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

\wedgekTM\toM

of the tangent bundle, i.e.

X

assigns to each point

p\inM

it assigns a

k

-
vector
in

ΛkTpM

.

The set of all multivector fields of degree

k

on

M

is denoted by

ak{X}k(M):=\Gamma(\wedgekTM)

or by
k
T
\rmpoly

(M)

.

Particular cases

k=0

one has

ak{X}0(M):=l{C}infty(M)

;

k=1

, one has

ak{X}1(M):=ak{X}(M)

, i.e. one recovers the notion of vector field;

k>dim(M)

, one has

ak{X}k(M):=\{0\}

, since

\wedgekTM=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)=opluskak{X}k(M)

is a graded vector space.

Furthermore, there is a wedge product

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

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

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.