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

[D_a, D_b] = –F_abⁱ T_i ,

with T_i the generators of the Lie algebra; Explicitly,

F = dA + A A = DA = g g^–1 = *,   with    = d + ;

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

F_abⁱ = _a A_bⁱ – _b A_aⁱ + f ⁱ_jk A_a^j A_b^k .

@ References: Goldberg 62; in Regge in(84) [useful remarks].

Bianchi Identities > s.a. einstein equation; regge calculus.
1. DT = R , 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 [D_b, *F ^ab] = 0; For a Riemannian connection,

_[a R_bc]d^e = 0 ,   _a R_bcd^a + _b R_cd + _c R_bd = 0 ,   and   ^a G_ab = 0 ,

where the second and the third ones are the "contracted'' one and the "twice contracted" one, respectively.
@ References: Einstein CJM(50) [in gravitation]; Pravda et al CQG(04)gq [vacuum Weyl tensor, types III and N, arbitrary D].

Specific Bundles > see riemann tensor.

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

Main page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 25 may 2008