Some Confusion in Evans' "Cartan Geometry"

Gerhard W. Bruhn, Darmstadt University of Technology

1. Application of exterior covariant derivatives

Let's start with some well-known basics of Cartan geometry: The moving frame compatibility relation

(C)                 ∂_μ q_ν^a − Γ_μ^λ_ν q_λ^a + ω_μ^a_b q_ν^b = 0         |         dx^μ Ù dx^ν

gives rise to introduce the 1-forms

q^a := q_ν^a dx^ν         and         ω^a_b := ω_μ^a_b dx^μ

[2, (J.36-37)], [4, p.239 (5.14)]. The forms q^a are vector-valued. However, note that the forms ω^a_b are not tensorial since under LLT's they transform inhomogeneously [1, (3.134)], [2, (J.15)], [4, p.230 (1.11)].

We have

d Ù q^a + ω^a_b Ù q^b = q_λ^a Γ_μ^λ_ν dx^μ Ù dx^ν = 2 q_λ^a Γ_[μ^λ_ν] dx^μ Ù dx^ν = q_λ^a T_μ^λ_ν dx^μ Ù dx^ν
= T_μ^a_ν dx^μ Ù dx^ν =: T^a ,

thus, as a direct consequence of (C) we obtain [1, (3.137)], [2, (J.28)]

(C1)                 T^a = d Ù q^a + ω^a_b Ù q^b .         (1st Maurer-Cartan structure equation)

Application of dÙ to (C1) yields

                                d Ù T^a = 0 + d Ù (ω^a_c Ù q^c)
                                = (d Ù ω^a_c) Ù q^c − ω^a_b Ù (d Ù q^b)
                                = (d Ù ω^a_c) Ù q^c − ω^a_b Ù (T^b − ω^b_c Ù q^c)
                                = (d Ù ω^a_c + ω^a_b Ù ω^b_c) Ù q^c − ω^a_b Ù T^b

or

                                d Ù T^a + ω^a_b Ù T^b = (d Ù ω^a_c + ω^a_b Ù ω^b_c) Ù q^c .

Introducing here [1, (3.138)], [2, (J.29)], [4, p.240 (6.10)]

(C2)         R^a_b := d Ù ω^a_b + ω^a_c Ù ω^c_b     (2nd Maurer-Cartan structure equation)

we obtain [1, (3.140)], [2, (J.31)]

(C3)                 d Ù T^a + ω^a_b Ù T^b = R^a_b Ù q^b .         (Ricci, 1st Bianchi identity)

Applying dÙ to (C2) yields

                                d Ù R^a_b = 0 + d Ù (ω^a_c Ù ω^c_b) = (d Ù ω^a_c) Ù ω^c_b − ω^a_c Ù (d Ù ω^c_b)
                                             = (R^a_c − ω^a_d Ù ω^d_c) Ù ω^c_b − ω^a_c Ù (R^c_b − ω^c_d Ù ω^d_b) ,

thus

(C4)                 d Ù R^a_b = − ω^c_b Ù R^a_c − ω^a_c Ù R^c_b .         (2nd Bianchi identity)

We introduce the exterior covariant derivative D Ù of p-forms X^a, X_b and X^a_b respectively ([1, (3.136)], [2, (J.31)]) by

                        D Ù X^a := d Ù X^a + ω^a_c Ù X^c ,
(D)
                        D Ù X^a_b := d Ù X^a_b + ω^a_c Ù X^c_b + ω^c_b Ù X^a_c

to obtain for the 2-form R^a_b [3, (19.24a)]

(C4')             D Ù R^a_b = 0 .                 (2nd Bianchi identity)

(C1) can be rewritten as

(C1')             D Ù q^a = T^a .                 (1st Maurer-Cartan structure equation)

                R^a_b := d Ù ω^a_b + ω^a_c Ù ω^c_b = d Ù ω^a_b + (ω^a_c Ù ω^c_b − ω^c_b Ù ω^a_c) + ω^c_b Ù ω^a_c

or

(C2')         D Ù ω^a_b = R^a_b + ω^a_c Ù ω^c_b .         (2nd Maurer-Cartan structure equation)

(D) applied to (C3) yields [3, (19.24a)]

(C3')             D Ù T^a = R^a_b Ù q^b .                                 (Ricci, 1st Bianchi identity)

evidently now assuming another meaning of the "exterior covariant derivative".

(D')             D' Ù X^a_b = d Ù X^a_b + ω^a_c Ù X^c_b .

Evans' readers are warned about the possible confusions that could arise from the inconsistency (D)//(D') of tacitly using two different meanings of DÙ in [5].

The question whether D'Ù can be considered as covariant is anwered by Arkadiusz Jadczyk in [6].

2. Evans' Poincaré Lemma

The usual Poincaré Lemma

(PL)                 d Ù d = 0