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*_{ab}^{i} *T*_{i}
,

with *T*_{i} the generators of the Lie algebra; Explicitly,

*F* = d*A* + *A* ∧ *A* = *DA* = *g* Ω *g*^{–1} = *σ**Ω, with Ω =
d*ω* + *ω* ∧ *ω* ;

In components, if *f*^{ i}_{jk} are
the structure constants of the Lie algebra in that basis,

*F*_{ab}^{i} = ∂_{a} *A*_{b}^{i} – ∂_{b} *A*_{a}^{i} + *f*^{ i}_{jk} *A*_{a}^{j}* A*_{b}^{k}
.

@ __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* = *R* ∧ *θ*,
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.

@ __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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