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];
Burtscher a1901
[distributional version, for spacetimes with timelike singularities]; Vassallo a2101 [physical role, and the laws of nature].

@ __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];
Klitgaard & Loll PRD(18)-a1712,
PRD(18)-a1802 [quantum Ricci curvature];
Tee & Trugenberger a2102 [Ollivier-Ricci and Forman-Ricci curvatures for graphs]; > s.a. graph functions;
tilings [combinatorial].

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