Polyvector field

In differential geometry, a field in mathematics, a multivector field, polyvector field of degree ${\displaystyle k}$, or ${\displaystyle k}$-vector field, on a smooth manifold ${\displaystyle M}$, is a generalization of the notion of a vector field on a manifold.

Definition

A multivector field of degree ${\displaystyle k}$ is a global section ${\displaystyle X}$ of the kth exterior power ${\displaystyle \wedge ^{k}TM\to M}$ of the tangent bundle, i.e. ${\displaystyle X}$ assigns to each point ${\displaystyle p\in M}$ it assigns a ${\displaystyle k}$-vector in ${\displaystyle \Lambda ^{k}T_{p}M}$.

The set of all multivector fields of degree ${\displaystyle k}$ on ${\displaystyle M}$ is denoted by ${\displaystyle {\mathfrak {X}}^{k}(M):=\Gamma (\wedge ^{k}TM)}$ or by ${\displaystyle T_{\rm {poly}}^{k}(M)}$.

Particular cases

• If ${\displaystyle k=0}$ one has ${\displaystyle {\mathfrak {X}}^{0}(M):={\mathcal {C}}^{\infty }(M)}$;
• If ${\displaystyle k=1}$, one has ${\displaystyle {\mathfrak {X}}^{1}(M):={\mathfrak {X}}(M)}$, i.e. one recovers the notion of vector field;
• If ${\displaystyle k>\mathrm {dim} (M)}$, one has ${\displaystyle {\mathfrak {X}}^{k}(M):=\{0\}}$, since ${\displaystyle \wedge ^{k}TM=0}$.

Algebraic structures

The set ${\displaystyle {\mathfrak {X}}^{k}(M)}$ of multivector fields is an ${\displaystyle \mathbb {R} }$-vector space for every ${\displaystyle k}$, so that ${\displaystyle {\mathfrak {X}}^{\bullet }(M)=\bigoplus _{k}{\mathfrak {X}}^{k}(M)}$ is a graded vector space.

Furthermore, there is a wedge product

$${\displaystyle \wedge$$ :{\mathfrak {X}}^{k}(M)\times {\mathfrak {X}}^{l}(M)\to {\mathfrak {X}}^{k+l}(M)}

which for ${\displaystyle k=0}$ and ${\displaystyle l=1}$ recovers the standard action of smooth functions on vector fields. Such product is associative and graded commutative, making ${\displaystyle ({\mathfrak {X}}^{\bullet }(M),\wedge )}$ into a graded commutative algebra.

Similarly, the Lie bracket of vector fields extends to the so-called Schouten-Nijenhuis bracket

$${\displaystyle [\cdot ,\cdot ]:{\mathfrak {X}}^{k}(M)\times {\mathfrak {X}}^{l}(M)\to {\mathfrak {X}}^{k+l-1}(M)}$$

which is ${\displaystyle \mathbb {R} }$-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 ${\displaystyle ({\mathfrak {X}}^{\bullet }(M),\wedge ,[\cdot ,\cdot ])}$ into a Gerstenhaber algebra.

Comparison with differential forms

Since the tangent bundle is dual to the cotangent bundle, multivector fields of degree ${\displaystyle k}$ are dual to ${\displaystyle k}$-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 ${\displaystyle (k,0)}$-tensor field is a differential ${\displaystyle k}$-form, a ${\displaystyle (0,1)}$-tensor field is a vector field, and a ${\displaystyle (0,k)}$-tensor field is ${\displaystyle k}$-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 ${\displaystyle (0,k)}$, except in the context of the geometric algebra (see also Clifford algebra).^[1][2][3]

See also

• Multivector
• Blade (geometry)