Contents

24 2 Differential geometry without a metricwith components u a ;b as yet unspecified. The directional covariant derivativeis given by the vector∇vu =(u a ;bv b )e a . (2.66)The covariant derivative of the basis vector e a in the direction of the basisvector e b can be expanded in terms of basis vectors:∇ b e a =Γ c abe c , Γ c ab = 〈ω c , ∇ b e a 〉. (2.67)For consistency of (2.67) with the Leibniz rule applied to (2.11), the covariantderivative of a dual basis {ω a } is given by∇ b ω a = −Γ a cbω c . (2.68)The coefficients Γ c ab, called the connection coefficients, relate the bases atdifferent points of M, and they have to be imposed as an extra structureon M. We restrict ourselves to covariant derivatives satisfying∇uv −∇vu =[u, v] (2.69)for two arbitrary vectors u and v. This relation is equivalent to the equation2Γ c [ab] = −D c ab, (2.70)where the commutation coefficients are defined by (2.6). In a coordinatebasis, the connection coefficients Γ c ab have a symmetric index pair (ab).Therefore a covariant derivative satisfying (2.69) is called symmetric (ortorsionfree).Using the symmetry axiom (2.70), we may replace the partial derivativesin (2.40) and (2.62) for the components of, respectively, the exteriorderivative and the Lie derivative by covariant derivatives, so that thecommas can be replaced by semicolons.Once the connection coefficients are prescribed, the components u a ;c ofthe covariant derivative of u in the direction of the basis vector e c arecompletely determined,∇ c u = ∇ c (u a e a )=(u a |c +Γ a dcu d )e a = u a ;ce a ,(2.71a)and the components of the covariant derivative ∇T of a tensor (2.19) areT a 1···a r b1···b s;c =(T a 1···a r b1···b s) |c +Γ a 1 dc T d···ar b 1···b s+ ···+Γ ar dcT a 1···d b1···b s−Γ d b 1 cT a 1···a r d···bs −···−Γ d b scT a 1···a r b1···d,(2.71b)where the symbol f |a ≡ e a (f) =f ,i e a i has been used. Note that (2.71b)is valid for a general basis {e a }.