Polyvector field

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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 ∧kTM→M of the tangent bundle, i.e. X assigns to each point p∈M it assigns a k-vector in ΛkTpM.

The set of all multivector fields of degree k on M is denoted by 𝔛k(M):=Γ(∧kTM) or by Tpolyk(M).

Particular cases

  • If k=0 one has 𝔛0(M):=𝒞∞(M);
  • If k=1, one has 𝔛1(M):=𝔛(M), i.e. one recovers the notion of vector field;
  • If k>dim(M), one has 𝔛k(M):={0}, since ∧kTM=0.

Algebraic structures

The set 𝔛k(M) of multivector fields is an ℝ-vector space for every k, so that 𝔛∙(M)=⨁k𝔛k(M) is a graded vector space.

Furthermore, there is a wedge product

∧:𝔛k(M)×𝔛l(M)→𝔛k+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 (𝔛∙(M),∧) into a graded commutative algebra.

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

[⋅,⋅]:𝔛k(M)×𝔛l(M)→𝔛k+l−1(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 (𝔛∙(M),∧,[⋅,⋅]) 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

References

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



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