Wolfram Alpha:

```Ricci Decomposition
-------------------

The Ricci Decomposition is a way of breaking up the the Riemann
tensor it 3 different parts as follws:

Rabcd = Sabcd + Eabcd + Cabcd

Eabcd is related to the RICCI TENSOR, Rab, as follows:

Where,

Sab = Rab - (1/n)gabR ... the traceless Ricci tensor.

We can define Rab as the contraction of the first and third
indeces of Rdacb (the 1st and 4th or 2nd and 4th are also
possible) as follows:

Rab = gcdRdacb

= Rcacb

Note: gdaRdabc = gbcRdabc = 0.

Eabcd is semi-traceless because it is built from the metric and
the traceless Ricci tensor.

Sabcd is related to the CURVATURE SCALAR, R, as follows:

Sabcd = (R/n(n - 1))(gacgdb - gadgcb)

Where R is the metric contraction gabRab.

Cabcd is a the WEYL TENSOR.

Notes:

Properties of Rabcd:

- Rabcd = Rcdab ... symmetric

= -Rbacd ... antisymmetric

= -Rabdc ... antisymmetric

- Rabcd + Radbc + Racdb = 0 ... 1st BIANCHI IDENTITY.

- ∇eRabcd + ∇cRabde + ∇dRabec = 0 ... 2nd BIANCHI IDENTITY.

Properties of Rab:

- Using the property that Rdabc = Rbcda

Rba = gdcRcbda ... the metric is symmetric.

Therefore, Rab = Rba ... symmetric

- Rab is associated with changes in the volume of a body due to
tidal forces it feels when moving along a geodesic.

Properties of Cabcd:

- Cabcd ... same symmetry properties as Rabcd

- Cabcd is completely traceless. It is associated with changes
in the shape of a body due to the tidal forces it feels when
moving along a geodesic.

The 3 Tensors
-------------

-----------------------------------------------------
|     Riemann     |   Ricci   |  R  |      Weyl       |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Dimension     |   4 |  3  |  2  |  4  |  3  |  2  |  4  |  3  |  2  |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Total Comps.  | 256 | 81  | 16  | 16  |  9  |  1  |     |     |     |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Indep. Comps. |  20 |  6  |  1  | 10  |  6  |  1  | 10  |  0  |  0  |
-----------------------------------------------------
Notes:

-  The Riemann tensor is the sum of the Weyl and Ricci tensors.

-  In 4D it is both necessary and sufficient for the Riemann tensor
to vanish for the manifold to be flat.  If the Ricci tensor vanishes
the space is said to be RICCI FLAT.

-  In 3D it is both necessary and sufficient for the Ricci tensor
to vanish for the manifold to be flat.

-  The Ricci tensor can describe everything about 3D curvature and all
the 81 elements of the Riemann tensor can be calculated from the 6
independent components of the Ricci tensor.

-  In 2D it is both necessary and sufficient for the Scalar tensor
to vanish for the manifold to be flat.

-  Independent components are calculated from:

Riemann = n2(n2 - 1)/12
Weyl = n2(n2 - 1)/12 - n(n + 1)/2 for n > 2
```