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* = ∇_{X}∇_{Y}
*Z* − ∇_{Y}∇_{X}
*Z* − ∇_{[X,Y]} *Z*, or

(∇_{a}∇_{b}
− ∇_{b}∇_{a})
*V*^{ c}
= −*R*_{abd}^{c}
*V*^{ d}
+ *T*^{ d}_{ab}
∇_{d }*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),

*R*_{abm}^{n}
= ^{0}*R*_{abm}^{n}
− 2 ^{0}∇_{[a}
Γ^{n}_{b]m}−
2 Γ^{n}_{p[a}
Γ^{p}_{b]m} .

* __Symmetries__:
*R*_{abcd}
= *R*_{cdab},
*R*_{abcd}
= *R*_{[ab][cd]} (if it comes from a metric),
*R*_{[abc]}^{d}
− *T*_{[ab}^{e}
*T*_{c]e}^{d}
− ∇_{[a}*T*_{bc]
}^{d} = 0;
Because of these symmetries, in *n* dimensions it has
\(1\over12\)*n*^{2}
(*n*^{2}−1) components.

* __Two dimensions__:
There is only scalar curvature, *R _{abcd}*
=

*

*

@

>

>

**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*
= *R*_{ab}
*g*^{ab}; Two
important quadratic ones are the square of the Ricci tensor
*R*_{ab}
*R*^{ab}
and the Kretschmann invariant *R*_{abcd}
*R*^{abcd}; In a vacuum spacetime, they
can be expressed in terms of *C*_{abcd}.

* __Karlhede invariant__:
The invariant *R*^{ abcd;e}
*R*_{abcd;e} formed
from the contraction of covariant derivatives of the Riemann tensor;
> s.a. schwarzschild spacetime.

* __Special cases__: Semi-symmetric
space, one where ∇_{[m}
∇_{n]}
*R*_{abcd} = 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
[∇_{m}∇_{n}
*R*_{abcd} = 0];
Åman JPCS(11)-a1006 [semi-symmetric spaces];
> s.a. lorentzian geometry.

@ __Derived quantities__: Gilkey 01 [natural operators];
Palatnik qp/03-wd
[*τ*^{i}_{j;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*:= *R*_{abcd}
*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(*K*_{ab}
− \(\kappa\)* q*_{ab}) = 0,
i.e., the eigenvalues of the mixed tensor
*K*_{a}^{b}.

* __Gaussian curvature__:
For a 2-manifold, *K* = \(\kappa_1^~\kappa_2^~\)
(= det *K*_{a}^{b})
= 2/*r*_{1}*r*_{2}
= *R*/2, where *r*_{i}
= principal radii of curvature of *S*, and *R* is the
scalar curvature; For example, for a 2-sphere *R*
= 2/*r*^{2}, 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/*r*_{1}
+ 1/*r*_{2}
= \(1\over2\)(\(\kappa_1^~+\kappa_2^~\)), up to coefficients
(∝ trace of second fundamental form,
tr *K*_{a}^{b}).

@ __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 *R*_{ab}, *R*
and *G*_{ab}].

@ __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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