This is a brief reference list of some formulae I have found useful.
Connection coefficients: (MTW §8.5, 10.4)
Recall . In a coordinate basis, the are termed Christoffel symbols. In a non-coordinate basis, they include nonzero “commutation coefficients” (MTW §8.4)
Geodesic equation for affine parameter : (MTW §8.5)
If the parameter is not affine, a more general formula is needed.
Covariant derivative. Suppose is a scalar, a vector, and and vector fields. Then: (MTW §8.5, 8.7, 10.3)
If is a 4-velocity, the last line equates to (c.f. Schutz §3.3)
Coordinate basis and dual basis: These are dual as bases (Schutz §3.3):
but not dual as individual vectors, i.e. and are not dual in general, instead: