Editing Hodge star operator (section)

{{Short description|Exterior algebraic map taking tensors from p forms to n-p forms}}
In [[mathematics]], the '''Hodge star operator''' or '''Hodge star''' is a [[linear map]] defined on the [[exterior algebra]] of a [[Dimension (vector space)|finite-dimensional]] [[orientation (mathematics)|oriented]] [[vector space]] endowed with a [[Degenerate bilinear form|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 {{math|''n''}}-dimensional vector space, the Hodge star is a one-to-one mapping of {{math|''k''}}-vectors to {{math|(''n – k'')}}-vectors; the dimensions of these spaces are the [[binomial coefficient]]s <math>\tbinom nk = \tbinom{n}{n - k}</math>.

The [[Natural transformation|naturalness]] of the star operator means it can play a role in differential geometry when applied to the cotangent [[Vector bundle|bundle]] of a [[pseudo-Riemannian manifold]], and hence to  [[Differential form|differential {{math|''k''}}-forms]]. This allows the definition of the codifferential as the Hodge adjoint of the [[exterior derivative]], leading to the [[Laplace–Beltrami operator#Laplace–de Rham operator|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 manifold|closed]] Riemannian manifold.