2 The bundle of Weyl structures

This section works in the setting of general parabolic geometries. We assume that the reader is familiar with the the basic concepts and only briefly collect what we need about parabolic geometries and Weyl structures. Then we define the bundle of Weyl structures and identify some of the geometric structures that are naturally induced on its total space. We then prove existence of a canonical connection and explain how these structures can be used as an equivalent encoding of the theory of Weyl structures.

2.1 Parabolic geometries

The basic ingredient needed to specify a type of parabolic geometry is a semisimple Lie algebra \(\mathfrak g\) that is endowed with a so-called \(|k|\)-grading. This is a decomposition \[\mathfrak g=\mathfrak g_{-k}\oplus\dots\oplus\mathfrak g_{-1}\oplus\mathfrak g_0\oplus\mathfrak g_1\oplus\dots\oplus\mathfrak g_k\] of \(\mathfrak g\) into a direct sum of linear subspaces such that

• \([\mathfrak g_i,\mathfrak g_j]\subset\mathfrak g_{i+j}\), where we agree that \(\mathfrak g_\ell=\{0\}\) for \(|\ell|>k\).

• No simple ideal of \(\mathfrak g\) is contained in the subalgebra \(\mathfrak g_0\).

• The subalgebra \(\mathfrak p_+=\mathfrak g_1\oplus\dots\oplus\mathfrak g_k\) is generated by \(\mathfrak g_1\).

In particular, this implies that the Lie subalgebra \(\mathfrak g_0\) naturally acts on each of the spaces \(\mathfrak g_i\) via the restriction of the adjoint action. Moreover, \(\mathfrak p:=\mathfrak g_0\oplus\mathfrak p_+\) is a Lie subalgebra of \(\mathfrak g\), which turns out to be a parabolic subalgebra in the sense of representation theory.

Such \(|k|\)-gradings can be easily described in terms of the structure theory of semisimple Lie algebras, see Section 3.2 of [16]. In particular, it turns out that any parabolic subalgebra is obtained in this way and, essentially, the classification of gradings is equivalent to the classification of parabolic subalgebras. Further, the decomposition \(\mathfrak p=\mathfrak g_0\oplus\mathfrak p_+\) is the reductive Levi decomposition, so it is a semi-direct product, \(\mathfrak p_+\) is the nilradical of \(\mathfrak p\), and the subalgebra \(\mathfrak g_0\) is reductive. Of course, also \(\mathfrak g_-:=\mathfrak g_{-k}\oplus\dots\oplus\mathfrak g_{-1}\) is a Lie subalgebra of \(\mathfrak g\), which is nilpotent by the grading property. It turns out that \(\mathfrak g_-\) and \(\mathfrak p_+\) are isomorphic.

Next, one chooses a Lie group \(G\) with Lie algebra \(\mathfrak g\). Then the normalizer of \(\mathfrak p\) in \(G\) has Lie algebra \(\mathfrak p\), and one chooses a closed subgroup \(P\subset G\) lying between this normalizer and its connected component of the identity. The subgroup \(P\) naturally acts on \(\mathfrak g\) and \(\mathfrak p\) via the adjoint action. More generally, one puts \(\mathfrak g^i:=\oplus_{j\geq i}\mathfrak g_j\) to define a filtration of \(\mathfrak g\) by linear subspaces that is invariant under the adjoint action of \(P\). This makes \(\mathfrak g\) into a filtered Lie algebra in the sense that \([\mathfrak g^i,\mathfrak g^j]\subset\mathfrak g^{i+j}\).

Having made these choices, there is the concept of a parabolic geometry of type \((G,P)\) on a manifold \(M\) of dimension \(\dim(G/P)\). This is defined as a Cartan geometry \((p:\mathcal G\to M,\omega)\) of type \((G,P)\), which means that \(p:\mathcal G\to M\) is a principal \(P\)-bundle and that \(\omega\in\Omega^1(\mathcal G,\mathfrak g)\) is a Cartan connection. This in turn means that \(\omega\) is equivariant for the principal right action, so \((r^g)^*\omega=\operatorname{Ad}(g^{-1})\circ\omega\), reproduces the generators of fundamental vector fields, and that \(\omega(u):T_u\mathcal G\to\mathfrak g\) is a linear isomorphism for each \(u\in\mathcal G\). In addition, one requires two conditions on the curvature of \(\omega\), which are called regularity and normality, which we don’t describe in detail.

While Cartan geometries provide a nice uniform description of parabolic geometries, this should be viewed as the result of a theorem rather than a definition. To proceed towards more common descriptions of the geometries, one first observes that the Lie group \(P\) can be decomposed as a semi-direct product. On the one hand, the exponential map restricts to a diffeomorphism from \(\mathfrak p_+\) onto a closed normal subgroup \(P_+\subset P\). On the other hand, one defines a closed subgroup \(G_0\subset P\) as consisting of those elements, whose adjoint action preserves the grading of \(\mathfrak g\) and observes that this has Lie algebra \(\mathfrak g_0\). Then the inclusion of \(G_0\) into \(P\) induces an isomorphism \(G_0\to P/P_+\).

Using this, one can pass from the Cartan geometry \((p:\mathcal G\to M,\omega)\) to an underlying structure by first forming the quotient \(\mathcal G_0:=\mathcal G/P_+\), which is a principal \(G_0\)-bundle. Moreover, for each \(i=-k,\dots,k\), there is a smooth subbundle \(T^i\mathcal G\subset T\mathcal G\) consisting of those tangent vectors that are mapped to \(\mathfrak g^i\subset\mathfrak g\) by \(\omega\). Since \(T^1\mathcal G\) is the vertical bundle of \(\mathcal G\to\mathcal G_0\), these subbundles descend to a filtration \(\{T^i\mathcal G_0:i=-k,\dots,0\}\) of \(T\mathcal G_0\). Moreover, for each \(i<0\), the component of \(\omega\) in \(\mathfrak g_i\) descends define a smooth section of the bundle \(L(T^i\mathcal G_0,\mathfrak g_i)\) of linear maps, so this can be viewed as a partially defined \(\mathfrak g_i\)-valued differential form.

The simplest case here is \(k=1\), for which the geometries in question are often referred to as AHS structures. In this case, one obtains a \(\mathfrak g_{-1}\)-valued one-form \(\theta\) on \(\mathcal G_0\), which is \(G_0\)-equivariant and whose kernel in each point is the vertical subbundle. This means that \((p_0:\mathcal G_0\to M,\theta)\) in this case simply is a classical first order structure corresponding to the adjoint action of \(G_0\) on \(\mathfrak g_{-1}\) (which turns out to be infinitesimally effective). According to a result of Kobayashi and Nagano (see [24]), the resulting class of structures for simple \(\mathfrak g\) is very peculiar, since these are the only irreducible first order structures of finite type, for which the first prolongation is non-trivial. This class contains important examples, like conformal structures, almost quaternionic structures, and almost Grassmannian structures.

For general \(k\), there is an interpretation of \(\mathcal G_0\) and the partially defined forms as a filtered analogue of a first order structure. This involves a filtration of the tangent bundle \(TM\) by smooth subbundles \(T^iM\) for \(i=-k,\dots,-1\) with prescribed (non–)integrability properties together with a reduction of structure group of the associated graded vector bundle to the tangent bundle. This leads to examples like hypersurface-type CR structures, in which the filtration is equivalent to a contact structure, while the reduction of structure group is defined by an almost complex structure on the contact subbundle. Further important example of such structures are path geometries, quaternionic contact structures and various types of generic distributions.

Except for two cases, the Cartan geometry can be uniquely (up to isomorphism) recovered from the underlying structure (see Section 3.1 of [16]), and indeed this defines an equivalence of categories. So in this case, one has two equivalent descriptions of the structure. The two exceptional cases are projective structures and a contact analogue of those. In these cases, the underlying structure contains no information respectively describes only the contact structure, and one in addition has to choose an equivalence class of connections in order to describe the structure. Still, these fit into the general picture with respect to Weyl structures, which we discuss next.

2.2 Weyl structures

These provide the basic tool to explicitly translate between the description of a parabolic geometry as a Cartan geometry and the picture of the underlying structure. So let us suppose that \((p:\mathcal G\to M,\omega)\) is a Cartan geometry of type \((G,P)\) and that \(p_0:\mathcal G_0\to M\) is the underlying structure described in 2.1. The original definition of a Weyl structure used in [15] is as a \(G_0\)-equivariant section \(\sigma\) of the natural projection \(q:\mathcal G\to \mathcal G_0=\mathcal G/P_+\). One shows that such sections always exist globally and by definition, they provide reductions of the principal \(P\)-bundle \(p:\mathcal G\to M\) to the structure group \(G_0\subset P\). As a representation of \(G_0\), the Lie algebra \(\mathfrak g\) splits as \(\mathfrak g_-\oplus\mathfrak g_0\oplus\mathfrak p_+\) (and indeed further according to the \(|k|\)-grading). Thus, the pullback \(\sigma^*\omega\) splits accordingly into a sum of three \(G_0\)-equivariant one-forms with values in \(\mathfrak g_-\), \(\mathfrak g_0\) and \(\mathfrak p_+\), respectively, which then admit nice interpretations in terms of the underlying structure. The \(\mathfrak g_0\)-component defines a principal connection on \(\mathcal G_0\), which induces the Weyl connections on associated bundles. The component in \(\mathfrak p_+\) descends to a one-form on \(M\) with values in the associated graded to the cotangent bundle \(T^*M\), which is the Rho-tensor associated to the Weyl structure. The \(\mathfrak g_-\)-component also descends to \(M\) and provides an isomorphism between the tangent bundle \(TM\) and its associated graded bundle. For the structures we consider in this article, this component coincides with the soldering form that identifies \(\mathcal G_0\) as a reduction of structure group of \(TM\).

As observed in [21], any reduction of \(p:\mathcal G\to M\) to the structure group \(G_0\subset P\) comes from a Weyl structure. This is because the composition of \(q\) with the principal bundle morphism defining such a reduction clearly is an isomorphism of \(G_0\)-principal bundles. Thus one could equivalently define a Weyl structure as such a reduction of structure group and then observe that this defines a \(G_0\)-equivariant section of \(q:\mathcal G\to\mathcal G_0\). It is a classical result that reductions of \(\mathcal G\) to the structure group \(G_0\) can be equivalently described as smooth sections of the associated bundle with fiber \(P/G_0\). This motivates the following definition from [21].

The correspondence between Weyl structures and smooth sections of \(\pi:A\to M\) can be easily made explicit. Given a \(G_0\)-equivariant section \(\sigma:\mathcal G_0\to\mathcal G\) one considers the map sending \(u_0\in \mathcal G_0\) to the class of \((\sigma(u_0),eG_0)\) in \(\mathcal G\times_P(P/G_0)\), where \(e\in P\) is the neutral element. By construction, the resulting smooth map \(\mathcal G_0\to A\) is constant on the fibers of \(p_0:\mathcal G_0\to M\) and thus descends to a smooth map \(s:M\to A\), which is a section of \(\pi\) by construction. Conversely, a section \(s\) of \(\pi\) corresponds to a smooth, \(G_0\)-equivariant map \(f:\mathcal G\to P/G_0\) characterized by the fact that \(s(x)\) is the class of \((u,f(u))\) for each \(u\) in the fiber of \(\mathcal G\) over \(x\). But then \(f^{-1}(eG_0)\) is a smooth submanifold of \(\mathcal G\) on which the projection \(q:\mathcal G\to\mathcal G_0\) restricts to a \(G_0\)-equivariant diffeomorphism. The inverse of this diffeomorphism gives the Weyl structure determined by \(s\).

From the definition, we can verify that the bundle of Weyl structures is similar to an affine bundle. This will also provide the well known affine structure on Weyl structures in our picture. To formulate this, recall first that the parabolic subgroup \(P\subset G\) is a semi-direct product of the subgroup \(G_0\subset P\) and the normal subgroup \(P_+\subset P\). In particular, any element \(g\in P\) can be uniquely written as \(g_0g_1\) with \(g_0\in G_0\) and \(g_1\in P_+\), compare with Theorem 3.1.3 of [16], and of course \(g_0g_1=(g_0g_1g_0^{-1})g_0\) provides the corresponding decomposition in the opposite order.

To connect to the well-known affine structure on the set of Weyl structures, we observe two alternative ways to express things using the exponential map. On the one hand, we have observed above that \(\exp:\mathfrak p_+\to P_+\) is a diffeomorphism. Thus we can write \(h(u)=\exp(\Upsilon(u))\) and equivariancy of \(h\) is equivalent to \(\Upsilon(u\cdot(g_0g_1))=\operatorname{Ad}(g_0)^{-1}(\Upsilon(u))\). On the other hand, in the proof of Theorem 3.1.3 of [16] it shown that also \((Z_1,\dots,Z_k)\mapsto\exp(Z_1)\cdots\exp(Z_k)\) defines a diffeomorphism \(\mathfrak g_1\oplus\dots\oplus \mathfrak g_k\to P_+\). Correspondingly, we can write \(h(u)=\exp(\Upsilon_1(u))\cdots\exp(\Upsilon_k(u))\) where \(\Upsilon_i:\mathcal G\to\mathfrak g_i\) is a smooth map for each \(i=1,\dots,k\). Again, equivariancy of \(h\) translates to \(\Upsilon_i(u\cdot(g_0g_1))=\operatorname{Ad}(g_0)^{-1}(\Upsilon_i(u))\) for each \(i\).

There is also a nice global way to express the affine structure. The filtration of \(TM\) induced by a parabolic geometry dualizes to a filtration of the cotangent bundle \(T^*M\) and we can form the associated graded bundle \(\operatorname{gr}(T^*M)\). The general theory implies that this can be realized as \(\mathcal G\times_{P}\operatorname{gr}(\mathfrak p_+)\cong\mathcal G_0\times_{G_0}\mathfrak p_+\).

2.3 The basic geometric structures on \(A\)

It was shown in [21] that the parabolic geometry \((p:\mathcal G\to M,\omega)\) gives rise to a connection on the tangent bundle \(TA\) of \(A\). The argument used to obtain this connection is rather intricate: There is the opposite parabolic subgroup \(P^{op}\) to \(P\) which corresponds to the Lie subalgebra \(\mathfrak g_-\oplus\mathfrak g_0\subset\mathfrak g\) and one considers the homogeneous space \(G/P^{op}\). Restricting the \(G\)-action to \(P\) and forming the associated bundle \(\mathcal G\times_P(G/P^{op})\) the Cartan connection \(\omega\) induces a natural affine connection on the total space of this bundle. It is then easy to see that \(P\cap P^{op}=G_0\), so acting with \(P\) on \(eP^{op}\) defines a \(P\)-equivariant open embedding \(A\to\mathcal G\times_P(G/P^{op})\), thus providing a connection on \(TA\) as claimed. Our first main result provides a more conceptual description of this connection, which directly implies compatibility with several additional geometric structures on \(A\).

This result also gives us a basic supply of natural vector bundles on \(A\), namely the vector bundles associated to the principal bundle \(\mathcal G\to A\) via representations of \(G_0\). Moreover, the principal connection on that bundle coming from \(\omega_0\) gives rise to an induced linear connection on each of these associated bundles. We will denote all these induced connections by \(D\). Given an associated bundle \(E\to A\), we can view \(D\) as an operator \(D:\Gamma(E)\to\Gamma(T^*A\otimes E)\). Of course the splitting \(TA=L^-\oplus L^+\) from Proposition 2.4 induces an analogous splitting of \(T^*A\), which allows us to split \(D\) into two partial connections \(D=D^-\oplus D^+\). Here \(D^\pm:\Gamma(E)\to\Gamma((L^\pm)^*\otimes E)\). Viewing \(D\) as a covariant derivative, \(D^\pm\) is defined by differentiating only in directions of the corresponding subbundle of \(TA\).

2.4 Relations between natural vector bundles

Recall that the natural vector bundles for the parabolic geometry \((p:\mathcal G\to M,\omega)\) are the associated vector bundles of the form \(\mathcal VM=\mathcal G\times_P\mathbb V\) for representations \(\mathbb V\) of \(P\). Throughout this article, we will only consider the case that the center \(Z(G_0)\) of the subgroup \(G_0\subset P\) acts diagonalizably on \(\mathbb V\). Together with the fact that \(G_0\) is reductive, this implies that \(\mathbb V\) is completely reducible as a representation of \(G_0\). One important subclass of natural bundles is formed by completely reducible bundles that correspond to representations of \(P\) on which the subgroup \(P_+\subset P\) acts trivially, which is equivalent to complete reducibility as a representation of \(P\). On the other hand, there are tractor bundles which correspond to restrictions to \(P\) of representations of \(G\).

Any representation \(\mathbb V\) of \(P\) can naturally be endowed with a \(P\)-invariant filtration of the form \(\mathbb V=\mathbb V^0\supset\mathbb V^1\supset\dots\supset\mathbb V^N\) as follows (see Section 3.2.12 of [16]). The smallest component \(\mathbb V^N\) consists of those elements, on which \(\mathfrak p_+\) acts trivially under the infinitesimal action. The larger components are characterized iteratively by the fact that \(v\in\mathbb V^j\) if and only if it is sent to \(\mathbb V^{j+1}\) by the action of any element of \(\mathfrak p_+\). Then one defines the associated graded representation \(\operatorname{gr}(\mathbb V):=\oplus_{i=0}^N\operatorname{gr}_i(\mathbb V)\) with \(\operatorname{gr}_i(\mathbb V):=(\mathbb V^i/\mathbb V^{i+1})\) and \(\mathbb V^{N+1}=\{0\}\). By construction, this is a completely reducible representation of \(P\).

As an important special case, consider the restriction of the adjoint representation of \(G\) to \(P\). Then it turns out that, up to a shift in degree, the canonical \(P\)-invariant filtration is exactly the filtration \(\{\mathfrak g^i\}\) derived from the \(|k|\)-grading of \(\mathfrak g\) as in 2.1. In particular, this implies that \(\mathfrak g^2=[\mathfrak p_+,\mathfrak p_+]\) and similarly, the higher filtrations components form the lower central series of \(\mathfrak p_+\). Using this it is easy to see that the natural filtration on any representation \(\mathbb V\) of \(P\) has the property that \(\mathfrak g^i\cdot\mathbb V^j\subset\mathbb V^{i+j}\) for all \(i,j\geq 0\) under the infinitesimal representation of \(\mathfrak p=\mathfrak g^0\). This readily implies that there is a natural action of the associated graded \(\operatorname{gr}(\mathfrak p)\) on \(\operatorname{gr}(\mathbb{V} )\), which is compatible with the grading. Since the filtration of \(\mathfrak p\) is induced by the (non-negative part of the) grading on \(\mathfrak g\), we can identify \(\operatorname{gr}(\mathfrak p)\) with \(\mathfrak p\) via the inclusion of \(\mathfrak g_i\) into \(\mathfrak g^i\). Altogether, we get, for each \(i,j\geq 0\), bilinear maps \(\mathfrak g_i\times\operatorname{gr}_j(\mathbb V)\to\operatorname{gr}_{i+j}(\mathbb V)\), which are \(P\)-equivariant (with trivial action of \(P_+\)) by construction.

As a representation of the subgroup \(G_0\subset P\), the associated graded \(\operatorname{gr}(\mathbb V)\) is isomorphic to \(\mathbb V\). Indeed, we have observed above that \(\mathbb V\) is completely reducible as a representation of \(G_0\), so the same holds for each of the subrepresentations \(\mathbb V^j\subset\mathbb V\). In particular, there always is a \(G_0\)-invariant complement \(\mathbb V_j\) to the invariant subspace \(\mathbb V^{j+1}\subset\mathbb V^j\) and we put \(\mathbb V_N=\mathbb V^N\). By construction, we on the one hand get \(\mathbb V\cong\oplus\mathbb V_j\) and on the other hand \(\mathbb V_j\cong\mathbb V^j/\mathbb V^{j+1}\) which implies that claimed statement. Otherwise put, one can interpret the passage from \(\mathbb V\) to \(\operatorname{gr}(\mathbb V)\) as keeping the restriction to \(G_0\) of the \(P\)-action on \(\mathbb V\) and extending this by the trivial action of \(P_+\) to a new action of \(P\).

The construction of the associated graded has a direct counterpart on the level of associated bundles. Putting \(\mathcal VM:=\mathcal G\times_P\mathbb V\to M\), any of the filtration components \(\mathbb V^i\) defines a smooth subbundle \(\mathcal V^iM:=\mathcal G\times_P\mathbb V^i\to M\). Thus \(\mathcal VM\) is filtered by the smooth subbundles \(\mathcal V^iM\) and we can form the associated graded vector bundle \(\operatorname{gr}(\mathcal VM)=\oplus (\mathcal V^iM/\mathcal V^{i+1}M)\). It is easy to see that this can be identified with the associated bundle \(\mathcal G\times_P\operatorname{gr}(\mathbb V)\). However, the fact that \(\mathbb V\) and \(\operatorname{gr}(\mathbb V)\) are isomorphic as representations of \(G_0\) does not have a geometric interpretation without making additional choices. Hence on the level of associated bundles, it is very important to carefully distinguish between a filtered vector bundle and its associated graded.

Any representation \(\mathbb V\) of \(P\) defines a representation of \(G_0\) by restriction. Hence denoting by \(\pi:A\to M\) the bundle of Weyl structures, \(\mathbb V\) also gives rise to a natural vector bundle over \(A\) that we denote by \(\mathcal VA:=\mathcal G\times_{G_0}\mathbb V\to A\). Some information is lost in that process, however, for example \(\mathcal G\times_{G_0}\mathbb V\cong\mathcal G\times_{G_0}\operatorname{gr}(\mathbb V)\) for any representation \(\mathbb V\) of \(P\). Next, sections of \(\mathcal VA\to A\) can be naturally identified with smooth functions \(\mathcal G\to\mathbb V\) that are \(G_0\)-equivariant. Similarly, sections of \(\mathcal VM\to M\) are in bijective correspondence with smooth functions \(\mathcal G\to\mathbb V\), which are \(P\)-equivariant. Thus we see that there is a natural inclusion of \(\Gamma(\mathcal VM\to M)\) as a linear subspace of \(\Gamma(\mathcal VA\to A)\). We will denote this by putting a tilde over the name of a section of \(\mathcal VM\to M\) in order to indicate the corresponding section of \(\mathcal VA\to A\). So both the sections of \(\mathcal VM\to M\) and of its associated graded vector bundle can be interpreted as (different) subspaces of the space of sections of \(\mathcal VA\to A\).

Now we can describe the relations of bundles and sections explicitly.

2.5 The Weyl connections

We next describe the interpretation of Weyl connections in our picture. At the same time, we obtain a nice description of the Rho-corrected derivative associated to a Weyl structure, that was first introduced in [10], see Section 5.1.9 of [16] for a discussion. The Rho-corrected derivative comes from a principal connection on \(\mathcal G\) determined by an equivariant section \(\sigma:\mathcal G_0\to\mathcal G\). One takes the component of \(\omega\) in \(\mathfrak p\) along the image of \(\sigma\) and extends it equivariantly to a principal connection. The name "Rho-corrected derivative" comes from the explicit formula of this derivative in terms of Weyl connection an the Rho-tensor. To obtain our description, we first observe that the pullback operation from part (3) of Theorem 2.5 clearly extends to differential forms with values in a natural vector bundle. Let \(\mathbb V\) be a representation of \(P\) with corresponding natural bundles \(\mathcal VM\to M\) and \(\mathcal VA\to A\). Then one can pull back a \(\mathcal VA\)-valued \(k\) form \(\varphi\) on \(A\) along a section \(s:M\to A\) to a \(\operatorname{gr}(\mathcal VM)\)-valued \(k\)-form \(s^*\varphi\) on \(M\) in an obvious way.

2.6 The universal Rho-tensor

Using the pullback of bundle valued forms, we can also describe the Rho tensor in our picture. Recall that we use the convention of [15] and [16] for Rho tensors in the setting of general parabolic geometries, which differ by sign from the standard conventions for projective and conformal structures.

2.7 Curvature and torsion quantities

The curvature \(K\in\Omega^2(\mathcal G,\mathfrak g)\) of the Cartan connection \(\omega\) is defined by \(K(\xi,\eta)=d\omega(\xi,\eta)+[\omega(\xi),\omega(\eta)]\) for \(\xi,\eta\in\mathfrak X(\mathcal G)\). Since \(K\) is horizontal and \(P\)-equivariant, it can be interpreted as \(\kappa\in\Omega^2(M,\mathcal AM)\), where \(\mathcal AM=\mathcal G\times_P\mathfrak g\) is the adjoint tractor bundle. In the same way, we can interpret it as a two-form on \(A\) with values in the associated bundle \(\mathcal G\times_{G_0}\mathfrak g\). Since \(K\) is horizontal over \(M\), it follows that this two form vanishes upon insertion of one tangent vector from \(L^+\subset TA\). In view of the \(G_0\)-invariant decomposition \(\mathfrak g=\mathfrak g_-\oplus\mathfrak g_0\oplus\mathfrak p_+\) we can decompose that two-form further. To do this, we denote by \(\operatorname{End}_0(TA)\) the associated bundle \(\mathcal G\times_{G_0}\mathfrak g_0\). Via the adjoint action, this can naturally be viewed as a subbundle of \(L(TA,TA)\).

The following result follows directly from the definitions.

These quantities are related to data associated to the Weyl structure determined by \(s\) in Section 5.2.9 of [16] and these results can be easily recovered in the current context.

On the level of \(A\), the best way to interpret the components of the Cartan curvature is via the torsion and curvature of the canonical connection \(D\). This interpretation will also be crucial for the analysis of the intrinsic geometric structure on \(A\) in 3 below. To formulate the result, we need a bit more notation. The Lie bracket is a \(G\)-equivariant, skew symmetric bilinear map \(\mathfrak g\times\mathfrak g\to\mathfrak g\). Now we can restrict this to entries from \(\mathfrak g_-\oplus\mathfrak p_+\) and then decompose the values according to \(\mathfrak g=(\mathfrak g_-\oplus\mathfrak p_+)\oplus\mathfrak g_0\), and the result will still be \(G_0\)-equivariant. The first component induces a two-form on \(A\) with values in \(TA\) which we denote by \(\{\ ,\ \}\). Similarly, the \(\mathfrak g_0\)-component of the bracket defines a two-form \(\{\ ,\ \}_0\) on \(A\) with values in \(\operatorname{End}_0(TA)\). Using this, we formulate

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