Contents

26 2 Differential geometry without a metricThe general rules (2.71) for the components of the covariant derivative ofa tensor imply the formulaR a bcd =Γ a bd|c − Γ a bc|d +Γ e bdΓ a ec − Γ e bcΓ a ed − D e cdΓ a be. (2.79)In a coordinate basis, the last term vanishes. The components (2.79) ofthe curvature tensor satisfy the symmetry relationsR a bcd = −R a bdc, R a [bcd] =0. (2.80)The covariant derivatives of the curvature tensor obey the BianchiidentitiesR a b[cd;e] =0. (2.81)By contraction we obtain the identitiesR a bcd;a +2R b[c;d] =0, (2.82)where the components R bd of the Ricci tensor are defined byR bd ≡ R a bad. (2.83)If a vector is parallelly transported round a closed curve, the initialand final vectors will in general not be equal: this phenomenon is calledholonomy. For infinitesimally small curves the holonomy is given by anintegral of the curvature tensor over an area enclosed by the curve, andconversely this gives an alternative way to define curvature.A compact and efficient method for calculating the components (2.79)with respect to a general basis is provided by Cartan’s procedure. Definingthe curvature 2-forms Θ a b byΘ a b ≡ 1 2 Ra bcdω c ∧ ω d , (2.84)equation (2.79) is completely equivalent to the second Cartan equationdΓ a b + Γ a c ∧ Γ c b = Θ a b, (2.85)which gives an algorithm for the calculation of the curvature from the connection.We collect the relations between the various quantities in Fig. 2.3.In this notation the Bianchi identities (2.81) are the components ofd 2 Γ a b =dΘ a b − Θ a c ∧ Γ c b + Γ a c ∧ Θ c b =0. (2.86)