W_{ijhk} = R_{ijhk} - 1/(n - 2)*(g_{ih} R_{jk} - g_{jh} R_{ik} - g_{ik} R_{jh} + g_{jk} R_{ih}) + 1/(n - 1)(n - 2)*(g_{ih} g_{jk} - g_{jh} g_{ik})*R.

Example 1.

First create a 3 dimensional manifold M and show that the Weyl tensor of a randomly defined metric g1 is zero.

Calculate the Christoffel symbols.

Calculate the curvature tensor.

Calculate the Weyl tensor.

Example 3.

Define a 4 dimensional manifold and a metric g2.

Calculate the Weyl tensor directly from the metric  g2

We check the various properties of the Weyl tensor. First we check that it is skew-symmetric in its 1st and 2nd indices, and also in its 3rd and 4th indices.

Check the 1st Bianchi identity.

Check that W2 is trace-free on the indices 1 and 3.

Finally we check the conformal invariance of the Weyl tensor by computing the Weyl tensor W3 for g3 = f(y, z)*g2 and comparing W3 with f(y, z)*W2

DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], ContractIndices, CurvatureTensor, Physics[Riemann], InverseMetric, Physics[g_], SymmetrizeIndices