (d{x}^{\mu }\wedge d{x}^{\nu }{)}^{\pm }:=\frac{1}{2}(d{x}^{\mu }\wedge d{x}^{\nu }\mp i\star (d{x}^{\mu }\wedge d{x}^{\nu }))}$$take${\displaystyle \pm i}$ as the eigenvalue for Hodge star operator, i.e.,$${\displaystyle \star (d{x}^{\mu }\wedge d{x}^{\nu }{)}^{\pm }=\pm i(d{x}^{\mu }\wedge d{x}^{\nu }{)}^{\pm },}$$and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in bothmathematical andphysical perspectives, making contacts to the use of thetwo-spinor language in modern physics such asspinor-helicity formalism ortwistor theory.

The Hodge star is conformally invariant onn-forms on a2n-dimensional vector space${\displaystyle V}$, i.e. if${\displaystyle g}$ is a metric on${\displaystyle V}$ and${\displaystyle \lambda >0}$, then the induced Hodge stars$${\displaystyle {\star }_g,{\star }_{\lambda g}:{\mathrm{\Lambda }}^{n}V\to {\mathrm{\Lambda }}^{n}V}$$are the same.

The combination of the${\displaystyle \star }$ operator and theexterior derivatived generates the classical operatorsgrad,curl, anddiv onvector fields in three-dimensional Euclidean space. This works out as follows:d takes a 0-form (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). For a 0-form${\displaystyle f=f(x,y,z)}$, the first case written out in components gives:$${\displaystyle df=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}dx+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}dy+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}dz.}$$

The scalar productidentifies 1-forms with vector fields as${\displaystyle dx\mapsto (1,0,0)}$, etc., so that${\displaystyle df}$ becomes$\mathrm{grad}f=\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x},\frac{\mathrm{\partial }f}{\mathrm{\partial }y},\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right)$.

In the second case, a vector field${\displaystyle \mathbf{F}=(A,B,C)}$ corresponds to the 1-form${\displaystyle \phi =Adx+Bdy+Cdz}$, which has exterior derivative:$${\displaystyle d\phi =\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}-\frac{\mathrm{\partial }B}{\mathrm{\partial }z}\right)dy\wedge dz+\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }x}-\frac{\mathrm{\partial }A}{\mathrm{\partial }z}\right)dx\wedge dz+\left(\frac{\mathrm{\partial }B}{\mathrm{\partial }x}-\frac{\mathrm{\partial }A}{\mathrm{\partial }y}\right)dx\wedge dy.}$$

Applying the Hodge star gives the 1-form:$${\displaystyle \star d\phi =\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}-\frac{\mathrm{\partial }B}{\mathrm{\partial }z}\right)dx-\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }x}-\frac{\mathrm{\partial }A}{\mathrm{\partial }z}\right)dy+\left(\frac{\mathrm{\partial }B}{\mathrm{\partial }x}-\frac{\mathrm{\partial }A}{\mathrm{\partial }y}\right)dz,}$$which becomes the vector field$\mathrm{curl}\mathbf{F}=\left(\frac{\mathrm{\partial }C}{\mathrm{\partial }y}-\frac{\mathrm{\partial }B}{\mathrm{\partial }z},-\frac{\mathrm{\partial }C}{\mathrm{\partial }x}+\frac{\mathrm{\partial }A}{\mathrm{\partial }z},\frac{\mathrm{\partial }B}{\mathrm{\partial }x}-\frac{\mathrm{\partial }A}{\mathrm{\partial }y}\right)$.

In the third case,${\displaystyle \mathbf{F}=(A,B,C)}$ again corresponds to${\displaystyle \phi =Adx+Bdy+Cdz}$. Applying Hodge star, exterior derivative, and Hodge star again:

One advantage of this expression is that the identityd² = 0, which is true in all cases, has as special cases two other identities: (1)curl gradf = 0, and (2)div curlF = 0. In particular,Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression${\displaystyle \star d\star }$ (multiplied by an appropriate power of −1) is called thecodifferential; it is defined in full generality, for any dimension, further in the article below.

One can also obtain theLaplacianΔf = div grad f in terms of the above operations:$${\displaystyle \mathrm{\Delta }f=\star d\star df=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}.}$$

The Laplacian can also be seen as a special case of the more generalLaplace–deRham operator${\displaystyle \mathrm{\Delta }=d\delta +\delta d}$ where in three dimensions,${\displaystyle \delta =(-1{)}^k\star d\star }$ is the codifferential for${\displaystyle k}$-forms. Any function${\displaystyle f}$ is a 0-form, and${\displaystyle \delta f=0}$ and so this reduces to the ordinary Laplacian. For the 1-form${\displaystyle \phi }$ above, the codifferential is${\displaystyle \delta =-\star d\star }$ and after some straightforward calculations one obtains the Laplacian acting on${\displaystyle \phi }$.

Applying the Hodge star twice leaves ak-vector unchangedup to a sign: for${\displaystyle \eta \in {\bigwedge }^{k}V}$ in ann-dimensional spaceV, one has

${\displaystyle \star \star \eta =(-1{)}^{k(n-k)}s\eta ,}$

wheres is the parity of thesignature of the scalar product onV, that is, the sign of thedeterminant of the matrix of the scalar product with respect to any basis. For example, ifn = 4 and the signature of the scalar product is either(+ − − −) or(− + + +) thens = −1. For Riemannian manifolds (including Euclidean spaces), we always haves = 1.

The above identity implies that the inverse of${\displaystyle \star }$ can be given as

Ifn is odd thenk(n −k) is even for anyk, whereas ifn is even thenk(n −k) has the parity ofk. Therefore:

wherek is the degree of the element operated on.

For ann-dimensional orientedpseudo-Riemannian manifoldM, we apply the construction above to eachcotangent space${\displaystyle {\text{T}}_p^{\ast }M}$ and its exterior powers$\stackrel{k}{\bigwedge }{\text{T}}_p^{\ast }M$, and hence to the differentialk-forms$\zeta \in {\mathrm{\Omega }}^k(M)=\mathrm{\Gamma }\left(\stackrel{k}{\bigwedge }{\text{T}}^{\ast }M\right)$, theglobal sections of thebundle$\stackrel{k}{\bigwedge }{\mathrm{T}}^{\ast }M\to M$. The Riemannian metric induces a scalar product on$\stackrel{k}{\bigwedge }{\text{T}}_p^{\ast }M$ at each point${\displaystyle p\in M}$. We define theHodge dual of ak-form${\displaystyle \zeta }$, defining${\displaystyle \star \zeta }$ as the unique (n –k)-form satisfying$${\displaystyle \eta \wedge \star \zeta =⟨\eta ,\zeta ⟩\omega }$$for everyk-form${\displaystyle \eta }$, where${\displaystyle ⟨\eta ,\zeta ⟩}$ is a real-valued function on${\displaystyle M}$, and thevolume form${\displaystyle \omega }$ is induced by the pseudo-Riemannian metric. Integrating this equation over${\displaystyle M}$, the right side becomes the${\displaystyle L^2}$ (square-integrable)scalar product onk-forms, and we obtain:$${\displaystyle {\int }_M\eta \wedge \star \zeta ={\int }_M⟨\eta ,\zeta ⟩\omega .}$$

More generally, if${\displaystyle M}$ is non-orientable, one can define the Hodge star of ak-form as a (n –k)-pseudo differential form; that is, a differential form with values in thecanonical line bundle.

We compute in terms oftensor index notation with respect to a (not necessarily orthonormal) basis$\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right\}$ in a tangent space${\displaystyle V=T_{p}M}$ and its dual basis${\displaystyle \{d{x}_1,\dots ,d{x}_n\}}$ in${\displaystyle V^{\ast }=T_p^{\ast }M}$, having the metric matrix$(g_{ij})=\left(⟨\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}⟩\right)$ and its inverse matrix${\displaystyle (g^{ij})=(⟨d{x}^i,d{x}^j⟩)}$. The Hodge dual of a decomposablek-form is:$${\displaystyle \star \left(d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}\right)=\frac{\sqrt{\left|det[g_{ij}]\right|}}{(n-k)!}{g}^{i_1{j}_1}\cdots g^{i_k{j}_k}{\epsilon }_{j_1\dots j_n}d{x}^{j_{k+1}}\wedge \cdots \wedge d{x}^{j_n}.}$$

Here${\displaystyle {\epsilon }_{j_1\dots j_n}}$ is theLevi-Civita symbol with${\displaystyle {\epsilon }_{1\dots n}=1}$, and weimplicitly take the sum over all values of the repeated indices${\displaystyle j_1,\dots ,j_n}$. The factorial${\displaystyle (n-k)!}$ accounts for double counting, and is not present if the summation indices are restricted so that. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces toLorentzian manifolds.

An arbitrary differential form can be written as follows:

The factorial${\displaystyle k!}$ is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component${\displaystyle {\alpha }_{i_1,\dots ,i_k}}$ so that the Hodge dual of the form is given by$${\displaystyle \star \alpha =\frac{1}{(n-k)!}(\star \alpha {)}_{i_{k+1},\dots ,i_n}d{x}^{i_{k+1}}\wedge \cdots \wedge d{x}^{i_n}.}$$

Using the above expression for the Hodge dual of${\displaystyle d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_k}}$, we find:^[3]$${\displaystyle (\star \alpha {)}_{j_{k+1},\dots ,j_n}=\frac{\sqrt{\left|det[g_{ab}]\right|}}{k!}{\alpha }_{i_1,\dots ,i_k}{g}^{i_1{j}_1}\cdots g^{i_k{j}_k}{\epsilon }_{j_1,\dots ,j_n}.}$$

Although one can apply this expression to any tensor${\displaystyle \alpha }$, 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$\omega =\star 1\in \stackrel{n}{\bigwedge }{V}^{\ast }$ is given by:$${\displaystyle \omega =\sqrt{\left|det[g_{ij}]\right|}d{x}^1\wedge \cdots \wedge d{x}^n.}$$

The most important application of the Hodge star on manifolds is to define thecodifferential${\displaystyle \delta }$ on${\displaystyle k}$-forms. Let$${\displaystyle \delta =(-1{)}^{n(k+1)+1}s\star d\star =(-1{)}^k{\star }^{-1}d\star }$$where${\displaystyle d}$ is theexterior derivative or differential, and${\displaystyle s=1}$ for Riemannian manifolds. Then$${\displaystyle d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)}$$while$${\displaystyle \delta :{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k-1}(M).}$$

The codifferential is not anantiderivation on the exterior algebra, in contrast to the exterior derivative.

The codifferential is theadjoint of the exterior derivative with respect to the square-integrable scalar product:$${\displaystyle ⟨⟨\eta ,\delta \zeta ⟩⟩=⟨⟨d\eta ,\zeta ⟩⟩,}$$where${\displaystyle \zeta }$ is a${\displaystyle k}$-form and${\displaystyle \eta }$ a${\displaystyle (k-1)}$-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:$${\displaystyle 0={\int }_{M}d(\eta \wedge \star \zeta )={\int }_M\left(d\eta \wedge \star \zeta +(-1{)}^{k-1}\eta \wedge \star {\star }^{-1}d\star \zeta \right)=⟨⟨d\eta ,\zeta ⟩⟩-⟨⟨\eta ,\delta \zeta ⟩⟩,}$$provided${\displaystyle M}$ has empty boundary, or${\displaystyle \eta }$ or${\displaystyle \star \zeta }$ has zero boundary values. (The proper definition of the above requires specifying atopological vector space that is closed and complete on the space of smooth forms. TheSobolev space is conventionally used; it allows the convergent sequence of forms${\displaystyle {\zeta }_i\to \zeta }$ (as${\displaystyle i\to \mathrm{\infty }}$) to be interchanged with the combined differential and integral operations, so that${\displaystyle ⟨⟨\eta ,\delta {\zeta }_i⟩⟩\to ⟨⟨\eta ,\delta \zeta ⟩⟩}$ and likewise for sequences converging to${\displaystyle \eta }$.)

Since the differential satisfies${\displaystyle d^2=0}$, the codifferential has the corresponding property$${\displaystyle {\delta }^2=(-1{)}^n{s}^2\star d\star \star d\star =(-1{)}^{nk+k+1}{s}^3\star d^2\star =0.}$$

TheLaplace–deRham operator is given by$${\displaystyle \mathrm{\Delta }=(\delta +d{)}^2=\delta d+d\delta }$$and lies at the heart ofHodge theory. It is symmetric:$${\displaystyle ⟨⟨\mathrm{\Delta }\zeta ,\eta ⟩⟩=⟨⟨\zeta ,\mathrm{\Delta }\eta ⟩⟩}$$and non-negative:$${\displaystyle ⟨⟨\mathrm{\Delta }\eta ,\eta ⟩⟩\ge 0.}$$

The Hodge star sendsharmonic forms to harmonic forms. As a consequence ofHodge theory, thede Rham cohomology is naturally isomorphic to the space of harmonick-forms, and so the Hodge star induces an isomorphism of cohomology groups$${\displaystyle \star :H_{\mathrm{\Delta }}^k(M)\to H_{\mathrm{\Delta }}^{n-k}(M),}$$which in turn gives canonical identifications viaPoincaré duality ofH^k(M) with itsdual space.

In coordinates, with notation as above, the codifferential of the form${\displaystyle \alpha }$ may be written as$${\displaystyle \delta \alpha =-\frac{1}{k!}{g}^{ml}\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_l}{\alpha }_{m,i_1,\dots ,i_{k-1}}-{\mathrm{\Gamma }}_{ml}^j{\alpha }_{j,i_1,\dots ,i_{k-1}}\right)d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_{k-1}},}$$where here${\displaystyle {\mathrm{\Gamma }}_{ml}^j}$ denotes theChristoffel symbols of$\left\{\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}\right\}$.

In analogy to thePoincare lemma forexterior derivative, one can define its version for codifferential, which reads^[4]

If${\displaystyle \delta \omega =0}$for${\displaystyle \omega \in {\mathrm{\Lambda }}^k(U)}$, where${\displaystyle U}$is astar domain on a manifold, then there is${\displaystyle \alpha \in {\mathrm{\Lambda }}^{k+1}(U)}$such that${\displaystyle \omega =\delta \alpha }$.

A practical way of finding${\displaystyle \alpha }$ is to use cohomotopy operator${\displaystyle h}$, that is a local inverse of${\displaystyle \delta }$. One has to define ahomotopy operator^[4]

${\displaystyle H\beta ={\int }_0^1\mathcal{K}⌟\beta {|}_{F(t,x)}{t}^{k}dt,}$

where${\displaystyle F(t,x)=x_0+t(x-x_0)}$ is the linear homotopy between its center${\displaystyle x_0\in U}$ and a point${\displaystyle x\in U}$, and the (Euler) vector${\displaystyle \mathcal{K}=\sum _{i=1}^n(x-x_0{)}^i{\mathrm{\partial }}_{x^i}}$ for${\displaystyle n=\dim (U)}$ is inserted into the form${\displaystyle \beta \in {\mathrm{\Lambda }}^{\ast }(U)}$. We can then define cohomotopy operator as^[4]

${\displaystyle h:\mathrm{\Lambda }(U)\to \mathrm{\Lambda }(U),h:=\eta {\star }^{-1}H\star }$,

where${\displaystyle \eta \beta =(-1{)}^k\beta }$ for${\displaystyle \beta \in {\mathrm{\Lambda }}^k(U)}$.

The cohomotopy operator fulfills (co)homotopy invariance formula^[4]

${\displaystyle \delta h+h\delta =I-S_{x_0},}$

where${\displaystyle S_{x_0}={\star }^{-1}{s}_{x_0}^{\ast }\star }$, and${\displaystyle s_{x_0}^{\ast }}$ is thepullback along the constant map${\displaystyle s_{x_0}:x\to x_0}$.

Therefore, if we want to solve the equation${\displaystyle \delta \omega =0}$, applying cohomotopy invariance formula we get

${\displaystyle \omega =\delta h\omega +S_{x_0}\omega ,}$ where${\displaystyle h\omega \in {\mathrm{\Lambda }}^{k+1}(U)}$ is a differential form we are looking for, and "constant of integration"${\displaystyle S_{x_0}\omega }$ vanishes unless${\displaystyle \omega }$ is a top form.

Cohomotopy operator fulfills the following properties:^[4]${\displaystyle h^2=0,\delta h\delta =\delta ,h\delta h=h}$. They make it possible to use it to define^[4]anticoexact forms on${\displaystyle U}$ by${\displaystyle \mathcal{Y}(U)=\{\omega \in \mathrm{\Lambda }(U)|\omega =h\delta \omega \}}$, which together withexact forms${\displaystyle \mathcal{C}(U)=\{\omega \in \mathrm{\Lambda }(U)|\omega =\delta h\omega \}}$ make adirect sum decomposition^[4]

${\displaystyle \mathrm{\Lambda }(U)=\mathcal{C}(U)\oplus \mathcal{Y}(U)}$.

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and theprojector operators on the summands fulfillsidempotence formulas:^[4]${\displaystyle (h\delta {)}^2=h\delta ,(\delta h{)}^2=\delta h}$.

These results are extension of similar results for exterior derivative.^[5]

• Differential forms
• Riemannian geometry
• Duality theories
• Differential operators

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