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In differential geometry, a field in mathematics, a multivector field, polyvector field of degree , or -vector field, on a smooth manifold , is a generalization of the notion of a vector field on a manifold.
Definition
editA multivector field of degree is a global section of the kth exterior power of the tangent bundle, i.e. assigns to each point it assigns a -vector in .
The set of all multivector fields of degree on is denoted by or by .
Particular cases
edit- If one has ;
- If , one has , i.e. one recovers the notion of vector field;
- If , one has , since .
Algebraic structures
editThe set of multivector fields is an -vector space for every , so that is a graded vector space.
Furthermore, there is a wedge product
which for and recovers the standard action of smooth functions on vector fields. Such product is associative and graded commutative, making into a graded commutative algebra.
Similarly, the Lie bracket of vector fields extends to the so-called Schouten-Nijenhuis bracket
which is -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 into a Gerstenhaber algebra.
Comparison with differential forms
editSince the tangent bundle is dual to the cotangent bundle, multivector fields of degree are dual to -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 -tensor field is a differential -form, a -tensor field is a vector field, and a -tensor field is -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 , except in the context of the geometric algebra (see also Clifford algebra).[1][2][3]
See also
editReferences
edit- ^ 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.
- ^ Artin, Emil, 1898-1962. (1988) [1957]. Geometric algebra. New York: Interscience Publishers. ISBN 9781118164518. OCLC 757486966.
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: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) - ^ 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.