In differential geometry, a field in mathematics, a multivector field, polyvector field of degree
-vector field, on a smooth manifold
, is a generalization of the notion of a vector field on a manifold.
A multivector field of degree
is a global section
of the kth exterior power
of the tangent bundle, i.e.
assigns to each point
it assigns a
The set of all multivector fields of degree
{\displaystyle T_{\rm {poly}}^{k}(M)}
The set
of multivector fields is an
is a graded vector space.
Furthermore, there is a wedge product
recovers the standard action of smooth functions on vector fields.
Such product is associative and graded commutative, making
into a graded commutative algebra.
Similarly, the Lie bracket of vector fields extends to the so-called Schouten-Nijenhuis bracket
-bilinear, graded skew-symmetric and satisfies the graded version of the Jacobi identity.
Furthermore, it satisfies a graded version of the Leibniz identity, i.e. it is compatible with the wedge product, making the triple
into a Gerstenhaber algebra.
Since the tangent bundle is dual to the cotangent bundle, multivector fields of degree
-forms, and both are subsumed in the general concept of a tensor field, which is a section of some tensor bundle, often consisting of exterior powers of the tangent and cotangent bundles.
-tensor field is a differential
-tensor field is a vector field, and a
-tensor field is
-vector field.
While differential forms are widely studied as such in differential geometry and differential topology, multivector fields are often encountered as tensor fields of type
, except in the context of the geometric algebra (see also Clifford algebra).