In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry.
Let (M,g) be a Riemannian or pseudo-Riemannian n-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention
Rijkl=glp(\partiali\Gamma
p-\partial | |
j\Gamma |
q); | |
ik |
\operatorname{Rm}(W,X,Y,Z)=g(\nablaW\nablaXY-\nablaX\nablaWY-\nabla[W,X]Y,Z).
Zjk=Rjk-
1 | |
n |
Rgjk,
\begin{align} Sijkl&=
R | |
n(n-1) |
(gilgjk-gikgjl)\\ Eijkl&=
1 | |
n-2 |
(Zilgjk-Zjlgik-Zikgjl+Zjkgil)\\ Wijkl&=Rijkl-Sijkl-Eijkl. \end{align}
Rijkl=Sijkl+Eijkl+Wijkl.
Terminological note. The tensor W is called the Weyl tensor. The notation W is standard in mathematics literature, while C is more common in physics literature. The notation R is standard in both, while there is no standardized notation for S, Z, and E.
Each of the tensors S, E, and W has the same algebraic symmetries as the Riemann tensor. That is:
\begin{align} Sijkl&=-Sjikl=-Sijlk=Sklij\\ Eijkl&=-Ejikl=-Eijlk=Eklij\\ Wijkl&=-Wjikl=-Wijlk=Wklij\end{align}
\begin{align} Sijkl+Sjkil+Skijl&=0\\ Eijkl+Ejkil+Ekijl&=0\\ Wijkl+Wjkil+Wkijl&=0. \end{align}
gilWijkl=0.
(In fewer than three dimensions, every manifold is locally conformally flat, whereas in three dimensions, the Cotton tensor measures deviation from local conformal flatness.)
One may check that the Ricci decomposition is orthogonal in the sense that
SijklEijkl=SijklWijkl=EijklWijkl=0,
Tijkl=gipgjqgkrglsTpqrs.
RijklRijkl=SijklSijkl+EijklEijkl+WijklWijkl.
This orthogonality can be represented without indices by
\langleS,E\rangleg=\langleS,W\rangleg=\langleE,W\rangleg=0,
2. | |
|\operatorname{Rm}| | |
g |
One can compute the "norm formulas"
\begin{align} SijklSijkl&=
2R2 | |
n(n-1) |
\\ EijklEijkl&=
4RijRij | - | |
n-2 |
4R2 | |
n(n-2) |
\\ WijklWijkl&=RijklRijkl-
4RijRij | + | |
n-2 |
2R2 | |
(n-1)(n-2) |
\end{align}
\begin{align} gilSijkl&=
1 | |
n |
Rgjk\\ gilEijkl&=Rjk-
1 | |
n |
Rgjk\\ gilWijkl&=0. \end{align}
Mathematically, the Ricci decomposition is the decomposition of the space of all tensors having the symmetries of the Riemann tensor into its irreducible representations for the action of the orthogonal group . Let V be an n-dimensional vector space, equipped with a metric tensor (of possibly mixed signature). Here V is modeled on the cotangent space at a point, so that a curvature tensor R (with all indices lowered) is an element of the tensor product V⊗V⊗V⊗V. The curvature tensor is skew symmetric in its first and last two entries:
R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)
R(x,y,z,w)=R(z,w,x,y),
S2Λ2V
b:S2Λ2V\toΛ4V
b(R)(x,y,z,w)=R(x,y,z,w)+R(y,z,x,w)+R(z,x,y,w).
The space in S2Λ2V is the space of algebraic curvature tensors. The Ricci decomposition is the decomposition of this space into irreducible factors. The Ricci contraction mapping
c:S2Λ2V\toS2V
c(R)(x,y)=\operatorname{tr}R(x, ⋅ ,y, ⋅ ).
(h{~\wedge circ~}k)(x,y,z,w)=h(x,z)k(y,w)+h(y,w)k(x,z)-h(x,w)k(y,z)-h(y,z)k(x,w)
If n ≥ 4, then there is an orthogonal decomposition into (unique) irreducible subspaces
where
SV=Rg{~\wedge circ~}g
R
EV=g{~\wedge circ~}
2 | |
S | |
0V |
CV=\kerc\cap\kerb.
The parts S, E, and C of the Ricci decomposition of a given Riemann tensor R are the orthogonal projections of R onto these invariant factors, and correspond (respectively) to the Ricci scalar, the trace-removed Ricci tensor, and the Weyl tensor of the Riemann curvature tensor. In particular,
R=S+E+C
|R|2=|S|2+|E|2+|C|2.
The Ricci decomposition can be interpreted physically in Einstein's theory of general relativity, where it is sometimes called the Géhéniau-Debever decomposition. In this theory, the Einstein field equation
Gab=8\piTab
Tab