Editing Differential form (section)

=== Riemannian manifold ===
On a [[Riemannian manifold]], or more generally a [[pseudo-Riemannian manifold]], the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles.  This makes it possible to convert vector fields to covector fields and vice versa.  It also enables the definition of additional operations such as the [[Hodge star operator]] <math>\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)</math> and the [[Hodge star#Codifferential|codifferential]] <math>\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)</math>, which has degree {{math|−1}} and is [[Differential operator#Adjoint of an operator|adjoint]] to the exterior differential {{math|''d''}}.

==== Vector field structures ====
On a pseudo-Riemannian manifold, {{math|1}}-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.

Firstly, each (co)tangent space generates a [[Clifford algebra]], where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the [[metric tensor|metric]]. This algebra is ''distinct'' from the [[exterior algebra]] of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in [[geometric algebra]].

Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of [[differential operator]]s they generate is the [[Weyl algebra]] and is a noncommutative ("quantum") deformation of the ''symmetric'' algebra in the vector fields.