Example 1.
First create a vector bundle with base coordinates and fiber coordinates .
Define a spacetime metric on .
Define an orthonormal frame on with respect to the metric .
Calculate the solder form from the frame F.
Calculate the spin-connection for the solder form .
Example 2.
Define a rank 1 spinor . Calculate the covariant derivative of . Calculate the directional derivatives of .
Example 3.
Check that the covariant derivative of vanishes. Because is a spin-tensor, two connections are required. Calculate the Christoffel connection for the metric .
Define an epsilon spinor and check that its covariant derivative vanishes.
Example 4.
Calculate the curvature spin-tensor for the spin-connection Gamma2.
The curvature tensor for the Christoffel connection can be expressed in terms of the curvature spin-tensor and the bivector solder forms by the identity
Let's check this formula for the Christoffel connection Gamma1 and the spin-connection Gamma2. First calculate the curvature tensor for Gamma1.
Calculate the complex conjugate of the spinor curvature F.
Calculate the bivector soldering forms S and barS.
The first term on the right-hand side of (*) is
The second term on the right-hand side of (*) is