Curvature of a Connection  

In General > s.a. gauge theory; holonomy; Operad.
* Idea: Curvature is a property of a connection on a bundle over some manifold, and manifests itself in changes in the elements of the bundle upon parallel transport around closed curves in the manifold (holonomy); Algebraically, it manifests itself as non-commutativity of covariant derivatives.
$ Def: The curvature tensor is the Lie-algebra-valued 2-form F defined by

[Da, Db] = –Fabi Ti ,

with Ti the generators of the Lie algebra; Explicitly,

F = dA + AA = DA = g Ω g–1 = σ*Ω,   with   Ω = dω + ωω ;

In components, if  f ijk are the structure constants of the Lie algebra in that basis,

Fabi = ∂a Abi – ∂b Aai + f ijk Aaj Abk .

@ References: Goldberg 62; in Regge in(84) [useful remarks]; Gilkey et al DG&A(11) [universal curvature identities]; Gilkey et al 12.
> Online resources: see Wikipedia page.

Bianchi Identities > s.a. einstein equation; regge calculus.
1. DT = Rθ, where T = torsion and R = curvature; For a Riemannian connection, T = 0, this means R a[bcd] = 0.
2. If R is the curvature of any principal fiber bundle, DR = 0, or [Db, *F ab] = 0; For a Riemannian connection,

[a Rbc]de = 0 ,   ∇a Rbcda + ∇b Rcd + ∇c Rbd = 0 ,   and   ∇a Gab = 0 ,

where the second and the third ones are the "contracted'' one and the "twice contracted" one, respectively.
@ General references: Einstein CJM(50) [in gravitation]; Pravda et al CQG(04)gq [vacuum Weyl tensor, types III and N, arbitrary D]; Loinger S&S-phy/07; Pommaret a1603, a1706-ln [for the Riemann and Weyl tensors].
@ In Regge calculus: in Regge NC(61); in Miller FP(86); Bezerra CQG(88); Hamber & Kagel CQG(04)gq/01; Gentle et al CQG(09)-a0807; Williams CQG(12) [contracted].

Specific Bundles > see riemann tensor.

Generalizations
@ Discrete: Korepanov n.SI/00 [tetrahedra in deformed euclidean space]; Roberts & Ruzzi TAG-m.AT/06 [over posets]; > s.a. graph functions; tilings [combinatorial].


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