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].

@ __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 & Bičαk 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]; Εman 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__:
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__: Εman 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 a1510
[in terms of scalar field propagators].

@ __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].

> __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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2017