Hodge dual

In mathematics, the Hodge star operator or Hodge star is a linear map introduced by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's Hodge dual. In mathematics, the Hodge star operator or Hodge star is a linear map introduced by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. The result when applied to an element of the algebra is called the element's Hodge dual. For example, in 3-dimensional Euclidean space, every oriented plane has a unique normal vector, and every vector can be used to define a plane perpendicular to that vector. The Hodge star can be thought of as generalizing this relationship: in general, for an n-dimensional vector space, the Hodge star maps k-vectors to (n − k)-vectors and vice versa. Suppose that n is the dimension of the oriented inner product space and k is an integer such that 0 ≤ k ≤ n, then the Hodge star operator establishes a one-to-one mapping from the space of k-vectors to the space of (n − k)-vectors. The image of a k-vector under this mapping is called its Hodge dual. The former space, of k-vectors, has dimension ( n k ) {displaystyle { binom {n}{k}}} , while the latter has dimension ( n n − k ) {displaystyle { binom {n}{n-k}}} , which are equal by the symmetry of the binomial coefficients. Equal-dimensional vector spaces are always isomorphic, but not necessarily in a natural or canonical way. In this case, however, Hodge duality exploits the nondegenerate symmetric bilinear form, hereafter referred to as the inner product (though it might not be positive definite), and a choice of orientation to single out a unique isomorphism, which parallels the combinatorial symmetry of binomial coefficients. This in turn induces an inner product on the space of k-vectors. The naturalness of this definition means the duality can play a role in differential geometry. The first interesting case is on three-dimensional Euclidean space V. In this context the relevant row of Pascal's triangle reads and the Hodge star sets up an isomorphism between the two three-dimensional spaces, which are V and the image space of the exterior product acting on pairs of vectors in V. See § Examples for details. In this case the Hodge star allows the definition of the cross product of traditional vector calculus in terms of the exterior product. While the properties of the cross product are special to three dimensions, the Hodge star applies to an arbitrary number of dimensions. Since the space of alternating linear forms in k arguments on a vector space is naturally isomorphic to the dual of the space of k-vectors over that vector space, the Hodge star can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge star can then be extended to a vector bundle. Thus a context in which the Hodge star is very often seen is the exterior algebra of the cotangent bundle, the space of differential forms on a manifold, where it can be used to construct the codifferential from the exterior derivative, and thus the Laplace–de Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds. The Hodge star operator on a vector space V with an inner product is a linear operator on the exterior algebra of V, mapping k-vectors to (n − k)-vectors where n = dim V, for 0 ≤ k ≤ n. It has the following property, which defines it completely: given two k-vectors α, β, where ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } denotes the inner product on k-vectors and ω is the preferred unit n-vector.:15 The inner product ⟨ ⋅ , ⋅ ⟩ {displaystyle langle cdot ,cdot angle } on k-vectors is extended from that on V by requiring that for any decomposable k-vectors α = α 1 ∧ ⋯ ∧ α k {displaystyle alpha =alpha _{1}wedge cdots wedge alpha _{k}} and β = β 1 ∧ ⋯ ∧ β k {displaystyle eta =eta _{1}wedge cdots wedge eta _{k}} it equals the determinant:14