Polyvector field

A Polyvector field within Mathematics topology is concerned with the properties of a geometric object. A multivector field, polyvector field of degree k , or k-vector field, on a manifold ${\displaystyle M}$, is a generalization of the notion of a vector field on a manifold.

Whereas a vector field ${\displaystyle X\in \Gamma (TM)}$is a global section of tangent bundle, which assigns to each point on the manifold a tangent vector ${\displaystyle X_{p}\in T_{p}M}$, a multivector field is a section of the kth exterior power of the tangent bundle, ${\displaystyle \Lambda ^{k}TM}$, and to each point ${\displaystyle p\in M}$it assigns a k-vector in ${\displaystyle \Lambda ^{k}T_{p}M}$. Just as the smooth sections of the tangent bundle (vector fields) make up a vector space, the space of smooth k-vector fields over M make up a vector space ${\displaystyle \Gamma (\Lambda ^{k}TM)}$.

Furthermore, since the tangent bundle is dual to the cotangent bundle, multivector fields of degree k are dual to k-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. A (k,0)-tensor field is a differential k-form, a (0,1)-tensor field is a vector field, and a (0,k)-tensor field is k-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 (0,k), except in the context of the geometric algebra (see also Clifford algebra).[1][2][3]