Riemann
Curvature Tensor| Riemann
Curvature Tensor |
In General > s.a. tetrads.
* Idea: In the case of
a tangent vectors to a manifold, curvature also manifests itself in other ways,
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 
nb]m– 2
np[a pb]m .
* Symmetries: Rabcd
= Rcdab, Rabcd = R[ab][cd] (if
it comes from a metric), R[abc]d – T[abe Tc]ed – [aTbc]d =
0; Because of these symmetries, in n dimensions it has (1/12) 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]b – gb[c Rd]a) – R ga[c gd]b.
* Four dimensions: There
is also Weyl curvature in general, Rabcd = Cabcd
+ ga[cRd]b – gb[cRd]a – 
R
ga[c gd]b.
@ Identities: Dianyan PRD(87); > s.a. curvature [Bianchi identities].
> Special types of metrics:
see spherical symmetry; metric types.
Derived Quantities and Invariants > s.a. bel
tensor;
Cotton Tensor; Einstein
Tensor; Lovelock
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 Kretschman invariant, I:= Rabcd Rabcd;
In
a
vacuum spacetime, they can be expressed in terms of Cabcd.
@ Scalar invariants: 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,
a0709-in;
Lim & Carminati JMP(07)
[minimal set and syzygies].
@ Special cases: Pravda & Bicak gq/01-MG9
[algebraically
special]; Schmidt gq/01-GR14
[indistinguishable spacetimes]; Cherubini al al IJMPD(02)gq/03 [second-order,
and black holes]; Pelavas et al JMP(05)gq
[vanishing 0th- and first-order invariants]; Deser & Ryzhov CQG(05)gq [static
spherical, any D]; Senovilla m.DG/05-in
[mn Rabcd =
0]; Page a0806 [vanishing scalar invariants]; > s.a. lorentzian
geometry.
@ Derived quantities: Palatnik qp/03-wd
[
ij;i =
0, second-order in
Riemann]; > s.a. spectral geometry.
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 aY bX cY d;
> for applications, see orientation [Synge's
theorem] and the Hopf
Conjecture.
* Principal curvatures:
For a two-surface S in R3, 
i
are the roots of the equation det(Kab – qab)
= 0, i.e., the eigenvalues of the mixed tensor Kab.
* Gaussian curvature: R =
2/r1r2 = 12
(= det Kab),
where ri = principal radii
of curvature of S; 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" 
[0,2
],
r = radius of "small loop", a = "big radius".
* Mean curvature: k =
1/r1
+ 1/r2 = 
(1
+ 2),
up
to coefficients (
trace
of second fundamental form, tr Kab).
@ Sectional curvature: Geroch GRG(76);
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: 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).
@ Symmetries and diffeomorphisms: Swift GRG(94);
Duggal & Sharma 99.
@ Related topics: Hestenes IJTP(86)
[Clifford algebra method]; Hall & Kay JMP(88);
Santander AJP(92)
[measurement];
Colding AM(97)
[R and volume]; Schmidt gq/04 [gravitoelectromagnetism
and decompositions]; Sharipov a0709 [and spinor curvature tensor].
> Effects and measurements:
see atomic
physics [interferometry]; lorentz symmetry
in physics.
> Generalized: see curvature; differential
geometry, types
of manifolds.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
15 jun 2008