# Covariant derivative of a covariant vector

In one of my books of general relativity the covariant derivative of a covariant vector is defined by using Christoffel symbols as

$$D_cX_a = \partial_c X_a - \Gamma^a_{bc} X_b$$

For my feeling this notation is somehow strange, because indices are at the same time up- and downstairs.

In all my other texts I find

$$D_cX_a = \partial_c X_a - \Gamma^b_{ac} X_b$$

which is more logically in regard to the convention of summing over up- and downstair indices.

Is the first line wrong in my book or do I miss something?

• Which book? Which page? Aug 8, 2021 at 19:30

As you noted, index heights alone refute one answer, but let's check the other one doesn't have a coefficient/sign error, which such an analysis wouldn't detect. One definition of Christoffel symbols is $$(D_c-\partial_c)Y^a=\Gamma_{cb}^aY^b$$ so$$0=(D_c-\partial_c)(Y^aX_a)=\color{blue}{\Gamma_{ca}^bY^aX_b}+Y^a(D_c-\partial_c)X_a\implies(D_c-\partial_c)X_a=-\Gamma_{ca}^bX_b,$$where the indices $$a,\,b$$ in the blue term have been respectively relabelled $$b,\,a$$ for easier comparison of $$Y^a$$ coefficients.