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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hodge star operator or Hodge star is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
defined on the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of a
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
endowed with a
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
. 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 In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...
of two basis vectors, and its Hodge dual is the
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
given by their
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
; 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 coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s \tbinom nk = \tbinom. The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent
bundle Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction pat ...
of a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, and hence to differential -forms. This allows the definition of the codifferential as the Hodge adjoint of the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of a vector field may be realized as the codifferential opposite to the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
operator, and the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
on a function is the divergence of its gradient. An important application is the
Hodge decomposition In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
of differential forms on a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
Riemannian manifold.


Formal definition for ''k''-vectors

Let be an -dimensional
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
with a nondegenerate symmetric bilinear form \langle \cdot,\cdot \rangle, referred to here as an inner product. This induces an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on ''k''-vectors for 0 \le k \le n, by defining it on decomposable -vectors \alpha = \alpha_1 \wedge \cdots \wedge \alpha_k and \beta = \beta_1 \wedge \cdots \wedge \beta_k to equal the Gram determinant
Harley Flanders Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory, linear algebra, electrical networks, scientific computing. ...
(1963) ''Differential Forms with Applications to the Physical Sciences'',
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ...
: \langle \alpha, \beta \rangle = \det \left( \left\langle \alpha_i, \beta_j \right\rangle _^k\right) extended to \bigwedge^V through linearity. The unit -vector \omega\in^V is defined in terms of an oriented
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
\ of as: :\omega := e_1\wedge\cdots\wedge e_n. The Hodge star operator is a linear operator on the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of , mapping -vectors to ()-vectors, for 0 \le k \le n. It has the following property, which defines it completely: :\alpha \wedge ( \beta) = \langle \alpha,\beta \rangle \,\omega for every pair of -vectors \alpha,\beta\in ^V . Dually, in the space ^V^*of -forms (alternating -multilinear functions on V^n), the dual to \omega is the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
\det, the function whose value on v_1\wedge\cdots\wedge v_n is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the n\times n matrix assembled from the column vectors of v_i in e_i-coordinates. Applying \det to the above equation, we obtain the dual definition: :\det(\alpha \wedge \beta) = \langle \alpha,\beta \rangle . or equivalently, taking \alpha = \alpha_1 \wedge \cdots \wedge \alpha_k, \beta = \beta_1 \wedge \cdots \wedge \beta_k, and \star\beta = \beta_1^\star \wedge \cdots \wedge \beta_^\star: : \det\left(\alpha_1\wedge \cdots \wedge\alpha_k\wedge\beta_1^\star\wedge \cdots \wedge\beta_^\star\right) \ = \ \det\left(\langle\alpha_i, \beta_j\rangle\right). This means that, writing an orthonormal basis of -vectors as e_I \ = \ e_\wedge\cdots\wedge e_ over all subsets I = \ of \, the Hodge dual is the ()-vector corresponding to the complementary set \bar = setminus I = \left\: : e_I = (-1)^ e_\bar , where (-1)^ is the
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
of the permutation \sigma(I) = i_1 \cdots i_k \bar i_1 \cdots \bar i_. Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
on the exterior algebra \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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
and a numerical scaling factor. Specifically, a non-zero decomposable -vector w_1\wedge\cdots\wedge w_k\in \textstyle\bigwedge^ V corresponds by the
Plücker embedding In mathematics, the Plücker map embeds the Grassmannian \mathbf(k,V), whose elements are ''k''-dimensional subspaces of an ''n''-dimensional vector space ''V'', in a projective space, thereby realizing it as an algebraic variety. More precisely ...
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 In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
, 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\cdots\wedge w_k) \,=\, u_1\wedge\cdots\wedge u_, where u_1,\ldots,u_ 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_ must form an oriented basis of . A general -vector is a linear combination of decomposable -vectors, and the definition of the 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 \begin \, 1 &= dx \wedge dy \\ \, dx &= dy \\ \, dy &= -dx \\ ( dx \wedge dy ) &= 1 . \end On the complex plane regarded as a real vector space with the standard
sesquilinear form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
as the metric, the Hodge star has the remarkable property that it is invariant under
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
changes of coordinate. If is a holomorphic function of , then by the
Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differ ...
we have that and . In the new coordinates \alpha \ =\ p \,dx + q \,dy \ =\ \left( p \frac + q \frac \right) \,du + \left( p \frac + q \frac \right) \,dv \ =\ p_1 du + q_1 \, dv , so that \begin \alpha &= -q_1 \,du + p_1 \,dv \\ pt &= - \left( p \frac + q \frac \right) du + \left(p \frac + q \frac \right) dv \\ pt &= -q \left( \frac du + \frac dv \right) + p \left( \frac du + \frac dv \right) \\ pt &= -q\,dx + p\, dy, \end proving 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-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
s often used in
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, one finds that \begin \,dx &= dy \wedge dz \\ \,dy &= dz \wedge dx \\ \,dz &= dx \wedge dy. \end The Hodge star relates the exterior and cross product in three dimensions: (\mathbf \wedge \mathbf) = \mathbf \times \mathbf \qquad (\mathbf \times \mathbf ) = \mathbf \wedge \mathbf . Applied to three dimensions, the Hodge star provides an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
between
axial vector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
s and
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a d ...
s, so each axial vector is associated with a bivector and vice versa, that is: \mathbf = \mathbf, \ \ \mathbf = \mathbf. The Hodge star can also be interpreted as a form of the geometric correspondence between an axis and an infinitesimal rotation around the axis, with speed equal to the length of the axis vector. An inner product on a vector space V gives an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
V\cong V^*\! identifying V with its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
, and the vector space L(V,V) is naturally isomorphic to the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
V^*\!\!\otimes V\cong V\otimes V. Thus for V = \mathbb^3, the star mapping \textstyle \star:V\to\bigwedge^\! V \subset V\otimes V takes each vector \mathbf to a bivector \star \mathbf \in V\otimes V, which corresponds to a linear operator L_ : V\to V. Specifically, L_ is a skew-symmetric operator, which corresponds to an
infinitesimal rotation In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end ...
: that is, the macroscopic rotations around the axis \mathbb are given by the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
\exp(t L_). With respect to the basis dx, dy, dz of \R^3, the tensor dx\otimes dy corresponds to a coordinate matrix with 1 in the dx row and dy column, etc., and the wedge dx\wedge dy \,=\, dx\otimes dy - dy\otimes dx is the skew-symmetric matrix \scriptscriptstyle\left begin \,0\!\!_&_\!\!1_&_\!\!\!\!0\!\!\!\!\!\!_ \\[-.5em\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!_ \\[-.5em.html" ;"title=".5em.html" ;"title="begin \,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! \\[-.5em">begin \,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! \\[-.5em\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\! \\[-.5em">.5em.html" ;"title="begin \,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! \\[-.5em">begin \,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! \\[-.5em\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\! \\[-.5em\,0\!\! & \!\!0\!\! & \!\!\!\!0\!\!\!\!\!\! \end\!\!\!\right], etc. That is, we may interpret the star operator as: \mathbf = a\,dx + b\,dy + c\,dz \quad\longrightarrow \quad \star \ \cong\ L_ \ = \left[\begin 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end\right]. Under this correspondence, cross product of vectors corresponds to the commutator Lie algebra, Lie bracket of linear operators: L_ = L_ L_ - L_ L_.


Four dimensions

In case n=4, the Hodge star acts as an
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
of the second exterior power (i.e. it maps 2-forms to 2-forms, since ). If the signature of the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
is all positive, i.e. on a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, then the Hodge star is an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
. If the signature is mixed, i.e.,
pseudo-Riemannian In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which t ...
, 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 In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
is oriented as \varepsilon_ = 1. For
one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
s, \begin \star dt &= -dx \wedge dy \wedge dz \,, \\ \star dx &= -dt \wedge dy \wedge dz \,, \\ \star dy &= dt \wedge dx \wedge dz \,, \\ \star dz &= -dt \wedge dx \wedge dy \,, \end while for
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, \begin \star (dt \wedge dx) &= - dy \wedge dz \,, \\ \star (dt \wedge dy) &= - dz \wedge dx \,, \\ \star (dt \wedge dz) &= - dx \wedge dy \,, \\ \star (dx \wedge dy) &= dt \wedge dz \,, \\ \star (dx \wedge dz) &= - dt \wedge dy \,, \\ \star (dy \wedge dz) &= dt \wedge dx \,. \end These are summarized in the index notation as \begin \star (dx^\mu) &= \eta^ \varepsilon_ \frac dx^\nu \wedge dx^\rho \wedge dx^\sigma \,,\\ \star (dx^\mu \wedge dx^\nu) &= \eta^ \eta^ \varepsilon_ \frac dx^\rho \wedge dx^\sigma \,. \end 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 \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^\mu \wedge dx^\nu)^ := \frac \big( dx^\mu \wedge dx^\nu \mp i \star (dx^\mu \wedge dx^\nu) \big) take \pm i as the eigenvalue for Hodge star operator, i.e., \star (dx^\mu \wedge dx^\nu)^ = \pm i (dx^\mu \wedge dx^\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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physical Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or
twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena ...
.


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 \lambda > 0 , then the induced Hodge stars \star_g, \star_ \colon \Lambda^n V \to \Lambda^n V are the same.


Example: Derivatives in three dimensions

The combination of the \star operator and the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
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 = \frac \, dx + \frac \, dy + \frac \, dz. The inner product identifies 1-forms with vector fields as dx \mapsto (1,0,0), etc., so that df becomes \operatorname f = \left(\frac, \frac, \frac\right). In the second case, a vector field \mathbf F = (A,B,C) corresponds to the 1-form \varphi = A\,dx + B\,dy + C\,dz, which has exterior derivative: d\varphi = \left(\frac - \frac\right) dy\wedge dz + \left(\frac - \frac\right) dx\wedge dz + \left( - \frac\right) dx\wedge dy. Applying the Hodge star gives the 1-form: \star d\varphi = \left( - \right) \, dx - \left( - \right) \, dy + \left( - \right) \, dz, which becomes the vector field \operatorname\mathbf = \left( \frac - \frac,\, -\frac + \frac,\, \frac - \frac \right). In the third case, \mathbf F = (A,B,C) again corresponds to \varphi = A\,dx + B\,dy + C\,dz. Applying Hodge star, exterior derivative, and Hodge star again: \begin \star\varphi &= A\,dy\wedge dz-B\,dx\wedge dz+C\,dx\wedge dy, \\ d &= \left(\frac+\frac+\frac\right)dx\wedge dy\wedge dz, \\ \star d &= \frac+\frac+\frac = \operatorname\mathbf. \end 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 Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
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 mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in terms of the above operations: \Delta f =\star d= \frac + \frac + \frac. The Laplacian can also be seen as a special case of the more general Laplace–deRham operator \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 for its sign: for \eta\in ^k V in an -dimensional space , one has : \eta = (-1)^ s \eta , where is the parity of the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the inner product on , that is, the sign of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
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 ^: ~ ^ V &\to ^ V \\ \eta &\mapsto (-1)^ \!s \eta \end If is odd then is even for any , whereas if is even then has the parity of . Therefore: :^ = \begin s & n \text \\ (-1)^k s & n \text \end where is the degree of the element operated on.


On manifolds

For an ''n''-dimensional oriented
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
''M'', we apply the construction above to each
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
\text^*_p M and its exterior powers \bigwedge^k\text^*_p M, and hence to the differential ''k'' -forms \zeta\in\Omega^k(M) = \Gamma\left(\bigwedge^k\text^*\!M\right), the global sections of the
bundle Bundle or Bundling may refer to: * Bundling (packaging), the process of using straps to bundle up items Biology * Bundle of His, a collection of heart muscle cells specialized for electrical conduction * Bundle of Kent, an extra conduction pat ...
\bigwedge^k \mathrm^*\! M\to M. The Riemannian metric induces an inner product on \bigwedge^k \text^*_p M at each point p\in M. We define the Hodge dual of a ''k'' -form \zeta , defining \zeta as the unique (''n'' – ''k'')-form satisfying \eta\wedge \zeta \ =\ \langle \eta, \zeta \rangle \, \omega for every ''k''-form \eta , where \langle\eta,\zeta\rangle is a real-valued function on M, and the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
\omega is induced by the Riemannian metric. Integrating this equation over M, the right side becomes the L^2 (
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
) inner product on ''k''-forms, and we obtain: \int_M \eta\wedge \zeta \ =\ \langle\eta,\zeta\rangle\ \text. 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 In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
.


Computation in index notation

We compute in terms of
tensor index notation In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
with respect to a (not necessarily orthonormal) basis \left\ in a tangent space V = T_p M and its dual basis \ in V^* = T^*_p M, having the metric matrix (g_) = \left(\left\langle \frac, \frac\right\rangle\right) and its inverse matrix (g^) = (\langle dx^i, dx^j\rangle). The Hodge dual of a decomposable ''k''-form is: \star\left(dx^ \wedge \dots \wedge dx^\right) \ =\ \frac g^ \cdots g^ \varepsilon_ dx^ \wedge \dots \wedge dx^. Here \varepsilon_ is the
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some ...
with \varepsilon_ = 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_ < \dots < j_n. The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to
Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
s. An arbitrary differential form can be written as follows: \alpha \ =\ \frac\alpha_ dx^\wedge \dots \wedge dx^ \ =\ \sum_ \alpha_ dx^\wedge \dots \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_ so that the Hodge dual of the form is given by \star\alpha = \frac(\star \alpha)_ dx^ \wedge \dots \wedge dx^. Using the above expression for the Hodge dual of dx^ \wedge \dots \wedge dx^, we find: (\star \alpha)_ = \frac \alpha^\, \,\varepsilon_. 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 \omega = \star 1\in \bigwedge^n V^* is given by: \omega = \sqrt\;dx^1 \wedge \cdots \wedge dx^n .


Codifferential

The most important application of the Hodge star on manifolds is to define the codifferential \delta on ''k''-forms. Let \delta = (-1)^ s\ d = (-1)^\, ^ d where d is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
or differential, and s = 1 for Riemannian manifolds. Then d:\Omega^k(M)\to \Omega^(M) while \delta:\Omega^k(M)\to \Omega^(M). The codifferential is not an
antiderivation In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Le ...
on the exterior algebra, in contrast to the exterior derivative. The codifferential is the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of the exterior derivative with respect to the square-integrable inner product: \langle\!\langle\eta,\delta \zeta\rangle\!\rangle \ =\ \langle\!\langle d\eta,\zeta\rangle\!\rangle, where \zeta is a -form and \eta a -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 \ =\ \int_M d (\eta \wedge \zeta) \ =\ \int_M \left(d \eta \wedge \zeta - \eta \wedge (-1)^\,^ d \zeta\right) \ =\ \langle\!\langle d\eta,\zeta\rangle\!\rangle - \langle\!\langle\eta,\delta\zeta\rangle\!\rangle, provided ''M'' has empty boundary, or \eta or \star\zeta has zero boundary values. (The proper definition of the above requires specifying a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
that is closed and complete on the space of smooth forms. The
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
is conventionally used; it allows the convergent sequence of forms \zeta_i \to \zeta (as i \to \infty) to be interchanged with the combined differential and integral operations, so that \langle\!\langle\eta,\delta \zeta_i\rangle\!\rangle \to \langle\!\langle\eta,\delta \zeta\rangle\!\rangle and likewise for sequences converging to \eta.) Since the differential satisfies d^2 = 0, the codifferential has the corresponding property \delta^2 = s^2 d d = (-1)^ s^3 d^2 = 0. The Laplace–deRham operator is given by \Delta = (\delta + d)^2 = \delta d + d\delta and lies at the heart of
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
. It is symmetric: \langle\!\langle\Delta \zeta,\eta\rangle\!\rangle = \langle\!\langle\zeta,\Delta \eta\rangle\!\rangle and non-negative: \langle\!\langle\Delta\eta,\eta\rangle\!\rangle \ge 0. The Hodge star sends
harmonic form In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
s to harmonic forms. As a consequence of
Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
, the
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
is naturally isomorphic to the space of harmonic -forms, and so the Hodge star induces an isomorphism of cohomology groups : H^k_\Delta (M) \to H^_\Delta(M), which in turn gives canonical identifications via
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact a ...
of with its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
. In coordinates, with notation as above, the codifferential of the form \alpha may be written as \delta \alpha=\ -\fracg^\left(\frac \alpha_ - \Gamma^j_ \alpha_ \right) dx^ \wedge \dots \wedge dx^, where here \Gamma^_ denotes the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
of \left\.


Poincare lemma for codifferential

In analogy to the Poincare lemma for
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, one can define its version for codifferential, which reads ''If'' \delta\omega=0 ''for'' \omega \in \Lambda^(U)'', where'' U ''is a
star domain In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...
on a manifold, then there is'' \alpha \in \Lambda^(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 operator In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain c ...
H\beta = \int_^ \mathcal\lrcorner\beta, _t^dt, where F(t,x)=x_+t(x-x_) is the linear homotopy between its center x_\in U and a point x \in U, and the (Euler) vector \mathcal=\sum_^(x-x_)^\partial_ for n=dim(U) is inserted into the form \beta \in \Lambda^(U). We can then define cohomotopy operator as h:\Lambda(U)\rightarrow \Lambda(U), \quad h:=\eta \star^H\star, where \eta \beta = (-1)^\beta for \beta \in \Lambda^(U). The cohomotopy operator fulfills (co)homotopy invariance formula \delta h + h\delta = I - S_, where S_=\star^s_^\star, and s_^is the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: in ...
along the constant map s_:x \rightarrow x_. Therefore, if we want to solve the equation \delta \omega =0, applying cohomotopy invariance formula we get \omega= \delta h\omega + S_\omega, where h\omega\in \Lambda^(U) is a differential form we are looking for, and 'constant of integration' S_\omega vanishes unless \omega is a top form. Cohomotopy operator fulfills the following properties: h^=0, \quad \delta h \delta =\delta, \quad h\delta h =h. They make it possible to use it to define ''anticoexact'' forms on U by \mathcal(U)=\, which together with exact forms \mathcal(U) =\ make a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
decomposition \Lambda(U)=\mathcal(U)\oplus \mathcal(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 Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
formulas: (h\delta)^=h\delta, \quad (\delta h)^=\delta h. These results are extension of similar results for exterior derivative.


Citations


References

* David Bleecker (1981) ''Gauge Theory and Variational Principles''. Addison-Wesley Publishing. . Chpt. 0 contains a condensed review of non-Riemannian differential geometry. * *
Charles W. Misner Charles W. Misner (; born June 13, 1932) is an American physicist and one of the authors of '' Gravitation''. His specialties include general relativity and cosmology. His work has also provided early foundations for studies of quantum gravity ...
, Kip S. Thorne,
John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in e ...
(1970) ''Gravitation''. W.H. Freeman. . A basic review of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
in the special case of four-dimensional
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. * Steven Rosenberg (1997) ''The Laplacian on a Riemannian manifold''. Cambridge University Press. . An introduction to the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
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. {{DEFAULTSORT:Hodge Dual Differential forms Riemannian geometry Duality theories