Differential
Forms| Differential
Forms |
In General > s.a. integration
on manifolds.
* Idea: Antisymmetric
tensors, for which Cartan developed a special formalism, terminology and
notation.
$ Def: A p-form is a totally antisymmetric covariant tensor field
of order p.
* Notation: The space of one-forms
on M is usually denoted by
T*M, that of p-forms by 
p T*M, pM
or 
pM (or
the same symbols with parentheses around M).
* And other structure:
Forms are a submodule of the algebra of all tensor fields on M; Together
with the exterior product, this forms the exterior algebra
or grassmann algebra of M; dim px(M)
= n!/p!(n–p)!
* Closed form: A form
such that d =
0; Exact form: A form
for which there is
another form 
such
that =
d.
@ General references: Flanders 63; HCartan 67; Von Westenholz 81; Bott & Tu
82; Darling 94; Jancewicz gq/98 [visualization].
@ Related topics: in de Rham 60, S5 [of odd kind]; Torre CQG(95)gq/94 [closed,
classification]; Hitchin m.DG/00 [3-forms
in 6D and 7D];
> s.a. grassmann structures.
@ Chiral p-forms: Bengtsson & Kleppe IJMPA(97)ht/96;
Pasti et al PRD(97)
[covariant actions].
@ In physics: Schleifer AJP(83)
[electrodynamics]; Petrova
mp/01 [conservation
laws], mp/05 [equations
of mathematical physics]; Rodrigues a0712 [detailed].
(Hodge) Dual
$ Def: Given a p-form f on
an n-manifold M with
volume form 
and metric g (with
p 
n),
its dual is the (n–p)-form
*fc.. d :=
(1/p!) f a.. b a.. bc.. d .
* Properties: It follows that
**f = (–1)s+p(n–p) f.
* For a curvature tensor: The left and right duals are, respectively,
*Rabcd := 
abmn Rmncd and R*abcd:=
Rabmn mncd .
@ And conformal structure: Dray et al JMP(89); Harnett JMP(91).
@ For affine manifold with torsion: Saa JGP(95).
Volume Form > s.a. types od
field theories.
$ Def: For an n-dimensional manifold M, a nowhere-vanishing n-form .
* And connections: A volume
form is compatible with a connection if for all X 
TM, 
X =
(Da Xa)
(for the divergence
theorem to apply).
* Example: The volume
form defined
by a metric g is =
|g|1/2 dx1 
...
dxn;
It is compatible with the Christoffel symbols, but not with a Riemann-Cartan
connection, X =
(Da Xa –
2 Ta Xa)
, where Ta:= Tbab.
@ References: Cartier et al in(01)mp/00 [characterization].
Lie-Algebra Valued Forms
* Canonical form: Given
a Lie group G, the canonical (Maurer-Cartan)
form on G is the Lie algebra-valued 1-form :
TG →
=
TeG
defined by (vg)
= Lg–1' vg; Theorem:
Rg* =
Ad(g–1) · .
* Maurer-Cartan structure
equation:
If {I} is a basis for
the dual * of the Lie
algebra of a group G, i.e. for the
left-invariant 1-forms on G, then
dI
= – CIJK J
K .
Other Special Types
* Closed form: One whose exterior derivative vanishes.
* Exact form: One which can
be expressed as the exterior derivative of
another form.
* Poincaré's lemma: On
a contractible space, any closed form is exact
(e.g., not in R2\{0}, non-contractible).
@ References: in Flanders 63; in Nash & Sen 83; Farber & Schütz Top(06)
[closed 1-forms
with at most one zero].
Related Concepts and Techniques > s.a. exterior
algebra and calculus; deformation
quantization; field theories.
@ Fractional order: Cottrill-Shepherd & Naber JMP(01)mp/03, mp/03;
Tarasov JPA(05).
@ Discrete: Richter et al CQG(07)gq/06 [and
spherical symmetry
in general relativity]; Dowker JGP(07)
[on sphere tessellations]; Dolotin et al a0704 [based
on simplicial complexes]; Richter & Frauendiener a0805 [Gowdy solutions, numerical].
@ Other generalizations: Nurowski & Robinson CQG(01)
[and spacetime geometry]; Robinson
IJTP(03)
[and gauge theories], JMP(03)
[and general relativity], JPA(07)
[rev, and applications]; Chatterjee et al a0706 [negative
forms, and path space forms]; > s.a. clifford
calculus.
@ Other topics: Frauendiener talk(03).
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
19 may 2008