Polyvector field explained
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
A multivector field of degree
is a global section
of the k
th 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 ak{X}k(M):=\Gamma(\wedgekTM)
or by
. Particular cases
one has
;
, one has
, i.e. one recovers the notion of vector field;
, one has
, since
.Algebraic structures
The set
of multivector fields is an
-vector space for every
, so that
ak{X}\bullet(M)=opluskak{X}k(M)
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
(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
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
Notes and References
- 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.
- Book: Artin, Emil, 1898-1962.. Geometric algebra. 1988. 1957. Interscience Publishers. 9781118164518. New York. 757486966.
- Book: Snygg, John.. A new approach to differential geometry using Clifford's geometric algebra. 2012. Springer Science+Business Media, LLC. 9780817682835. New York. 769755408.