3+1 decomposition of terms Involving Covariant derivative

[Jitesh Kumar Singh 20136]

Hi,

I am tring to do 3+1 decomposition of the Einstein-Maxwell equations using Xtensor. In the decomposed equations, I am encountering terms like n[a]n[b]CD[−b][X[−a]], where:

• n[a] represents the normal vector to the spatial hypersurface,
• CD denotes the covariant derivative on the 4-manifold,
• X[a] is the 4-vector field.

My goal is to express this term in terms of the covariant derivative on the 3-manifold and the Lie derivative. However, I am unsure how to proceed with this transformation using Xtensor.

Thanks

Jitesh