Curvature of a Connection| 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 + A ∧ A = DA = g Ω g−1 = σ*Ω, with Ω = dω + ω ∧ ω ;
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 = 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 [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];
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].
main page
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