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  fa... b:= U  f(x) dnx ,   sometimes written   U  fa... b dva... b ,

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

fa... b = f(x) (dx1 ∧ ... ∧ dxn)a... b .

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

U  f (dnv):= U  f εa... 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 Aa... b dva... b ,   or   S Bm... n dSm... n ,

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

Stokes' Theorem > s.a. Gauss' Theorem; Wilson Loop.
* In 3D space:

∇ × A · ds = ∂(S) A · dl ;   2 a Ab dvab = ∂(S) Aa dva ;   V ∇ · A d3v = ∂(V) A · ds .

* In general:

U dω = ∂(U) ω ,   from which   Ua va = ∂(U) va εab...c = ∂(U) va na ;

Special cases are the 3D version, Gauss' theorem, and the Green identities (> see vector calculus).
* For a rank-two (antisymmetric) tensor:

S ∇a T [ab] dSb = –\(1\over2\)∂(S) T [ab] dSab .

@ 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-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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