Hodge star operator explained

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients

\tbinom nk = \tbinom{n}{n - k} .

The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential -forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.

Formal definition for k-vectors

Let be an -dimensional oriented vector space with a nondegenerate symmetric bilinear form

\langle ⋅ , ⋅ \rangle , referred to here as an inner product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive.) This induces an inner product on k-vectors for 0 \le k \le n , by defining it on decomposable -vectors \alpha = \alpha _{1} \wedge … \wedge \alpha _{k} and \beta = \beta _{1} \wedge … \wedge \beta _{k} to equal the Gram determinant^[1] \langle \alpha, \beta \rangle = \det \left( \left\langle \alpha _{i,} \beta _{j} \right\rangle ^{k\right)} ^{i,j =1}

extended to $\backslash bigwedge^V$ through linearity.

The unit -vector

\omega\in{ style w edge} ^{n} V is defined in terms of an oriented orthonormal basis \{e _{1,\ldots,e} _{n\}} of as:\omega : = e _{1\wedge … \wedge} e _{n.}

(Note: In the general pseudo-Riemannian case, orthonormality means

\langle e _{i,e} _{j\rangle \in\{\delta} _{ij} , - \delta _{ij} \} for all pairs of basis vectors.)The Hodge star operator is a linear operator on the exterior algebra of, mapping -vectors to -vectors, for 0 \le k \le n . It has the following property, which defines it completely:\alpha \wedge ({\star} \beta) = \langle \alpha,\beta \rangle \omega for all -vectors \alpha,\beta\in { style w edge} ^{k} V .

Dually, in the space

{ style w edge} ^{n} V ^{*} of -forms (alternating -multilinear functions on V ^{n} ), the dual to \omega is the volume form \det , the function whose value on v _{1\wedge … \wedge} v _{n} is the determinant of the n x n matrix assembled from the column vectors of v _{j} in e _{i} -coordinates. Applying \det to the above equation, we obtain the dual definition:\det(\alpha \wedge {\star} \beta) = \langle \alpha,\beta \rangle for all -vectors \alpha,\beta\in { style w edge} ^{k} V .

Equivalently, taking

\alpha = \alpha _{1} \wedge … \wedge \alpha _{k} , \beta = \beta _{1} \wedge … \wedge \beta _{k} , and \star\beta = \beta ^{\star} _{1} \wedge … \wedge \beta ^{\star} _{n - k} :\det\left(\alpha _{1\wedge} … \wedge\alpha _{k\wedge\beta} ^{\star\wedge} ^{1} … \wedge\beta ^{\star\right)} _{n - k} = \det\left(\langle\alpha _{i,} \beta _{j\rangle\right).}

This means that, writing an orthonormal basis of -vectors as

e _{I} = e _{i _{1}} \wedge … \wedge e _{i _{k}} over all subsets I = \{i _{1< … <i} _{k\}} of [n] = \{ 1 ,\ldots,n\} , the Hodge dual is the -vector corresponding to the complementary set \bar{I} = [n]\setminus I = \left\{\bar i _{1} < … < \bar i _{n-k} \right\} :{\star} e _{I} = s ⋅ t ⋅ e _{\bar{I}} ,

where

s\in\{ 1 , -1 \} is the sign of the permutation i _{1} … i _{k} \bar i _{1} … \bar i _{n-k} and t\in\{ 1 , -1 \} is the product\langle e _{i _{1}} ,e _{i _{1}} \rangle … \langle e _{i _{k}} ,e _{i _{k}} \rangle . In the Riemannian case, t =1 .

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

Geometric explanation

The Hodge star is motivated by the correspondence between a subspace of and its orthogonal subspace (with respect to the inner product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable -vector

w _{1\wedge … \wedge} w _{k\in} style w edge ^{k} V corresponds by the Plücker embedding to the subspace W with oriented basis w _{1,\ldots,w} _{k} , endowed with a scaling factor equal to the -dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of inner products \langle w _{i,} w _{j} \rangle ). The Hodge star acting on a decomposable vector can be written as a decomposable -vector:\star(w _{1\wedge … \wedge} w _{k)} = u _{1\wedge … \wedge} u _{n-k} , where u _{1,\ldots,u} _{n-k} form an oriented basis of the orthogonal space U = W ^{\perp} . Furthermore, the -volume of the u _{i} -parallelepiped must equal the -volume of the w _{i} -parallelepiped, and w _{1,\ldots,w} _{k,u} _{1,\ldots,u} _{n-k} must form an oriented basis of V .

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

Examples

Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering, the Hodge star on -forms is given by

On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If is a holomorphic function of, then by the Cauchy–Riemann equations we have that and . In the new coordinates$$\backslash alpha\; \backslash \; =\backslash \; p\; \backslash ,dx\; +\; q\; \backslash ,dy\; \backslash \; =\backslash \; \backslash left(p\; \backslash frac\; +\; q\; \backslash frac\; \backslash right)\; \backslash ,du\; +\; \backslash left(p\; \backslash frac\; +\; q\; \backslash frac\; \backslash right)\; \backslash ,dv\; \backslash \; =\backslash \; p\_1\; \backslash ,\; du\; +\; q\_1\; \backslash ,\; dv,$$so thatproving the claimed invariance.

Three dimensions

A common example of the Hodge star operator is the case, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R³ with the basis

dx, dy, dz of one-forms often used in vector calculus, one finds that

The Hodge star relates the exterior and cross product in three dimensions: $$(\backslash mathbf\; \backslash wedge\; \backslash mathbf)\; =\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash qquad\; (\backslash mathbf\; \backslash times\; \backslash mathbf)\; =\; \backslash mathbf\; \backslash wedge\; \backslash mathbf\; .$$ Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa, that is:^[2]

A = {\star} a , a = {\star} A .

The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. An inner product on a vector space

V gives an isomorphism V\cong V ^{*} identifying V with its dual space, and the vector space L(V,V) is naturally isomorphic to the tensor product V ^{* ⊗} V\cong V ⊗ V . Thus for V = R ^{3} , the star mapping $\backslash textstyle\; \backslash star\backslash colon\; V\backslash to\backslash bigwedge^\backslash !\; V\; \backslash subset\; V\backslash otimes\; V$ takes each vector v to a bivector \star v \in V ⊗ V , which corresponds to a linear operator L _{v} \colon V\to V . Specifically, L _{v} is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis v are given by the matrix exponential \exp(t L _{v} ) . With respect to the basis dx, dy, dz of \R ^{3} , the tensor dx ⊗ dy corresponds to a coordinate matrix with 1 in the dx row and dy column, etc., and the wedge dx\wedge dy = dx ⊗ dy - dy ⊗ dx is the skew-symmetric matrix \scriptscriptstyle\left[\begin{array}{rrr} 0 & 1 & 0 \\[ - . 5 em] -1 & 0 & 0 \\[ - . 5 em] 0 & 0 & 0 \end{array}\right] , etc. That is, we may interpret the star operator as: Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: L _{u x v} = L _{v} L _{u} - L _{u} L _{v} =- \left[L _{u} , L _{v} \right] .

Four dimensions

In case

n =4 , the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since). If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see below. This particular endomorphism property of 2-forms in four dimensions makes self-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 \pm 1 (or \pm i , depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where

n =4 with metric signature and coordinates (t,x,y,z) . The volume form is oriented as \varepsilon _{0123} = 1 . For one-forms,while for 2-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,

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

Note that the combinations$$(dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^\; :=\; \backslash frac\; \backslash big(dx^\backslash mu\; \backslash wedge\; dx^\backslash nu\; \backslash mp\; i\; \backslash star\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)\; \backslash big)$$take

\pm i as the eigenvalue for Hodge star operator, i.e.,$$\backslash star\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^\; =\; \backslash pm\; i\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^,$$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 both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

Conformal invariance

The Hodge star is conformally invariant on n forms on a 2n dimensional vector space V, i.e. if

g is a metric on V and λ > 0 , then the induced Hodge stars $$\backslash star\_g,\; \backslash star\_\; \backslash colon\; \backslash Lambda^n\; V\; \backslash to\; \backslash Lambda^n\; V$$are the same.

Example: Derivatives in three dimensions

The combination of the

\star operator and the exterior derivative generates the classical operators,, and on vector fields in three-dimensional Euclidean space. This works out as follows: 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 f = f(x,y,z) , the first case written out in components gives:$$df\; =\; \backslash frac\; \backslash ,\; dx\; +\; \backslash frac\; \backslash ,\; dy\; +\; \backslash frac\; \backslash ,\; dz.$$

The inner product identifies 1-forms with vector fields as

dx \mapsto ( 1 , 0 , 0 ) , etc., so that df becomes $\backslash operatorname\; f\; =\; \backslash left(\backslash frac,\; \backslash frac,\; \backslash frac\backslash right)$.

In the second case, a vector field

F = (A,B,C) corresponds to the 1-form \varphi = Adx + Bdy + Cdz , which has exterior derivative:$$d\backslash varphi\; =\backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; dy\backslash wedge\; dz\; +\backslash left(\backslash frac\; -\; \backslash frac\backslash right)\; dx\backslash wedge\; dz\; +\backslash left(-\; \backslash frac\backslash right)\; dx\backslash wedge\; dy.$$

Applying the Hodge star gives the 1-form:$$\backslash star\; d\backslash varphi\; =\; \backslash left(-\; \backslash right)\; \backslash ,\; dx\; -\; \backslash left(-\; \backslash right)\; \backslash ,\; dy\; +\; \backslash left(-\; \backslash right)\; \backslash ,\; dz,$$which becomes the vector field $\backslash operatorname\backslash mathbf\; =\; \backslash left(\backslash frac\; -\; \backslash frac,\backslash ,\; -\backslash frac\; +\; \backslash frac,\backslash ,\; \backslash frac\; -\; \backslash frac\backslash right)$.

In the third case,

F = (A,B,C) again corresponds to \varphi = Adx + Bdy + Cdz . Applying Hodge star, exterior derivative, and Hodge star again:

One advantage of this expression is that the identity, which is true in all cases, has as special cases two other identities: 1), and 2) . 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

\star d\star (multiplied by an appropriate power of -1) is called the codifferential; it is defined in full generality, for any dimension, further in the article below.

One can also obtain the Laplacian in terms of the above operations:$$\backslash Delta\; f\; =\backslash star\; d=\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac.$$

\Delta = d\delta + \delta d where \delta = ( -1 ) ^{k} \star d\star is the codifferential for k -forms. Any function f is a 0-form, and \delta f = 0 and so this reduces to the ordinary Laplacian. For the 1-form \varphi above, the codifferential is \delta = - \star d\star and after some straightforward calculations one obtains the Laplacian acting on \varphi .

Duality

Applying the Hodge star twice leaves a -vector unchanged except possibly for its sign: for

η \in { style w edge} ^{k} V in an -dimensional space, one has{\star} {\star} η = ( -1 ) ^{k(n-k)} s η ,

where is the parity of the signature of the inner product on, that is, the sign of the determinant of the matrix of the inner product with respect to any basis. For example, if and the signature of the inner product is either or then . For Riemannian manifolds (including Euclidean spaces), we always have .

The above identity implies that the inverse of

\star can be given as\begin{align} {\star} ^{-1} : ~ { style w edge} ^{k} V &\to { style w edge} ^{n-k} V \\ η &\mapsto ( -1 ) ^{k(n-k)} s {\star} η \end{align}

If is odd then is even for any, whereas if is even then has the parity of . Therefore:

{\star} ^{-1} = \begin{cases} s {\star} & n isodd \ ( -1 ) ^{k} s {\star} & n iseven \end{cases}

where is the degree of the element operated on.

On manifolds

T ^{*} _{p} M and its exterior powers $\backslash bigwedge^k\backslash text^*\_p\; M$, and hence to the differential k-forms $\backslash zeta\backslash in\backslash Omega^k(M)\; =\; \backslash Gamma\backslash left(\backslash bigwedge^k\backslash text^*\backslash !M\backslash right)$, the global sections of the bundle $\backslash bigwedge^k\; \backslash mathrm^*\backslash !\; M\backslash to\; M$. The Riemannian metric induces an inner product on $\backslash bigwedge^k\; \backslash text^*\_p\; M$ at each point p\in M . We define the Hodge dual of a k-form \zeta , defining {\star} \zeta as the unique (n – k)-form satisfying$$\backslash eta\backslash wedge\; \backslash zeta\; \backslash \; =\backslash \; \backslash langle\; \backslash eta,\; \backslash zeta\; \backslash rangle\; \backslash ,\; \backslash omega$$for every k-form η , where \langle η ,\zeta\rangle is a real-valued function on M , and the volume form \omega is induced by the pseudo-Riemannian metric. Integrating this equation over M , the right side becomes the L ^{2} (square-integrable) inner product on k-forms, and we obtain:$$\backslash int\_M\; \backslash eta\backslash wedge\; \backslash zeta\backslash \; =\backslash \; \backslash int\_M\; \backslash langle\backslash eta,\backslash zeta\backslash rangle\backslash \; \backslash omega.$$

More generally, if

M is non-orientable, one can define the Hodge star of a k-form as a (n – k)-pseudo differential form; that is, a differential form with values in the canonical line bundle.

Computation in index notation

We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis $\backslash left\backslash$ in a tangent space

V = T _{p} M and its dual basis \{dx _{1,\ldots,dx} _{n\}} in V ^{*} = T ^{*} _{p} M , having the metric matrix $(g\_)\; =\; \backslash left(\backslash left\backslash langle\; \backslash frac,\; \backslash frac\backslash right\backslash rangle\backslash right)$ and its inverse matrix (g ^{ij} ) = (\langle dx ^{i,} dx ^{j\rangle)} . The Hodge dual of a decomposable k-form is:$$\backslash star\backslash left(dx^\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^\backslash right)\; \backslash \; =\backslash \; \backslash frac\; g^\; \backslash cdots\; g^\; \backslash varepsilon\_\; dx^\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^.$$

Here

\varepsilon _{j _{1} ... j _{n}} is the Levi-Civita symbol with \varepsilon _{1} = 1 , and we implicitly take the sum over all values of the repeated indices j _{1,\ldots,j} _{n} . The factorial (n - k)! accounts for double counting, and is not present if the summation indices are restricted so that j _{k+1} < ... < j _{n} . The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds.

An arbitrary differential form can be written as follows:$$\backslash alpha\; \backslash \; =\backslash \; \backslash frac\backslash alpha\_\; dx^\backslash wedge\; \backslash dots\; \backslash wedge\; dx^\; \backslash \; =\backslash \; \backslash sum\_\; \backslash alpha\_\; dx^\backslash wedge\; \backslash dots\; \backslash wedge\; dx^.$$

The factorial

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 \alpha _{i _{1,} ... , i _{k}} so that the Hodge dual of the form is given by$$\backslash star\backslash alpha\; =\; \backslash frac(\backslash star\; \backslash alpha)\_\; dx^\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^.$$

Using the above expression for the Hodge dual of

dx ^{i _{1}} \wedge ... \wedge dx ^{i _{k}} , we find:^[3] $$(\backslash star\; \backslash alpha)\_\; =\; \backslash frac\; \backslash alpha\_\backslash ,g^\backslash cdots\; g^\; \backslash ,\backslash varepsilon\_\backslash ,\; .$$

Although one can apply this expression to any tensor

\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 $\backslash omega\; =\; \backslash star\; 1\backslash in\; \backslash bigwedge^n\; V^*$ is given by:$$\backslash omega\; =\; \backslash sqrt\backslash ;dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n\; .$$

Codifferential

The most important application of the Hodge star on manifolds is to define the codifferential

\delta on k -forms. Let$$\backslash delta\; =\; (-1)^\; s\backslash \; d\; =\; (-1)^\backslash ,\; ^\; d\; \backslash ,$$where d is the exterior derivative or differential, and s = 1 for Riemannian manifolds. Then$$d:\backslash Omega^k(M)\backslash to\; \backslash Omega^(M)$$while$$\backslash delta:\backslash Omega^k(M)\backslash to\; \backslash Omega^(M).$$

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

The codifferential is the adjoint of the exterior derivative with respect to the square-integrable inner product:$$\backslash langle\backslash !\backslash langle\backslash eta,\backslash delta\; \backslash zeta\backslash rangle\backslash !\backslash rangle\; \backslash \; =\backslash \; \backslash langle\backslash !\backslash langle\; d\backslash eta,\backslash zeta\backslash rangle\backslash !\backslash rangle,$$where

\zeta is a k -form and η a (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:$$0\; \backslash \; =\backslash \; \backslash int\_M\; d\; (\backslash eta\; \backslash wedge\; \backslash zeta)\; \backslash \; =\backslash \; \backslash int\_M\; \backslash left(d\; \backslash eta\; \backslash wedge\; \backslash zeta\; +\; (-1)^\backslash eta\; \backslash wedge\; \backslash ,^\; d\backslash ,\; \backslash zeta\backslash right)\; \backslash \; =\backslash \; \backslash langle\backslash !\backslash langle\; d\backslash eta,\backslash zeta\backslash rangle\backslash !\backslash rangle\; -\; \backslash langle\backslash !\backslash langle\backslash eta,\backslash delta\backslash zeta\backslash rangle\backslash !\backslash rangle,$$provided M has empty boundary, or η or \star\zeta 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 \zeta _{i} \to \zeta (as i \to inf ty ) to be interchanged with the combined differential and integral operations, so that \langle\langle η ,\delta \zeta _{i\rangle\rangle} \to \langle\langle η ,\delta \zeta\rangle\rangle and likewise for sequences converging to η .)

Since the differential satisfies

d ^{2} = 0 , the codifferential has the corresponding property$$\backslash delta^2\; =\; (-1)^n\; s^2\; d\; d\; =\; (-1)^\; s^3\; d^2\; =\; 0.$$

The Laplace–deRham operator is given by$$\backslash Delta\; =\; (\backslash delta\; +\; d)^2\; =\; \backslash delta\; d\; +\; d\backslash delta$$and lies at the heart of Hodge theory. It is symmetric:$$\backslash langle\backslash !\backslash langle\backslash Delta\; \backslash zeta,\backslash eta\backslash rangle\backslash !\backslash rangle\; =\; \backslash langle\backslash !\backslash langle\backslash zeta,\backslash Delta\; \backslash eta\backslash rangle\backslash !\backslash rangle$$and non-negative:$$\backslash langle\backslash !\backslash langle\backslash Delta\backslash eta,\backslash eta\backslash rangle\backslash !\backslash rangle\; \backslash ge\; 0.$$

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic -forms, and so the Hodge star induces an isomorphism of cohomology groups$$:\; H^k\_\backslash Delta\; (M)\; \backslash to\; H^\_\backslash Delta(M),$$which in turn gives canonical identifications via Poincaré duality of with its dual space.

In coordinates, with notation as above, the codifferential of the form

\alpha may be written as$$\backslash delta\; \backslash alpha=\backslash \; -\backslash fracg^\backslash left(\backslash frac\; \backslash alpha\_\; -\; \backslash Gamma^j\_\; \backslash alpha\_\; \backslash right)\; dx^\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^,$$where here \Gamma ^{j} _{ml} denotes the Christoffel symbols of $\backslash left\backslash$.

Poincare lemma for codifferential

In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads^[4]

If

\delta\omega =0 for \omega \in Λ ^{k} (U) , where U is a star domain on a manifold, then there is \alpha \in Λ ^{k+1} (U) such that \omega = \delta\alpha .

A practical way of finding

\alpha is to use cohomotopy operator h , that is a local inverse of \delta . One has to define a homotopy operatorH\beta = \int ^{1} _{0} l{K}\lrcorner\beta| _{F(t,x)} t ^{k} dt,

where

F(t,x) = x _{0} + t(x - x _{0} ) is the linear homotopy between its center x _{0} \in U and a point x \in U , and the (Euler) vector l{K}=\sum ^{n} _{i =1} (x - x _{0} ) ^{i} \partial _{x ^{i}} for n = \dim(U) is inserted into the form \beta \in Λ ^{*} (U) . We can then define cohomotopy operator ash: Λ (U) → Λ (U), h: =η \star ^{-1} H\star ,

where

η \beta = ( -1 ) ^{k} \beta for \beta \in Λ ^{k} (U) .

The cohomotopy operator fulfills (co)homotopy invariance formula

\delta h + h\delta = I - S _{x _{0}} ,

where

S _{x _{0}} = \star ^{-1} s ^{*} _{x _{0}} \star , and s ^{*} _{x _{0}} is the pullback along the constant map s _{x _{0}} :x → x _{0} .

Therefore, if we want to solve the equation

\delta \omega =0 , applying cohomotopy invariance formula we get\omega = \delta h\omega + S _{x _{0}} \omega, where h\omega\in Λ ^{k+1} (U) is a differential form we are looking for, and ″constant of integration″ S _{x _{0}} \omega vanishes unless \omega is a top form.

Cohomotopy operator fulfills the following properties:

h ^{2} =0 , \delta h \delta = \delta, h\delta h = h . They make it possible to use it to define anticoexact forms on U by l{Y}(U)=\{ \omega\in Λ (U)| \omega = h\delta \omega \} , which together with exact forms l{C}(U) = \{ \omega\in Λ (U)|\omega = \delta h\omega \} make a direct sum decomposition Λ (U) = l{C}(U) ⊕ l{Y}(U) .

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the projector operators on the summands fulfills idempotence formulas:

(h\delta) ^{2} = h\delta, (\delta h) ^{2} = \delta h .

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

References

• David Bleecker (1981) Gauge Theory and Variational Principles. Addison-Wesley Publishing. . Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
• Book: Jost. Jürgen. Jürgen Jost. Riemannian Geometry and Geometric Analysis. 2002. Springer-Verlag. 3-540-42627-2.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. . A basic review of differential geometry in the special case of four-dimensional spacetime.
• Steven Rosenberg (1997) The Laplacian on a Riemannian manifold. Cambridge University Press. . An introduction to the heat equation and the Atiyah–Singer theorem.
• Tevian Dray (1999) The Hodge Dual Operator. A thorough overview of the definition and properties of the Hodge star operator.

Notes and References

1. [Harley Flanders]
2. Book: Clifford Algebras and Spinors, Volume 286 of London Mathematical Society Lecture Note Series . Pertti Lounesto . https://books.google.com/books?id=E_xvJuA4M7QC&pg=PA39 . 39 . §3.6 The Hodge dual . 0-521-00551-5 . 2001 . 2nd . Cambridge University Press.
3. Book: Frankel, T.. The Geometry of Physics. Cambridge University Press. 2012. 978-1-107-60260-1. 3rd.
4. Kycia . Radosław Antoni . 2022-07-29 . The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics . Results in Mathematics . en . 77 . 5 . 182 . 10.1007/s00025-022-01646-z . 1420-9012. 2009.08542 . 221802588 .
5. Book: Edelen, Dominic G. B. . Applied exterior calculus . 2005 . 978-0-486-43871-9 . Revised . Mineola, N.Y. . 56347718.