Linear and Affine Connections  

On the Bundle of (Orthonormal) Frames > s.a. metric [examples]; tetrads [calculation of curvature].
$ Linear connection: A linear connection on a smooth manifold X is a connection on the principal fiber bundle F(X) of frames on X.
$ Levi-Civita, metric or Riemannian connection: Given the reduced bundle O(M) over M (i.e., a Riemannian metric g), it is the only connection such that (i) Γ has no torsion, Γmab = Γmba, and (ii) Parallel transport preserves the metric, gab;m = 0.
$ Riemann-Cartan connection: The 1-form

ωij := ΓijKij ,

where Γ is the Levi-Civita connection 1-form, and K the contorsion, with T i = θ jKji.
* Cartan structure equations: They describe the affine connection on a manifold through the torsion and curvature tensors (which in turn can be obtained from the connection); If T is the torsion 2-form, θ the coframe, Ω the curvature 2-form, and ω the connection,

dθ + ω ∧ θ = T ,   Ω = dω + ω ∧ ω .

> Online resources: For affine connections, see Wikipedia page and Encyclopedia of Mathematics page; for Levi-Civita connection, see Wikipedia page.

Christoffel Symbols
$ First kind: The Christoffel symbols of the first kind are defined as {ab, c} = \(1\over2\)(∂bgac + ∂agbc – ∂cgab).
$ Second kind: The Christoffel symbols of the second kind are defined as {a \atop bc} = gad {bc, d}.
* And connection: Given a choice of coordinates, the components of the linear connection compatible with a metric gab are expressed by

Γabc = \(\{{a \atop bc}\}\) = \(1\over2\)gmn (gnb,a + gan,bgab.n) .

* Gauge transformation: For an infinitesimal diffeomorphism generated by va, δΓmab = ∇abvmRm(ab)c vc.

On the Tangent Bundle > s.a. coordinates [Fermi transport]; torsion.
* History: Linear connections were invented by Cartan to geometrize Newtonian gravitation.
* Existence: They always exist on a manifold with a countable basis.
* Parallel transport: To define one, start with xM, a curve γ through x, and a lift of γ to the frame bundle, γF(M); Fix a vector XTx(M); Then the parallel transport of X is

γT(M)(t):= γF(M)(t) X.

* Covariant derivative: Defined in T(M) of Z = γT(M)(t) = bm (∂/∂xm) with respect to Y = cm (∂/∂xm) as

Y Z:= (d/dt)γT(M) = (dxm(t)/dt) DmγF(M)(t) X ,

where Dm is the covariant derivative in F(M); One finds

Y Z = cn bm;n ∂/∂xm ,   bm;n:= bm,nAmns bs,   Amns = Γmns .

* Flat connection: If the torsion and curvature both vanish, the connection is called flat, and there exist local coordinates in which the connection vanishes; Then, in other coordinates, Γ = h–1dh.
* On a Riemannian manifold: The connection can be written as

Γabc = \(\{{a \atop bc}\}\) + Kabc ,

where {a \atop bc} are the Christoffel symbols, Qabc = Ka[bc] the torsion, and Mabc = Ka(bc) the non-metricity.
* Weyl vector: The trace of the non-metricity, Ba:= Mabc gbc.

Special Types of Metrics > s.a. FLRW models; spherical symmetry; schwarzschild spacetime; symmetry.
@ References: Armstrong JGP(07) [Ricci-flat, classification]; Guadagnini et al a1710 [3D, flat].

References > s.a. Parallel Transport; torsion.
@ General: Levi-Civita RCMP(17).
@ Connection and metric: Schmidt CMP(73); Hall GRG(88); Edgar JMP(92); Thompson CQG(93), JGP(96); Cocos JGP(06) [condition for symmetric connection to be Levi-Civita]; Atkins mp/06; Atkins a0804 [connections that are only locally Levi-Civita]; > s.a. formulations of general relativity.
@ Lorentz connections: Hall & Lonie JPA(06)gq/05 [4D]; Pereira AIP(12)-a1210 [in general relativity and teleparallel gravity].
@ Non-metricity: Sobreiro & Vasquez Otoya BJP(10)-a0711 [in gravity theories]; Mol AACA(17)-a1406 [and general relativity]; Casanova et al MPLA(08)-a0712 [and torsion, Schouten classification]; Foster et al a1612 [constraints from searches for Lorentz violation]; Järv et al a1802 [non-metricity formulation of general relativity].
@ Non-commutative geometry: Dubois-Violette et al LMP(95)ht/94 [quantum plane]; Madore et al CQG(95)ht/94 [matrix geometries]; Madore CQG(96)ht/95; Mourad CQG(95)ht/94; Sitarz ht/95; Masson & Serié JMP(05) [invariant]; > s.a. geodesics.
@ And field theory: Barut et al HPA(93); in DeWitt & Molina-París MPLA(98)ht [Vilkovisky connection on space of histories].
@ On use in gravitational theories: Kofinas a1712.
@ Other generalizations: Milani & Shafei Deh Abad LMP(97)ht/96 [quantized manifolds, geodesics]; Pham MJM(15)-a1408 [higher affine connections].
> Other generalizations: see finsler geometry; Lagrange Spaces [non-linear connection]; newton-cartan theory [degenerate metrics].

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