Riemann Curvature Tensor  

In General > s.a. affine connections; curvature of a connection; tetrads.
* Idea: The Riemann tensor is the curvature tensor for an affine connection on a manifold; Like other curvatures, it measures the non-commutativity of parallel transport of objects, in this case tangent vectors (or dual vectors or tensors of higher rank), along two different paths between the same two points of the manifold; In this case, however, curvature also manifests itself in other ways related to the geometry of the manifold, like geodesic deviation, or the dependence of the volume of a ball on its radius, as the radius goes to zero.
$ Def: The Riemann tensor of a connection ∇ is defined by R(X, Y) Z = ∇XY Z − ∇YX Z − ∇[X,Y] Z, or

(∇ab − ∇ba) V c = −Rabdc V d + T dabd V c ;

Notice that the last term is absent if there is no torsion.
* Calculation: Use the tetrad formalism or, with a reference connection (e.g., from coordinates),

Rabmn = 0Rabmn − 2 0[a Γnb]m− 2 Γnp[a Γpb]m .

* Symmetries: Rabcd = Rcdab, Rabcd = R[ab][cd] (if it comes from a metric), R[abc]dT[abe Tc]ed − ∇[aTbc] d = 0; Because of these symmetries, in n dimensions it has \(1\over12\)n2 (n2−1) components.
* Two dimensions: There is only scalar curvature, Rabcd = R ga[c gd]b.
* Three dimensions: The curvature is determined by the Ricci tensor, Rabcd = 2 (ga[c Rd]bgb[c Rd]a) − R ga[c gd]b.
* Four dimensions: There is also Weyl curvature in general, Rabcd = Cabcd + ga[c Rd]bgb[c Rd]a − \(1\over3\) R ga[c gd]b.
@ Identities: Dianyan PRD(87); Navarro & Navarro JGP(14)-a1310; > s.a. curvature [Bianchi identities].
> Special types of metrics: see spherical symmetry; metric types [including self-dual].
> Online resources: see MathWorld page; Wikipedia page.

Derived Quantities and Invariants > s.a. bel tensor; Cotton Tensor; Einstein Tensor; Ricci Tensor; weyl tensor.
* 4D spacetime: In general, there are 14 independent real algebraic invariant local scalars; The only linear one is the scalar curvature R = Rab gab; Two important quadratic ones are the square of the Ricci tensor Rab Rab and the Kretschmann invariant Rabcd Rabcd; In a vacuum spacetime, they can be expressed in terms of Cabcd.
* Karlhede invariant: The invariant R abcd;e Rabcd;e formed from the contraction of covariant derivatives of the Riemann tensor; > s.a. schwarzschild spacetime.
* Special cases: Semi-symmetric space, one where ∇[mn] Rabcd = 0; > s.a. types of lorentzian geometry.
@ Scalar invariants: Karlhede et al GRG(82) [Karlhede invariant]; Carminati & McLenaghan JMP(91); Barvinsky et al JMP(94)gq; Harvey JMP(95); Zakhary & McIntosh GRG(97); Sneddon JMP(96), JMP(98), JMP(99) [identities]; Zakhary & Carminati JMP(01), Carminati et al JMP(02), Carminati & Zakhary JMP(02) [algebraic completeness]; Tapia gq/02 [differential invariants]; Siklos GRG(06); Hall & MacNay CQG(06) [curvature function]; Labbi m.DG/06-Hab, Sigma(07)-a0709-proc; Lim & Carminati JMP(07) [minimal set and syzygies]; Papadopoulos a1405 [insufficiency, pseudo-Riemannian geometries]; MacCallum a1504-proc [algebraically independent local invariants and their uses]; Musoke et al GRG(16)-a1511 [3D spacetimes, minimal set]; in Bray et al a2102 [from level sets of harmonic functions].
@ Vanishing scalar invariants: Pelavas et al JMP(05)gq [vanishing 0th- and first-order invariants]; Page CQG(09)-a0806; Hervik & Coley CQG(11)-a1008.
@ Constant scalar invariants: Coley & Hervik a1105 [and universal classical solutions].
@ Other special cases: Pravda & Bičák gq/01-MG9 [algebraically special]; Schmidt gq/01-GR14 [indistinguishable spacetimes]; Cherubini al al IJMPD(02)gq/03 [second-order, and black holes]; Deser & Ryzhov CQG(05)gq [static spherical, any D]; Senovilla AIP(06)m.DG/05 [∇mn Rabcd = 0]; Åman JPCS(11)-a1006 [semi-symmetric spaces]; > s.a. lorentzian geometry.
@ Derived quantities: Gilkey 01 [natural operators]; Palatnik qp/03-wd [τij;i = 0, second-order in Riemann tensor]; Lusanna & Villani a1401, a1401 [in the York canonical basis]; > s.a. Lovelock Tensor; spectral geometry; Riemann-Lovelock Curvature Tensor.

Quantities Associated with a Submanifold > s.a. extrinsic curvature; Submanifold; vector field; Weingarten Matrix.
$ Sectional curvatures: With respect to the 2-plane defined by the orthonormal vectors X and Y, K:= Rabcd X a Y b X c Y d; After parallel transport along a small loop in the 2-plane, K = (angle by which the vector rotates)/(area enclosed by loop); > for applications, see orientation [Synge's theorem] and the Hopf Conjecture.
* Principal curvatures: For a two-surface S in \(\mathbb R\)3, \(\kappa_i^~\) are the roots of the equation det(Kab − \(\kappa\) qab) = 0, i.e., the eigenvalues of the mixed tensor Kab.
* Gaussian curvature: For a 2-manifold, K = \(\kappa_1^~\kappa_2^~\) (= det Kab) = 2/r1r2 = R/2, where ri = principal radii of curvature of S, and R is the scalar curvature; For example, for a 2-sphere R = 2/r2, and for a 2-torus, R = (cos u) /[r (a + r cos u)], where u = coordinate around the "small loop" ∈ [0, 2π], r = radius of "small loop", a "big radius".
* Mean curvature: k = 1/r1 + 1/r2 = \(1\over2\)(\(\kappa_1^~+\kappa_2^~\)), up to coefficients (∝ trace of second fundamental form, tr Kab).
@ General references: Arnlind et al a1001 [in terms of Poisson brackets].
@ Sectional curvature: Geroch GRG(76); Cormack & Hall IJTP(79) [critical-point structure and Petrov classification]; Hall GRG(84); Hall & Rendall GRG(87), Hall IJGMP(06) [and general relativity].
@ Gaussian curvature: Guan & Spruck JDG(02) [constant].

References > s.a. causality; Collineation; metric; tests of general relativity; weyl tensor.
@ Interpretation: Synge AM(34) + GRG(09); Pirani APP(56) + GRG(09); Loveridge gq/04 [including Rab, R and Gab].
@ Metric from curvature: Ihrig GRG(76); Hall & McIntosh IJTP(83); Kazdan 85; Rendall CQG(88); Hall et al GRG(89); Bradley & Karlhede CQG(90); Edgar JMP(91); Quevedo GRG(92); Bradley & Marklund CQG(96); > s.a. lanczos tensor; lorentzian geometry.
@ Potential: Lanczos RMP(62); Massa & Pagani GRG(84); Edgar GRG(94); Andersson & Edgar gq/99; > s.a. lanczos tensor.
@ Classification: Åman et al GRG(91); Coley & Hervik a1011 [Lorentzian, using discriminating curvature invariants], IJGMP(11) [arbitrary dimensionality and signature]; Hervik CQG(11)-a1107 [the ε-property].
@ Symmetries and diffeomorphisms: Swift GRG(94); Duggal & Sharma 99; Hussain et al IJMPD(05)-a0812 [vs Weyl symmetries].
@ Curvature measurements: Santander AJP(92)sep [mechanical device for visualizing curvature and parallel transport]; Rojo et al CJP(09)-a0809 [mechanical device, parallelometer]; Saravani et al PRD(16)-a1510 [in terms of scalar field propagators]; Bonalda et al a1903 [using a quantum particle].
@ Related topics: Hestenes IJTP(86) [Clifford algebra method]; Hall & Kay JMP(88); Colding AM(97) [R and volume]; Schmidt gq/04 [gravitoelectromagnetism and decompositions; not PRD(11)]; Sharipov a0709 [and spinor curvature tensor]; Gies & Martini PRD(18)-a1802 [curvature bounds from physics].
> Effects and measurements: see atomic physics [interferometry]; information; lorentz symmetry in physics.
> Generalized: see curvature [including discrete]; differential geometry [including infinitesimal]; types of manifolds; types of metrics [distributional curvature].

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