An Atiyah Sequence for Noncommutative Principal Bundles

We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given $^*$-automorphism on the quantum base space lifts to a $^*$-automorphism on the quantum total space that commutes with the underlying structure group.


Introduction
Chern-Weil theory is an important tool for many disciplines such as analysis, geometry, and mathematical physics. For instance, it provides invariants of principal bundles and vector bundles by means of connections and curvature, and thus a way to measure their non-triviality. The Chern-Weil homomorphism of a smooth principal bundle q : P → M with structure group G is an algebra homomorphism from the algebra of polynomials that are invariant under the adjoint action of G on its Lie algebra, into the even de Rham cohomology H 2• dR (M, K). This map is achieved by evaluating an invariant polynomial on the curvature of a connection 1-form ω on P . The latter procedure involves lifting vector fields on M horizontally with respect to ω to G-equivariant vector fields on P . An important remark in this context is that connection 1-forms on P are in a bijective correspondence with C ∞ (M )-linear sections of the associated Lie algebra extension which is well-known as the so-called Atiyah sequence of the principal bundle P (see, e.g., [4,24]). About three decades after the seminal works of Chern, Weil, and Atiyah, Lecomte described in [26] a cohomological construction which generalizes the classical Chern-Weil homomorphism: Lecomte's construction associates characteristic classes to each Lie algebra extension, and the classical construction of Chern and Weil arises in this context from the Atiyah sequence above. The work presented here is an attempt towards a derivation-based Chern-Weil theory for noncommutative principal bundles. More precisely, our main objective is to provide a derivationbased generalization of the classical Atiyah sequence in equation (1.1) to the setting of noncommutative principal bundles. For this purpose, we focus on free C * -dynamical systems, which provide a natural framework for noncommutative principal bundles (see, e.g., [6,19,31,36] and references therein). Their structure theory and their relation to K-theory (see, e.g., [15, arXiv:2107.04653v3 [math.OA] 7 Mar 2022 20,34,35,36] and references therein) certainly appeal to operator algebraists and functional analysts. Additionally, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [22,23,28,37,38]) and mathematical physics (see, e.g., [7,10,13,14,18,21,25,41] and references therein).
For the sake of completeness, we bring to mind that the algebraic setting of Hopf-Galois extensions already comprises abstract notions of connections, curvature and characteristic classes in terms of a universal differential calculus (see, e.g., [8,9,12] and references therein). Furthermore, we recall that Neeb [29] associated Lie group extensions with projective modules that generalize the classical Atiyah sequence for vector bundles.
Finally, we wish to mention the recent theory of pseudo-Riemannian calculus introduced by Arnlind and Wilson in [3], which constitutes a derivation-based computational framework for Riemannian geometry over noncommutative algebras (see also [1,2]). It is our hope that this work will advance to the development of pseudo-Riemannian calculus.

Organization of the article
Let (A, G, α) be a free C * -dynamical system with fixed point algebra B. After some preliminaries, we review in Section 3 the notion of a factor system of (A, G, α), which is the key feature of our investigation. In fact, factor systems provide us with a natural framework for doing computations and constitute invariants for (A, G, α). In Section 4 we utilize factor systems to give a characterization of when a * -automorphism on B can be lifted to a * -automorphism on A that commutes with α (Theorem 4.1). Moreover, we use our findings to examine an "integrated" version of the Atiyah sequence for noncommutative principal bundles (equation (4.2)). Section 5 is devoted to the study of the special case when G is compact Abelian. Most notably, we get a characterization in terms of second group cohomology on the dual group of G with values in the unitary group of the center of B (Theorem 5.5). In addition, we are able to show that if A is commutative, then every * -automorphism on B lifts to A provided it leaves the class of A invariant (Corollary 5.7). In Section 6 we make use of the results of Section 4 to establish a lifting result for 1-parameter groups (Theorem 6.4). This is of particular interest in Section 7, where we finally present a generalization of the classical Atiyah sequence in equation (1.1) to the setting of free C * -dynamical systems. Last but not least, we discuss infinitesimal objects such as connections and curvature (Section 7.3).
Finally, we would like to mention that with little effort the arguments and the results in Sections 3, 4, and 6 extend to free actions of quantum groups (see [36]).

Preliminaries and notation
This preliminary section exhibits the most fundamental definitions and notations used in this article.
About groups. Let G be a compact group. We write Irr(G) for the set of equivalence classes of irreducible representations of G and denote by 1 ∈ Irr(G) the class of the trivial representation.
About Hilbert spaces. Let G be a compact group. Furthermore, let H σ , σ ∈ Irr(G), be a family of Hilbert spaces, and for each σ ∈ Irr(G) let T σ be an operators on H σ . Throughout this article, we shall freely utilize the fact that σ → H σ and σ → T σ can be extended to arbitrary finite-dimensional representations of G by taking direct sums with respect to irreducible subrepresentations.
About C * -dynamical systems. Let A be a unital C * -algebra and let G be a compact group that acts on A by * -automorphisms α g : A → A, g ∈ G, such that G × A → A, (g, x) → α g (x) is continuous. Throughout this article, we call such data a C * -dynamical system and denote it briefly by (A, G, α). Moreover, we typically write B := A G for the corresponding fixed point algebra.
The conditional expectation P 1 onto B allows us to define a definite right B-valued inner product on A by The completion of A with respect to the induced norm yields a right Hilbert B-module, which we denote by L 2 (A). The algebra A admits a faithful * -representation on L 2 (A) by adjointable operators given by λ : A → L L 2 (A) , λ(x)y := x · y, and consequently we may identify A with λ(A) ⊆ L L 2 (A) . Furthermore, for each g ∈ G we have a unitary operator U g on L 2 (A) defined for x ∈ A ⊆ L 2 (A) by U g x := α g (x). The map G g → U g ∈ U L 2 (A) is strongly continuous and implements the * -automorphisms α g , g ∈ G, via λ(α g (x)) = U g λ(x)U * g for all x ∈ A. Like every representation of G, the algebra A can be decomposed into its isotypic components, let us say, A(σ), σ ∈ Irr(G), which amounts to saying that their algebraic sum is a dense * -subalgebra of A. Furthermore, the isotypic components are pairwise orthogonal, right Hilbert B-submodules of L 2 (A) such that L 2 (A) = π∈Ĝ A(π).
has dense range with respect to the canonical C * -norm on C(G, A). This condition was originally introduced for actions of quantum groups on C * -algebras by Ellwood [19] and is known to be equivalent to Rieffel's saturatedness [32] and the Peter-Weyl-Galois condition [6]. One of the key tools used in this article is a characterization of freeness that we provided in [36,Lemma 3.3], namely that a C * -dynamical system (A, G, α) is free if and only if for each irreducible representation (σ, V σ ) of G there is a finite-dimensional Hilbert space H σ and an isometry s(σ) ∈ A ⊗ L(V σ , H σ ) satisfying α g (s(σ)) = s(σ) · σ g for all g ∈ G. A rich class of free actions is given by so-called cleft actions, which are characterized as follows: For each irreducible representation (σ, V σ ) of G there is a unitary element u(σ) ∈ A ⊗ L(V σ ) such that α g (u(σ)) = u(σ) · σ g for all g ∈ G (cf. [35,Definition 4.1]).
About 1-parameter groups. Let A be a unital C * -algebra and let (ϕ t ) t∈R be a 1-parameter group of * -automorphisms ϕ t ∈ Aut(A). We typically use the letter A ∞ to denote the smooth domain of (ϕ t ) t∈R , which is the set of elements x ∈ A such that R t → ϕ t (x) ∈ A is smooth. Moreover, we let stand for the corresponding * -derivation. About derivations. Let A be a unital * -algebra. We let der(A) stand for the Lie algebra of * -derivations of A. Furthermore, we write A skew ⊆ A for the subset of skew-adjoint elements, i.e., A skew := {a ∈ A : a * = −a}, and recall that each a ∈ A skew gives rise to a * -derivation

Factor systems
Let (A, G, α) be a free C * -dynamical system. Furthermore, for each σ ∈ Irr(G) let H σ be a finitedimensional Hilbert space and let s(σ) ∈ A⊗L(V σ , H σ ) an isometry satisfying α g s(σ) = s(σ)·σ g for all g ∈ G (cf. [36,Lemma 3.3]). For 1 ∈ Irr(G) we choose H 1 := C and let s(1) := 1 A . A key feature of (A, G, α) is the factor system associated with the isometries s(σ), σ ∈ Irr(G), (see [36,Definition 4.1]), which we now recall for the convenience of the reader. First, we put B := A G . Second, for expediency, we naturally extend σ → H σ and σ → s(σ) to arbitrary finite-dimensional representations σ of G by taking the direct sum with respect to irreducible subrepresentations. For each finite-dimensional representation σ of G we may then define the * -homomorphism and for each pair (σ, π) of finite-dimensional representations of G the element Then the following relations hold: for all finite-dimensional representations σ, π, ρ of G and b ∈ B (see [37,Lemma 4.3]). The triple (H, γ, ω) of the above families is referred to as the factor system of (A, G, α) associated with s(σ), σ ∈ Irr(G), or simply as a factor system of (A, G, α) when no explicit reference to the isometries is needed. We recall from [36] that, for σ ∈ Irr(G), the isotypic component A(σ) of A can be written as A(σ) = Tr(ys(σ)) : y ∈ B ⊗ L(H σ , V σ ) .
By construction, each factor system of (A, G, α) is a factor system for (B, G) and, by [36,Lemma 4.3], all factor systems of (A, G, α) are conjugated. In fact, given any unital C * -algebra B and any compact group G, we have shown in [36,Section 5] that the equivalence classes of free C * -dynamical systems (A, G, α) with fixed point algebra B are in 1-to-1 correspondence with conjugacy classes of factor systems of (B, G).

Lifting an automorphism
Let (A, G, α) be a free C * -dynamical system with fixed point algebra B and let β be a * -automorphism of B. In this section we address the question whether β can be lifted to a * -automorphismβ of A that commutes with all α g , g ∈ G. In the affirmative case we say that β lifts to A and thatβ is a lift of β.
To phrase our result we note that Aut(B) acts on the factor systems for (B, G). For β ∈ Aut(B) and a factor system (H, γ, ω) for (B, G) we may define a new factor system H, γ β , ω β for (B, G) by putting for all σ, π ∈ Irr(G). With this we give an answer to the above lifting problem by proving the following theorem: be a free C * -dynamical system with fixed point algebra B and let β be a * -automorphism of B. Then the following statements are equivalent: Remark 4.2. By the above theorem, a necessary condition for β to lift to A is that We split the proof of Theorem 4.1 into a sequence of lemmas. For a start we fix, for each σ ∈ Irr(G), a finite-dimensional Hilbert space H σ and an isometry s(σ) ∈ A⊗L(V σ , H σ ) satisfying α g (s(σ)) = s(σ)·σ g for all g ∈ G; for 1 ∈ Irr(G) we take H 1 := C and s(1) := 1 A . As in Section 3, we write (H, γ, ω) for the associated factor system. Our first result establishes the implication "(a)⇒(b)" of Theorem 4.1: If β lifts to A, then (H, γ, ω) and H, γ β , ω β are conjugated.
The task is now to prove the converse implication, "(b)⇒(a)", of Theorem 4.1. For this purpose, we consider partial isometries for all y ∈ B ⊗ L(H σ , V σ ). Taking direct sums gives a mapβ on the dense * -subalgebra A f . Due to the normalizations v(1) = 1 B and s(1) That is,β extends β. Furthermore, a few moments of thought show thatβ is bijective. In fact, its inverse is given by the direct sum of the mapsβ −1 : A(σ) → A(σ), σ ∈ Irr(G), defined bŷ . We proceed to establish further properties.
Lemma 4.4. The following assertions hold for the mapβ on A f : 2.β is multiplicative.
Proof . Since isotypic components are pairwise orthogonal, it suffices to show β (x 1 ),β(x 2 ) B = β x 1 , x 2 B for all x 1 , x 2 ∈ A(σ) and σ ∈ Irr(G). To this end, let σ ∈ Irr(G) and let x 1 , x 2 ∈ A(σ). By Lemma 4.4, we obtain We now decompose x * 1 x 2 as i∈I Tr(y i s(σ i )) for some mutually distinct representations σ i ∈ Irr(G) and y i ∈ B ⊗ L(H σ i , V σ i ) and note that there is i 0 ∈ I such that σ i 0 = 1, which is due to the fact that σ ⊗σ contains 1 as a subrepresentation. Hence By Lemma 4.5, the bijectivity ofβ, and the fact that A f is dense in L 2 (A), we may extendβ to a unitary map, let's say, U on L 2 (A). Consequently, there is a * -automorphism on A, for which we shall use the same letterβ by a slight abuse of notation, such that It is easily checked thatβ extends β and commutes with all α g , g ∈ G. Summarizing, we have shown the implication "(b)⇒(a)" of Theorem 4.1, which concludes the proof of this theorem.
We now turn to an application of our findings: The group admits a short exact sequence is the group of gauge transformations of (A, G, α) and Aut(B) [A] ⊆ Aut(B) is the group of * -automorphisms that lift to A. Theorem 4.1 states that Aut(B) [A] can be characterized in terms of a factor system (H, γ, ω) of (A, G, α) as Looking at equation (4.1), we easily see that different choices of v(σ), σ ∈ Irr(G), amount to different lifts. Hence the construction, in fact, shows that Gau(A) is isomorphic to the group which consists of all families of unitaries u(σ) ∈ γ σ (1 B ) B ⊗ L(H σ ) γ σ (1 B ), σ ∈ Irr(G), that lie in the commutant of γ σ (B) and satisfy the equation for all σ, π ∈ Irr(G).

The special case of a compact Abelian structure group
Let G be a compact Abelian group, let (A, G, α) be a free C * -dynamical system with fixed point algebra B, and let β be a * -automorphism of B. In the previous section we showed in Theorem 4.1 that β lifts to A if and only if (H, γ, ω) ∼ H, γ β , ω β for any factor system (H, γ, ω) of (A, G, α).
In this section we give another characterization of when β lifts to A in terms of second group cohomology on the dual groupĜ := Hom(G, T) with values in the group UZ(B) of unitaries in the center of B.
To begin with, let us fix, for each σ ∈Ĝ, a finite-dimensional Hilbert space H σ and an isometry s(σ) ∈ A ⊗ L(C, H σ ) satisfying α g (s(σ)) = σ(g) · s(σ) for all g ∈ G. For the trivial character, denoted by 0, we choose H 0 := C and s(0) := 1 A . Just as before, we write (H, γ, ω) for the associated factor system. We shall also make use of the so-called Fröhlich map ∆ :Ĝ → Aut(UZ(B)) associated with (A, G, α), which is for each σ ∈Ĝ defined by restricting the map We point out that, since all factor systems of (A, G, α) are conjugated, the Fröhlich map does not depend on the choice of the factor system. Given a liftβ of β, we have seen in the proof of Lemma 4 for all σ, π ∈Ĝ. A moment's thought shows that equation (5.2) can be rewritten as Our objective is now to give a group cohomological interpretation of the latter equation. For this purpose, let us for a moment assume that for all σ ∈Ĝ the * -homomorphisms γ σ and γ β σ are conjugated, i.e., there is a partial isometry v(σ) ∈ B ⊗ L(H σ ) satisfying equation (5.1). We freely use the fact that there is no loss of generality in assuming that γ σ (1 B ) and γ β σ (1 B ) are the initial and final projections of v(σ), respectively. Let us now consider the map u :Ĝ ×Ĝ → B defined by u(σ, π) := s(σ + π) * β −1 v(σ + π)ω(π, σ) * γ π v(σ) * v(π) * s(π)s(σ). (5.3) Lemma 5.1. The following assertions hold: 2. u(σ, π) is unitary for all σ, π ∈Ĝ.
This shows that u and u are cohomologous as asserted.
2. Let A = C(P ) and B = C(X) for some compact spaces P and X, respectively, and consider P as a topological principal G-bundle over X. Let h : X → X be the homeomorphism such that β(f ) = f • h for all f ∈ C(X). Then the "if" condition in Corollary 5.7 states that, for each σ ∈Ĝ, the vector bundles determined by γ σ (1 B ) and γ σ (1 B ) • h, respectively, are equivalent.

Lifting 1-parameter groups
Let (β t ) t∈R be a smooth 1-parameter group of * -automorphisms β t ∈ Aut(B) and let δ := Dβ t denote the corresponding * -derivation on its smooth domain B ∞ (cf. Section 2). In this section we investigate whether there is a smooth 1-parameter group β t t∈R of * -automorphismsβ t ∈ Aut(A) such that, for each t ∈ R,β t is a lift of β t . In the affirmative case we say that (β t ) t∈R lifts smoothly to A and that (β t ) t∈R is a smooth lift of (β t ) t∈R .
1. We recall from the classical theory of smooth principal bundles that every smooth 1-parameter group on the base manifold lifts (smoothly) to the total space of the principal bundle. For a compact Abelian group G this follows from Corollary 5.7, because, for each σ ∈Ĝ, the projections γ σ (1 B ) and βγ σ (1 B ) are obviously homotopic, and hence Murrayvon Neumann equivalent.
2. The example in [5,Section 4] shows that not all 1-parameter groups lift.
Let us fix, for each σ ∈ Irr(G), a finite-dimensional Hilbert space H σ and an isometry s(σ) ∈ A ⊗ L(V σ , H σ ) satisfying α g s(σ) = s(σ) · σ g for all g ∈ G; for 1 ∈ Irr(G) we take H 1 := C and s(1) := 1 A . Throughout the following, we make the standing assumption that the associated factor system (H, γ, ω) is smooth in the sense that and for all σ, π ∈ Irr(G).
Combining Lemmas 6.2 and 6.3, we have established: Theorem 6.4. Let (A, G, α) be a free C * -dynamical system with fixed point algebra B. Furthermore, let (β t ) t∈R be a smooth 1-parameter group of * -automorphisms β t ∈ Aut(B). Then the following statements are equivalent: (a) (β t ) t∈R lifts smoothly to A.

An Atiyah sequence for noncommutative principal bundles
In this section we generalize the classical Atiyah sequence in equation (1.1) to the setting of free C * -dynamical systems. In addition, we explain how this can be used to produce characteristic classes.
To this end, we consider a free C * -dynamical system (A, G, α) with fixed point algebra B and we fix a dense unital * -subalgebra B 0 ⊆ B. Again, for each σ ∈ Irr(G) we choose a finitedimensional Hilbert space H σ and an isometry s(σ) in A ⊗ L(V σ , H σ ) satisfying α g (s(σ)) = s(σ)(1 A ⊗ σ g ) for all g ∈ G; for 1 ∈ Irr(G), we take H 1 := C and s(1) := 1 A . We denote by (H, γ, ω) the associated factor system and assume that and for all σ, π ∈ Irr(G). Similar arguments as in [37,Section 5.1]

The associated Atiyah sequence
For a * -derivation δ on B 0 we say that δ lifts to A 0 if there is a * -derivationδ on A 0 that extends δ and commutes with all α g , g ∈ G. In this case we callδ a lift of δ. As an immediate consequence of Theorem 6.4 we obtain: Corollary 7.1. Let δ ∈ der(B 0 ). Then the following statements are equivalent: (a) δ lifts to A 0 .
Remark 7.2. Suppose that G is compact Abelian and that (A, G, α) is cleft. This is essentially the setting studied by Batty, Carey, Evans, and Robinson in [5], but without the factor system terminology. Combining Corollary 7.1 with the results from Section 5, we obtain a generalization of the main results in [5] to the setting of free actions of compact groups.
Our study revolves around the restricted gauge action α : T → Aut A 3 θ defined by α z (u 1 ) := u 1 , α z (u 2 ) := u 2 , α z (u 3 ) := z · u 3 for all z ∈ T. Its fixed point algebra is the quantum 2-torus A 2 θ generated by the unitaries u 1 and u 2 , where θ denotes the real skew-symmetric 2 × 2-matrix with upper right off-diagonal entry θ 12 . More generally, for each k ∈ Z, the corresponding isotypic component is A 3 θ (k) takes the form u k 3 A 2 θ . In particular, the C * -dynamical system A 3 θ , T, α is cleft and therefore free. The factor system associated with the unitaries u(k) := u k 3 , k ∈ Z, is given by the following data: For k ∈ Z we have γ k = τ z with z = λ k 31 , λ k 32 , 1 , and for k, l ∈ Z the cocycle ω(k, l) computes as 1 A 2 θ . Next, we look at the dense unital * -subalgebra B 0 of A 2 θ generated by all noncommutative polynomials in u 1 and u 2 . Then A 0 is given by the dense unital * -algebra of A 3 θ generated by all noncommutative polynomials in u 1 , u 2 , and u 3 , as is easily seen. Furthermore, it follows from [17,Section 3.4] that the * -derivations of B 0 split as a semidirect product of inner and outer * -derivations, i.e., der(B 0 ) ∼ = Inn(der(B 0 )) Out(der(B 0 )). If moreover θ 12 is irrational, then Out(der(B 0 )) is linearly generated by δ 1|B 0 and δ 2|B 0 . Obviously, δ 1|B 0 and δ 2|B 0 may be lifted to δ 1|A 0 and δ 2|B 0 , respectively. It is also clear that each inner * -derivation of B 0 lifts to A 0 . In consequence, der(B 0 ) (γ,ω) = der(B 0 ). Since we also have Z 1 Z, B skew 0 ∼ = B skew 0 via the evaluation map f → f (1), it follows that the associated Atiyah sequence reads as Remark 7.3). Finally, a moment's thought shows that this sequence is split, and so it does, unfortunately, only give trivial Chern-Weil-Lecomte classes. However, if needed, one may associate secondary characteristic classes as described in [40].

Associating connections and curvature
Connection 1-forms are the fundamental tool in the theory of smooth principal bundles and give rise to the notion of connections on associated vector bundles. Such a connection or, more precisely, its induced covariant derivative is an operator that can differentiate sections of each associated vector bundle along tangent directions in the base manifold.
In this section we discuss suitable generalizations of theses notions to the C * -algebraic setting of noncommutative principal bundles. In particular, we provide explicit formulas for connections and curvature on associated noncommutative vector bundles. To do this, we proceed as follows: For each finite-dimensional representation (σ, V σ ) of G we consider the associated natural inner product with left A 0 ⊗L(V σ )-valued inner product A 0 ⊗L(Vσ) ·, · and right B 0 -valued inner product ·, · B 0 defined on simple tensors by respectively. Notably, the bimodule structure and the inner products are related by the compatibility condition A 0 ⊗L(Vσ) x, y . z = x . y, z B 0 for all x, y, z ∈ Γ A 0 (V σ ).
The following result provides a criterion for ensuring that the space Γ A 0 (V σ ) admits a so-called standard right-module frame (see, e.g., [33,Section 2]). Lemma 7.7. Let (σ, V σ ) be a finite-dimensional representation of G. Then there are elements In particular, the reproducing formula x = d k=1 s k . s k , x B 0 holds for all x ∈ Γ A 0 (V σ ). If (A, G, α) is cleft, we find such elements with s k , s l B 0 = δ k,l · 1 B for all 1 ≤ k, l ≤ d.
Proof . Let e 1 , . . . , e d be an orthonormal bases of H σ and let s k ∈ A 0 ⊗ V σ , 1 ≤ k ≤ d, be the columns of s(σ) * ∈ A 0 ⊗ L(H σ , V σ ). Then s k ∈ Γ A 0 (V σ ) for all 1 ≤ k ≤ d, which is due to the fact that α g (s(σ)) = s(σ)(1 A ⊗ σ g ) for all g ∈ G. In addition, a moment's thought reveals that In what follows, we consider a fixed finite-dimensional representation (σ, V σ ) of G and elements s 1 , . . . , s d ∈ Γ A 0 (V σ ) as in Lemma 7.7. Lemma 7.8. Let (σ, V σ ) be a finite-dimensional representation of G. If δ : B 0 → B 0 is a * -derivation, then the linear map satisfies the following equations for all x, y ∈ Γ A 0 (σ) and b ∈ B 0 .
Proof . Let x, y ∈ Γ A 0 (σ) and b ∈ B 0 . Using first the right B 0 -linearity of ·, · B 0 , second the derivation property of δ, and finally the reproducing formula for x, we obtain Likewise, it may be concluded that (δ ( x, s k B 0 ) · s k , y B 0 + x, s k B 0 · δ ( s k , y B 0 )) = where for the second equality we have exploited the fact that δ is a * -derivation.
To proceed, we bring to mind that, given a finitely generated projective right B 0 -module E together with a connection ∇ : der(B 0 ) × E → E, its curvature R := R ∇ with respect to ∇ is the map defined by Lemma 7.11. Let (σ, V σ ) be a finite-dimensional representation of G. Then the curvature R σ := R ∇ σ of Γ A 0 (σ) with respect to ∇ σ takes the form for all δ 1 , δ 2 ∈ der(B 0 ) and x ∈ Γ A 0 (σ).
Remark 7.13. With a little more effort one can also find a similar formula for the curvature associated with the metric connection ∇ χ,σ .