Polyvector field

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

Definition

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

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

Particular cases

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

Algebraic structures

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

Furthermore, there is a wedge product

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

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

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

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

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

Comparison with differential forms

Since the tangent bundle is dual to the cotangent bundle, multivector fields of degree $k$ 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).

See also

• Multivector
• Blade (geometry)