Integration on Manifolds |

**In General**
> s.a. differential forms; lie groups.

* __Idea__: Integration on
an *n*-manifold (without a metric, with just an orientation)
is defined only for *n*-forms, by

∫_{U}
*f*_{a... b}:=
∫_{U} *f*(*x*)
d^{n}*x* , sometimes written
∫_{U}
*f*_{a... b}
d*v*^{a... b} ,

where *U* has to be covered by one (right-handed) coordinate chart
(otherwise we generalize the definition using partitions of unity) and

*f*_{a... b} = *f*(*x*)
(d*x*^{1} ∧ ...
∧ d*x*^{n})_{a...
b} .

* __For functions__: Choose a volume
element or measure (which could be defined by a metric), and define

∫_{U}
*f* (d^{n}*v*):=
∫_{U}
*f* *ε*_{a... b} .

* __For other objects__: Need even more
structure, except for some integrals over lower-dimensional submanifolds.

@ __In curved spacetime, for general relativity__:
DeBenedictis phy/98.

**On Submanifolds**

* __Idea__: Given a *p*-dimensional
submanifold Σ in an *n*-dimensional manifold *M*, the natural things
to integrate there are *p*-forms, but we could integrate on Σ an *n*-form
*f* defined on *M* by defining the result to be the integral of the *p*-form
*φ***f*, where *φ* is the embedding *φ*: Σ →
*M*.

* __Notation__: An integral over
a *p*-submanifold *S* can be indicated in one of two ways,

∫_{S}
*A*_{a... b}
d*v*^{a... b} , or
∫_{S}
*B*^{m... n}
d*S*_{m... n} ,

where *A* is a *p*-form on *S* and *B* an
(*n*−*p*)-th rank tensor with indices normal to *S*.

**Stokes' Theorem**
> s.a. Gauss' Theorem; holonomy
[for Levi-Civita connection, and curvature]; Wilson Loop.

* __In 3D space__:

∫_{S }∇ × *A*
· d*s* = ∫_{∂(S)} *A*
· d*l* ; 2 ∫_{S}
∇_{a}* A*_{b}
d*v*^{ab}
= ∫_{∂(S)}
*A*_{a} d*v*^{a} ;
∫_{V} ∇ ·
*A* d^{3}*v* =
∫_{∂(V)}
*A* · d*s* .

* __In general__:

∫_{U}
d*ω* =
∫_{∂(U)}
*ω* , from which
∫_{U}
∇_{a}
*v*^{a}
= ∫_{∂(U)}
*v*^{a}
*ε*_{ab...c}
= ∫_{∂(U)}
*v*^{a}
*n*_{a}
;

Special cases are the 3D version, Gauss' theorem, and the Green identities
(> see vector calculus).

* __For a rank-two (antisymmetric) tensor__:

\(\int_S\) ∇_{a}
*T*^{ [ab]}
d*S*_{b}
= \(-{1\over2}\int_{\partial S}\)
*T*^{ [ab]}
d*S*_{ab} .

@ __General references__: in Flanders 63;
Saslow EJP(07) [computationally based proof].

@ __Generalized__: Bralic ht/93 ["surfaceless"];
Mensky PLA(04)gq [in terms of path group in Minkowski space];
Meunier EJC(08) [combinatorial].

@ __Non-abelian__: Diakonov & Petrov hl/00,
JETP(01)ht/00 [including Yang-Mills and gravity];
Kondo ht/00-proc [Wilson loops];
Broda in(01)mp/00;
Broda & Duniec mp/01 [Wilson loops];
Mensky gq/02-conf [in terms of path group in Minkowski];
Matsudo & Kondo PRD(15)-a1509 [and implications for quark confinement];
> s.a. loops [gauge theories and loop variables];
path integrals for gauge theories.

> __Online resources__:
see Wikipedia page.

** Specific Results and Types of Manifolds** > s.a. lie groups
and examples; Reynolds Theorem.

@ __References__: Hannay & Nye JPA(04) [2-sphere, Fibonacci numerical integration];
Somogyi JMP(11)-a1101 [angular integrals in *d* dimensions];
Felder & Kazhdan a1608 [regularization of divergent integrals].

**Generalizations**

@ __Generalized integrals__: Yekutieli 15 [non-abelian multiplicative integration on surfaces].

> __Generalized manifolds__:
see differential geometry; fractals;
grassmann; operator theory;
Supermanifolds.

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send feedback and suggestions to bombelli at olemiss.edu – modified 10 apr 2020