Polyvector field explained

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 Λ ^{k} T _{p} M .

The set of all multivector fields of degree

k on M is denoted by ak{X} ^{k} (M) : = \Gamma (\wedge ^{k} TM) or by T ^{k} _{\rm poly} (M) .

Particular cases

• If

k =0 one has ak{X} ^{0} (M) : = l{C} ^{infty(M)} ;

• If

k =1 , one has ak{X} ^{1} (M) : = ak{X}(M) , i.e. one recovers the notion of vector field;

• If

k > dim (M) , one has ak{X} ^{k} (M) : = \{ 0 \} , since \wedge ^{k} TM = 0 .

Algebraic structures

The set

ak{X} ^{k} (M) of multivector fields is an R -vector space for every k , so that ak{X} ^{\bullet} (M) = o plus _{k} ak{X} ^{k} (M) is a graded vector space.

Furthermore, there is a wedge product

$$\backslash wedge:\; \backslash mathfrak^k\; (M)\; \backslash times\; \backslash mathfrak^l\; (M)\; \backslash to\; \backslash mathfrak^\; (M)$$

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 ( ak{X} ^{\bullet} (M), \wedge) into a graded commutative algebra.

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

$$[\backslash cdot,\backslash cdot]:\; \backslash mathfrak^k\; (M)\; \backslash times\; \backslash mathfrak^l\; (M)\; \backslash to\; \backslash mathfrak^\; (M)$$

which is

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 ( ak{X} ^{\bullet} (M), \wedge, [ ⋅ , ⋅ ]) 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).^{[1] [2] [3]}

See also

• Multivector
• Blade (geometry)

Notes and References

1. Book: Doran, Chris (Chris J. L.). Geometric algebra for physicists. 2007. Cambridge University Press. Lasenby, A. N. (Anthony N.), 1954-. 9780521715959. 1st pbk. ed. with corr. Cambridge. 213362465.
2. Book: Artin, Emil, 1898-1962.. Geometric algebra. 1988. 1957. Interscience Publishers. 9781118164518. New York. 757486966.
3. Book: Snygg, John.. A new approach to differential geometry using Clifford's geometric algebra. 2012. Springer Science+Business Media, LLC. 9780817682835. New York. 769755408.