Differential Forms |

**In General**
> s.a. exterior algebra and calculus;
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 Λ^{p} T**M*,
Λ^{p}*M*
or Ω^{p}*M*
(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 \(\Lambda^p{}_x(M) = n!/p!(n-p)!\)

@ __General references__: Flanders 63;
Cartan 67;
Von Westenholz 81;
Bott & Tu 82;
Darling 94;
Jancewicz gq/98 [visualization];
Morita 01 [geometry];
Ivancevic & Ivancevic a0807-ln;
Lessig a1206 [primer];
Guillemin & Haine 19.

@ __Related topics__: in de Rham 60 [of odd kind];
Hitchin m.DG/00 [3-forms in 6D and 7D];
> s.a. grassmann structures;
types of cohomology theories.

@ __Chiral p-forms__: Bengtsson & Kleppe IJMPA(97)ht/96;
Pasti et al PRD(97) [covariant actions];
Bonetti et al PLB(13) [Kaluza-Klein inspired action].

@

**(Hodge) Dual**

$ __Def__: Given a *p*-form *f* on
an *n*-manifold *M* with volume form *ε* and metric *g*
(with *p* ≤ *n*), its dual is the (*n*−*p*)-form

**f*_{c.. d}
:= (1/*p*!) *f*^{ a.. b}
ε_{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,

**R*_{abcd}
:= \(1\over2\)ε_{ab}^{mn}
*R*_{mncd} and
*R**_{abcd}:=
\(1\over2\)*R*_{abmn}
ε^{mn}_{cd} .

* __And physics__: The duality operation
applied to the electromagnetic (Faraday) field tensor (2-form) interchanges the electric
and magnetic fields; > s.a. duality.

@ __And conformal structure__: Dray et al JMP(89);
Harnett JMP(91).

@ __Related topics__: Saa JGP(95) [for affine manifold with torsion];
Klinker JGP(11) [generalized definition];
Sen a1903 [self-dual forms, dynamics].

**Volume Form** > s.a. types of field theories.

$ __Def__: For an *n*-dimensional
manifold *M*, a nowhere-vanishing *n*-form *ω*.

* __And connections__: A volume
form is compatible with a connection if for all *X* ∈ T*M*,
\(\cal L\)_{X }*ω* =
(*D*_{a} *X*^{a})
*ω* (for the divergence theorem to apply).

* __Example__: The volume
form ε defined by a metric *g* is ε =
|*g*|^{1/2} d*x*^{1}
∧ ... ∧ d*x*^{n};
It is compatible with the Christoffel symbols, but not with a Riemann-Cartan
connection, \(\cal L\)_{X}
*ω* = (*D*_{a}
*X*^{a} −
2 *T*_{a}
*X*^{a})
*ω*, where *T*_{a}:=
*T*_{ba}^{b}.

@ __General references__:
Cartier et al in(01)mp/00 [characterization].

@ __As a dynamical variable__:
Guendelman & Kaganovich a0811-conf;
Guendelman et al a1505-conf,
Benisty et al a1905-conf [in gravity and cosmology].

**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 *ω*: T*G* →
\(\cal G\) = T_{e}*G*
defined by *ω*(*v*_{g})
= L_{g}^{−1}*'*
*v*_{g}; __Theorem__:
R_{g}* *ω* =
Ad(*g*^{−1}) · *ω*.

* __Maurer-Cartan structure equation__:
If {*θ*^{I}} is a basis for the
dual \(\cal G\)* of the Lie algebra of a group *G*, i.e. for the left-invariant
1-forms on *G*, then

d*θ*^{I}
= −\(1\over2\)*C*^{I}_{JK}
*θ*^{J}
∧ *θ*^{K} .

**Other Special Types and Generalizations** > s.a. deformation
quantization; Star Product; superspace.

* __Closed form__: A form
*ω* whose exterior derivative vanishes, d*ω* = 0.

* __Exact form__: A form
*ω* which can be expressed as the exterior derivative
of another form, *ω* = d*θ*.

* __Poincaré's lemma__:
On a contractible space, any closed form is exact (e.g., not in
\({\mathbb R}^2\setminus\{0\}\), non-contractible).

@ __General references__: in Flanders 63;
in Nash & Sen 83.

@ __Closed forms__: Torre CQG(95)gq/94 [classification];
Farber 04;
Farber & Schütz Top(06) [closed 1-forms with at most one zero];
Burns a1906
[generalization of Poincaré lemma to divergence-free multivector fields].

@ __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 TMP(08)-a0704 [based on simplicial complexes];
Richter & Frauendiener JSC(10)-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],
CQG(09)
[integral calculus, Stokes' theorem, Chern-Simons and Einstein-Yang-Mills theories];
Chatterjee et al IJGMP(08)-a0706 [negative forms, and path-space forms];
Gallego a1207
[higher-order, and applications to electrodynamics];
Robinson a1312 [and gravitation];
> s.a. clifford calculus;
formulations of electromagnetism [pair and impair].

@ __Related topics__: Frauendiener talk(03).

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