Geometric theory of Weyl structures

1 Introduction

Parabolic geometries form a class of geometric structures that look very diverse in their standard description. This class contains important and well-studied examples like conformal and projective structures, non-degenerate CR structures of hypersurface type, path geometries, quaternionic contact structures, and various types of generic distributions. They admit a uniform conceptual description as Cartan geometries of type \((G,P)\) for a semisimple Lie group \(G\) and a parabolic subgroup \(P\subset G\) in the sense of representation theory. Such a geometry on a smooth manifold \(M\) is given by a principal \(P\)-bundle \(p : \mathcal{G} \to M\) together with a Cartan connection \(\omega \in \Omega^1(\mathcal{G},\mathfrak{g})\), which defines an equivariant trivialization of the tangent bundle \(T\mathcal G\). A standard reference for parabolic geometries is [16].

The group \(P\) can be naturally written as a semi-direct product \(G_0\ltimes P_+\) of a reductive subgroup \(G_0\) and a nilpotent normal subgroup \(P_+\). For a Cartan geometry \((p:\mathcal G\to M,\omega)\) the quotient \(\mathcal G_0:=\mathcal G/P_+\to M\) is a principal \(G_0\)-bundle, and some parts of \(\omega\) can be descended to that bundle. In the simplest cases, this defines a usual first order \(G_0\)-structure on \(M\), in more general situations a filtered analog of such a structure. Thus the Cartan geometry can be viewed as an extension of a first order structure. This reflects the fact that morphisms of parabolic geometries are in general not determined locally around a point by their 1-jet in that point, and the Cartan connection captures the necessary higher order information.

To work explicitly with parabolic geometries, one often chooses a more restrictive structure, say a metric in a conformal class, a connection in a projective class or a pseudo-Hermitian structure on a CR manifold, expresses things in terms of this choice and studies the effect of different choices. It turns out that there is a uniform way to do this that can be applied to all parabolic geometries, namely the concept of Weyl structures introduced in [15], see Chapter 5 of [16] for an improved exposition. Choosing a Weyl structure, one in particular obtains a linear connection on any natural vector bundle associated to a parabolic geometry, as well as an identification of higher order geometric objects like tractor bundles with more traditional natural bundles. The set of Weyl structures always forms an affine space modeled on the space on one-forms on the underlying manifold, and there are explicit formulae for how a change of Weyl structure affects the various derived quantities.

The initial motivation for this article were the results in [19] on projective structures. Such a structure on a smooth manifold \(M\) is given by an equivalence class \([\nabla]\) of torsion-free connections on its tangent bundle, where two connections are called equivalent if they have the same geodesics up to parametrization. While these admit an equivalent description as a parabolic geometry, the underlying structure \(\mathcal G_0\to M\) is the full frame bundle of \(M\) and thus contains no information. Hence there is the natural question, whether a projective structure can be encoded into a first order structure on some larger space constructed from \(M\). Indeed, in [19], the authors associate to a projective structure on an \(n\)-dimensional manifold \(M\) a certain rank \(n\) affine bundle \(A \to M\), whose total space can be canonically endowed with a neutral signature metric \(h\), as well as a non-degenerate \(2\)-form \(\Omega\). It turns out that the metric \(h\) is Einstein and \(\Omega\) is closed. Moreover, the pair \((h,\Omega)\) is related by an endomorphism of \(TA\) which squares to the identity map and its eigenbundles \(L^{\pm}\) are Lagrangian with respect to \(\Omega\). Equivalently, we may think of the pair \((h,\Omega)\) as an almost para-Kähler structure or as an almost bi-Lagrangian structure \((\Omega,L^+,L^-)\) on \(A\), see Section 3.1 for the formal definition and more details.

In addition, it is observed that the sections of \(A \to M\) are in bijective correspondence with the connections in the projective class. Consequently, all the submanifold notions of symplectic – and pseudo-Riemannian geometry can be applied to the representative connections of \([\nabla]\). This leads in particular to the notion of a minimal Lagrangian connection [28]. As detailed below, this concept has close relations to the concept of properly convex projective structures. These in turn provide a connection to the study of representation varieties and higher Teichmüller spaces, see [32] for a survey.

In an attempt to generalize these constructions to a larger class of parabolic geometries, we were led to a definition of \(A\to M\) that directly leads to an interpretation as a bundle of Weyl structures. This means that the space of sections of \(A\to M\) can be naturally identified with the space of Weyl structures for the geometry \((p:\mathcal G\to M,\omega)\). At some stage it was brought to our attention that a bundle of Weyl structures had been defined in that way already in the article [21] by M. Herzlich in the setting of general parabolic geometries. In this article, Herzlich gave a rather intricate argument for the existence of a connection on \(TA\) and used this to study canonical curves in parabolic geometries.

The crucial starting point for our results here is that a parabolic geometry \((p:\mathcal G\to M,\omega)\) can also naturally be interpreted as a Cartan geometry on \(A\) with structure group \(G_0\). This immediately implies that for any type of parabolic geometry, there is a canonical linear connection on any natural vector bundle over \(A\) as well as natural almost bi-Lagrangian structure on \(A\) that is compatible with the canonical connection. So in particular, we always obtain a non-degenerate two-form \(\Omega\in\Omega^2(A)\), a neutral signature metric \(h\) on \(TA\) as well as a decomposition \(TA=L^-\oplus L^+\) as a sum of Lagrangian subbundles.

Using the interpretation via Cartan geometries, it turns out that all elements of the theory of Weyl structures admit a natural geometric interpretation in terms of pulling back operations on \(A\) via the sections defined by a Weyl structure. This works for general parabolic geometries as shown in Section 2. In particular, we show that Weyl connections are obtained by pulling back the canonical connection on \(A\), while the Rho tensor (or generalized Schouten tensor) associated to a Weyl connection is given by the pullback of a canonical \(L^+\)-valued one-form on \(A\).

We believe that this interpretation of Weyl structures should be a very useful addition to the tool set available for the study of parabolic geometries. Indeed, working with the canonical geometric structures on \(A\) compares to the standard way of using Weyl structures, like working on a frame bundle compares to working in local frames.

For the second part of the article, we adopt a different point of view. From Section 3 on, we use the relation to Weyl structures as a tool for the study of the intrinsic geometric structure on \(A\) and its relation to non-linear invariant PDE. Our first main result shows that one has to substantially restrict the class of geometries in order to avoid getting into exotic territory. Recall that for a parabolic subgroup \(P\subset G\) the corresponding Lie subalgebra \(\mathfrak p\subset\mathfrak g\) can be realized as the non-negative part in a grading \(\mathfrak g=\oplus_{i=-k}^k\mathfrak g_i\) of \(\mathfrak g\), which is usually called a \(|k|\)-grading. There is a subclass of parabolic geometries that is often referred to as AHS structures, see e.g. [3, 4, 14], which is the case \(k=1\), see Section 2.2 and 14 for more details. This is exactly the case in which the underlying structure \(\mathcal G_0\to M\) is an ordinary first order \(G_0\)-structure. In particular, there is the notion of intrinsic torsion for this underlying structure. Vanishing of the intrinsic torsion is equivalent to the existence of a torsion free connection compatible with the structure and turns out to be equivalent to torsion-freeness of the Cartan geometry \((p:\mathcal G\to M,\omega)\). Using this background, we can formulate the first main result of Section 3, that we prove as Theorem 3.1:

Hence we restrict our considerations to torsion-free AHS structures from this point on. Apart from projective and conformal structures, this contains also Grassmannian structures of type \((2,n)\) and quaternionic structures, for which there are many non-flat examples. For several other AHS structures, torsion-freeness implies local flatness, but the locally flat case is of particular interest for us anyway. Our next main result, which we prove in Theorem 3.4, vastly generalizes [19]:

While one could prove the aforementioned Theorems on a case by case basis by using the techniques from [19], our arguments instead rely on a careful analysis of the properties of the curvature tensor of the induced connection on \(TA\). Following [28], we next initiate the study of Weyl structures via the geometry of submanifolds in \(A\). We call a Weyl structure \(s : M \to A\) of a torsion-free AHS structure Lagrangian if \(s : M \to (A,\Omega)\) is a Lagrangian submanifold. Likewise, \(s\) is called non-degenerate if \(s : M \to (A,h)\) is a non-degenerate submanifold. We show that a Weyl structure is Lagrangian if and only if its Rho tensor is symmetric and that it is non-degenerate if and only if the symmetric part of its Rho tensor is non-degenerate.

In Theorem 3.7 we characterize Lagrangian Weyl structures that lead to totally geodesic submanifolds \(s(M)\subset A\), which provides a connection to Einstein metrics and reductions of projective holonomy. If \(s\) in addition is non-degenerate, then there is a well defined second fundamental form of \(s(M)\) with respect to any linear connection on \(TA\) that is metric for \(h\) and we show that this admits a natural interpretation as a \(\binom12\)-tensor field on \(M\). In our next main result, Theorem 3.8, we give explicit formulae for the second fundamental forms of the canonical connection and the Levi-Civita connection of \(h\). These are universal formulae in terms of the Weyl connection, the Rho-tensor, and its inverse, which are valid for all torsion-free AHS structures. As an application, we are able to characterize non-degenerate Lagrangian Weyl structures that are minimal submanifolds in \((A,h)\) in terms of a universal PDE. Again, this is a vast generalization of [28], where merely the case of projective structures on surfaces was considered.

In Section 4 we connect our results to the study of fully non-linear invariant PDE on AHS structures. A motivating example arises from the work of E. Calabi. In [7], Calabi related complete affine hyperspheres to solutions of a certain Monge-Ampère equation. This Monge-Ampère equation, when interpreted correctly, is an invariant PDE that one can associate to a projective structure and it is closely linked to properly convex projective manifolds, see [26]. In Theorem 4.3, we relate Calabi’s equation to our equation for a minimal Lagrangian Weyl structure and as Corollary 4.4, we obtain:

The convention for the Rho tensor used here is chosen to be consistent with [16]. This convention is natural from a Lie theoretic viewpoint, but differs from the standard definition in projective – and conformal differential geometry by a sign. It should also be noted that a relation between properly convex projective manifolds and minimal Lagrangian submanifolds has been observed previously in [23, 22] (but not in the context of Weyl structures).

The notion of convexity for projective structures is only defined for locally flat structures. The above Corollary thus provides a way to generalize the notion of a properly convex projective structure to a class of projective structures that are possibly curved, namely those arising from a minimal Lagrangian Weyl structure. Going beyond projective geometry, this class of differential geometric structures is well-defined for all torsion-free AHS structures.

We conclude the article by showing that there are analogs of the projective Monge-Ampère equation for other AHS structures, and that these always can be described in terms of the Rho tensor, which provides a relation to submanifold geometry of Weyl structures. These topics will be studied in detail elsewhere.

Acknowledgments

AČ gratefully acknowledges support by the Austrian Science Fund (FWF): P 33559-N. A part of the research for this article was carried out while TM was visiting Forschungsinstitut für Mathematik (FIM) at ETH Zürich. TM thanks FIM for its hospitality and DFG for partial funding through the priority programme SPP 2026 “Geometry at Infinity”. We also thank M. Dunajski, J. Šilhan and V. Žadn`k for helpful discussions and the anonymous referee for interesting comments that helped improving the article significantly.

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 Section 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 Section 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 Section 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

3 The natural almost bi-Lagrangian structure

From here on, we take a different point of view. We study the geometry on the total space of the bundle of Weyl structures associated to a parabolic geometry from an intrinsic point of view, using the relation to parabolic geometries and Weyl structures as technical input. We shall see below that these structures become rather exotic in the case of general gradings, so we will restrict to parabolic geometries associated to \(|1|\)-gradings soon.

3.1 The almost bi-Lagrangian structure and torsion freeness

Consider a parabolic geometry \((p:\mathcal G\to M,\omega)\) of some type \((G,P)\) and let \(\pi:A\to M\) the associated bundle of Weyl structures. As we have noted in Section 2.2, the tangent bundle \(TA\) decomposes as \(L^-\oplus L^+\), where \(L^-=\mathcal G\times_{G_0}\mathfrak g_-\) and \(L^+=\mathcal G\times_{G_0}\mathfrak p_+\). It is also well known that \(\mathfrak g_-\) and \(\mathfrak p_+\) are dual as representations of \(G_0\) via the restriction of the Killing form of \(\mathfrak g\). Thus we obtain a non-degenerate pairing \(B\) mapping \(L^-\times L^+\) to the trivial real line bundle \(M\times\mathbb R\). This pairing can be extended as either a skew symmetric or a symmetric bilinear bundle map on \(TA\), thus defining \(\Omega\in\Omega^2(A)\) and \(h\in\Gamma(S^2T^*A)\). By construction, for each \(y\in A\) both values \(\Omega(y)\) and \(h(y)\) are non-degenerate bilinear forms on \(T_yA\) for which \(L^+_y\) and \(L^-_y\) are isotropic. The resulting structure \((\Omega,L^+,L^-)\) is called an almost bi-Lagrangian structure.

In particular, \(\Omega\in\Omega^2(A)\) is an almost symplectic structure and an obvious first question is when this structure is symplectic, i.e. when \(d\Omega=0\).

3.2 Local frames

From this point on, we restrict the discussion to torsion-free geometries of some type \((G,P)\) that corresponds to a \(|1|\)-grading of \(\mathfrak g\), so that by Theorem 3.1 \(\Omega\) defines a symplectic structure on \(A\). Recall from Section 2.4 that any vector field \(\xi\in\mathfrak X(M)\) determines a section \(\tilde\xi\in\Gamma(L^-)\), and since \(\mathfrak g_-\) is a completely reducible representation in the \(|1|\)-graded case, we get \(D^+\tilde\xi=0\). Similarly, a one-form \(\alpha\in\Omega^1(M)\) defines a section \(\tilde\alpha\in\Gamma(L^+)\) such that \(D^+\tilde\alpha=0\). We further know that \(L^-\cong\pi^*TM\) and \(L^+=\pi^*T^*M\). This implies that starting with local frames for \(TM\) and \(T^*M\) defined on some open set \(U\subset M\), the lifts form local frames for \(L^\pm\) defined on \(\pi^{-1}(U)\), so together, these form a local frame for \(TA\). One may in particular use dual local frames for \(TM\) and \(T^*M\) in which case the resulting local frame for \(TA\) is nicely adapted to the almost bi-Lagrangian structure and thus both to \(\Omega\) and to \(h\). As a preparation for the following computations, we next compute the Lie brackets of such sections.

In particular, we see that, while \(L^+\) always defines an involutive distribution, \(L^-\) is only involutive if the curvature component \(Y\) from Definition 2.9 vanishes identically. From the interpretation via the Cartan curvature, one easily concludes that this is equivalent to local vanishing of the Cartan curvature. Thus our structure is bi-Lagrangian (in the sense that both subbundles \(L^\pm\) are integrable) if and only if the initial parabolic geometry is locally flat.

3.3 The canonical metric

We next study the pseudo-Riemannian metric \(h\) induced on \(A\). By definition, the subbundles \(L^\pm\) are isotropic for \(h\), so this metric always has split signature \((n,n)\), where \(n=\dim(M)\). Our main next aim will be to prove that the metric \(h\) is always Einstein. As a first step in this direction, consider the canonical connection \(D\) and its curvature \(\rho\in\Omega^2(A,\operatorname{End}_0(TA))\) as described in Section 2.7.

Now by construction, the canonical connection \(D\) satisfies \(Dh=0\), so \(D\) is metric for \(h\). This implies that the Levi-Civita connection \(\nabla\) of \(h\) can be computed from \(D\) and its torsion \(\tau\). Indeed, we claim that for \(\xi,\eta,\zeta\in\mathfrak X(A)\), \(h(\nabla_\xi\eta,\zeta)\) is given by \[ h(D_\xi\eta,\zeta)-\tfrac12h(\tau(\xi,\eta),\zeta)+\tfrac12h(\tau(\xi,\zeta),\eta)+ \tfrac12h(\tau(\eta,\zeta),\xi).\](3.2) This evidently defines a linear connection \(\nabla\) on \(TA\). Moreover, the last three terms in (3.2) are visibly skew symmetric in \(\eta\) and \(\zeta\), whence the fact that \(D\) is metric with respect to \(h\) implies that \(\nabla\) is metric with respect to \(h\), too. On the other hand, since the last two terms in (3.2) are symmetric in \(\xi\) and \(\eta\), and \(\tau\) is the torsion of \(D\), one immediately verifies that \(\nabla\) is torsion-free. Let us write \(C\in\Gamma(\otimes^2T^*A\otimes TA)\) for the contorsion tensor between \(\nabla\) and \(D\), so \(C(\xi,\eta)=\nabla_\xi\eta-D_\xi\eta\) and the last three terms in (3.2) explicitly express \(h(C(\xi,\eta),\zeta)\). Using this, we prove the following result

3.4 Geometry of Weyl structures

Viewed as a section of \(\pi:A\to M\), any Weyl structure defines an embedding of \(M\) into \(A\), and we can now study this embedding via submanifold geometry related to the almost bi-Lagrangian structure. In particular, we can pull back the two-form \(\Omega\) and the pseudo-Riemannian metric to \(M\) along \(s\), and this naturally leads to the following definitions.

These properties can easily be characterized in terms of the Rho tensor.

To start the geometric study of Lagrangian Weyl structures, we can characterize when \(s\) has the property that the submanifold \(s(M)\subset A\) is totally geodesic.

The result of Theorem 3.7 is particularly interesting if \(s\) is non-degenerate. By Proposition 3.6 this implies that \(\mbox{\textsf{P}}^s\) defines a pseudo-Riemannian metric on \(M\) and since \(\nabla^s\) is torsion-free, \(\nabla^s\mbox{\textsf{P}}^s=0\) implies that \(\nabla^s\) is the Levi-Civita connection of \(\mbox{\textsf{P}}^s\). On the other hand, \(\mbox{\textsf{P}}^s\) is always related to the Ricci-type contraction of the curvature of \(\nabla^s\), see Section 4.1.1 of [16]. In particular, for projective structures, symmetry of \(\mbox{\textsf{P}}^s\) implies that it is a non-zero multiple of the Ricci curvature of \(\nabla^s\), see [2], so in this case \(\mbox{\textsf{P}}^s\) defines an Einstein metric on \(M\). The condition that a projective structure contains the Levi-Civita connection of an Einstein metric can be expressed as a reduction of projective holonomy, see [12] and [11].

For a non-degenerate Lagrangian Weyl structure \(s\), there is a well defined second fundamental form of \(s(M)\) with respect to any linear connection on \(TA\) which is metric for \(h\). Extending the result of Theorem 3.7 in this case, we can next explicitly compute the second fundamental forms for \(D\) and for the Levi-Civita connection. To formulate the result, we use abstract index notation as in the proof of Theorem 3.7.

Fix the section \(s:M\to A\) corresponding to a non-degenerate, Lagrangian Weyl structure. By non-degeneracy, the Rho tensor \(\mbox{\textsf{P}}_{ij}\) of \(s\) admits an inverse \(\mbox{\textsf{P}}^{ij}\in\Gamma(S^2TM)\) which is characterized by \(\mbox{\textsf{P}}^{ij}\mbox{\textsf{P}}_{jk}=\delta^i_k\). In the proof of Theorem 3.7, we have seen that \(T_{s(x)}s(M)\) is spanned by all elements of the form \((\eta^i,\mbox{\textsf{P}}_{ka}\eta^a)\) with \(\eta^i\in T_xM\). The definition of \(h\) readily implies that the normal space \(T^\perp_{s(x)}s(M)\) consists of all pairs of the form \((\eta^i,-\mbox{\textsf{P}}_{jk}\eta^k)\), so we can identify both the tangent and the normal space in \(s(x)\) with \(T_xM\) via projection to the first component. Correspondingly, the second fundamental form of \(s(M)\) (with respect to any connection on \(TA\) which is metric for \(h\)) can be viewed as a \(\binom12\)-tensor field on \(M\). We denote the second fundamental form of \(s : M \to (A,h)\) with respect to \(D\) by \(\mathrm{II}^s_{D}\) and with respect to the Levi-Civita connection of \(h\) by \(\mathrm{II}^s_h\).

4 Relations to non-linear invariant PDE

We conclude this article by discussing a relation between the geometry on the bundle \(A\) of Weyl structures and non-linear invariant PDE associated to AHS structures. We will mainly consider the prototypical example of a projectively invariant PDE of Monge-Ampère type. We briefly discuss analogs of this and other invariant non-linear PDE for specific types of AHS structures, but this will be taken up in detail elsewhere.

4.1 A tractorial description of \(A\)

We start by deriving an alternative description of the bundle \(A\to M\) of Weyl structures associated to an AHS structure \((p:\mathcal G\to M,\omega)\) based on tractor bundles. As mentioned in Section 2.4, these are bundles associated to representations of \(P\) that are restrictions of representations of \(G\). An important feature of these bundles is that they inherit canonical linear connections from the Cartan connection \(\omega\). Together with some algebraic ingredients, these form the basis for the machinery of BGG sequences that was developed in [17] and [9], which will provide input to some of the further developments. We have also met the canonical invariant filtration on representations of \(P\) in Section 2.4 and the corresponding filtration of associated bundles by smooth subbundles. For representations of \(G\), these admit a simpler description, that we derive first. In most of the examples we need below, this description is rather obvious, so readers not interested in representation theory aspects can safely skip the proof of this result.

Recall that for any \(|k|\)-grading on \(\mathfrak g\), there is a unique grading element \(E\), such that for \(i=-k,\dots,k\) the subspace \(\mathfrak g_i\) is the eigenspace with eigenvalue \(i\) for the adjoint action of \(E\). In particular, \(E\) has to lie in the center of the subalgebra \(\mathfrak g_0\). In the case of a \(|1|\)-grading, this center has dimension \(1\) and thus is spanned by \(E\).

Using this, we can now prove an alternative description of the bundle of Weyl structures that, as we shall see in the examples below, generalizes the construction of [19].

4.2 The projective Monge-Ampère equation

This is the prototypical example of the non-linear PDE that we want to study. In the setting of projective geometry, we have met the density bundles \(\mathcal E(k)\) for \(k>0\) in 28. We define \(\mathcal E(-k)\) to be the line bundle dual to \(\mathcal E(k)\) and use the convention that adding “\((k)\)” to the name of a bundle indicates a tensor product with \(\mathcal E(k)\) for \(k\in\mathbb Z\). The first step towards the construction of the projective Monge-Ampère equation is that there is a projectively invariant, linear, second order differential operator \(H:\Gamma(\mathcal E(1))\to \Gamma(S^2T^*M(1))\) called the projective Hessian. Indeed, this is the first operator in the BGG sequence determined by the standard cotractor bundle, see [13].

Now for a section \(\sigma\in\Gamma(\mathcal E(1))\), \(H(\sigma)\) defines a symmetric bilinear form on each tangent space of \(M\), and such a form has a well defined determinant. In projective geometry, this determinant admits an interpretation as a density as follows. In the setting of part (1) of 28, the top exterior power \(\Lambda^{n+1}\mathbb R^{(n+1)*}\) is a trivial representation, which implies that the bundle \(\Lambda^{n+1}\mathcal T^*M\) is canonically trivial. Identifying \(\mathcal T^*M\) with \(J^1\mathcal E(1)\), the jet exact sequence \(0\to T^*M(1)\to J^1\mathcal E(1)\to\mathcal E(1)\to 0\) implies that \(\Lambda^{n+1}\mathcal T^*M\cong\Lambda^nT^*M(n+1)\), so \(\Lambda^nT^*M\cong\mathcal E(-n-1)\). This isomorphism can be encoded as a tautological section of \(\Lambda^nTM(-n-1)\). To form the determinant of \(H(\sigma)\), one now takes the tensor product of two copies of this canonical section and of \(n\) copies of \(H(\sigma)\) and forms the unique (potentially) non-trivial complete contraction of the result (so the two indices of each copy of \(H(\sigma)\) have to be contracted into different copies of the tautological form). This shows that \(\det(H(\sigma))\) can be naturally interpreted as a section of \(\mathcal E(-n-2)\).

Assuming that \(\sigma\in\Gamma(\mathcal E(1))\) is nowhere vanishing, we can form \(\sigma^k\in\Gamma(\mathcal E(k))\) for any \(k\in\mathbb Z\), and hence \[ \det(H(\sigma))=\pm\sigma^{-n-2}\](4.1) is a projectively invariant, fully non-linear PDE on nowhere vanishing sections of \(\mathcal E(1)\). Observe that multiplying \(\sigma\) by a constant, the two sides of the equation scale by different powers of the constant, so allowing a constant factor instead of just a sign in the right hand side of the equation would only be a trivial modification.

4.3 Interpretation in terms of Weyl structures

Let us first observe that a nowhere vanishing section \(\sigma\in\mathcal E(1)\) uniquely determines a Weyl structure. In the language of Theorem 4.2 this can be either described as the structure corresponding to the flat connection on \(\mathcal E(1)\) determined by \(\sigma\) or as the one corresponding to the line in \(\mathcal T^*M\) spanned by \(S(\sigma)\), where \(S\) denotes the BGG splitting operator. From either interpretation it is clear that this Weyl structure remains unchanged if \(\sigma\) is multiplied by a non-zero constant. Alternatively, one can easily verify that any projective class on \(M\) contains a unique connection such that \(\sigma\) is parallel for the induced connection on \(\mathcal E(1)\).

To deal with non-flat cases in the following theorem, we use a concept of mean curvature tailored to the case of connections compatible with an almost bi-Lagrangian structure that was introduced in [18]. That article uses the terminology of (almost) para-Kähler structures, which is slightly different from ours, but it is easy to translate between the two.

A deep relation between solutions of the projective Monge-Ampère equation and properly convex projective structures was established in the works [25] by Labourie and [26] by Loftin. Recall that a projective manifold \((M,[\nabla])\) is called properly convex if it arises as a quotient of a properly convex open set \(\tilde{M}\subset \mathbb{RP}^n\) by a group \(\Gamma\) of projective transformations which acts discretely and properly discontinuously. The projective line segments contained in \(\tilde{M}\) project to \(M\) to become the geodesics of \([\nabla]\). Therefore, locally, the geodesics of a properly convex projective structure \([\nabla]\) can be mapped diffeomorphically to segments of straight lines, that is, \([\nabla]\) is locally flat. Combining the work of Labourie and Loftin with Theorem 4.3, we obtain:

4.4 Invariant non-linear PDE for other AHS structures

We conclude this article with some remarks on analogs of the projective Monge-Ampère equation for other AHS structures. The first observation is that a small representation theoretic condition is sufficient to obtain an analog of the projectively invariant Hessian, which again is closely related to the Rho tensor.

To formulate this, we need a bit of background. Suppose that \((G,P)\) corresponds to a \(|1|\)-grading of \(G\) and let \(G_0\subset P\) be the subgroup determined by the grading. Then this naturally acts on each \(\mathfrak g_i\), and there is an induced representation on \(S^2\mathfrak g_1\). We can decompose this representation into irreducibles and there is a unique component whose highest weight is twice the highest weight of \(\mathfrak g_1\). This is called the Cartan square of \(\mathfrak g_1\) and denoted by \(\circledcirc^2\mathfrak g_1\). It comes with a canonical \(G_0\)-equivariant projection \(\pi:\otimes^2\mathfrak g_1\to\circledcirc^2\mathfrak g_1\). For any parabolic geometry of type \((G,P)\), this induces a natural subbundle \(\circledcirc^2T^*M\subset S^2T^*M\) and a natural bundle map \(\pi:\otimes^2T^*M\to\circledcirc^2T^*M\).

Observe that 28 provides representations \(\mathbb V\) that satisfy the assumptions of the proposition for conformal and for almost Grassmannian structures. Hence for these two geometries an invariant Hessian is available. It is worth mentioning that, for conformal structures, \(\pi(\mbox{\textsf{P}}^\sigma)\) is the trace-free part of \(\mbox{\textsf{P}}^\sigma\).

It is also a general fact that the top-exterior power of \(T^*M\) is isomorphic to a positive, integral power of the dual \(\mathcal E^*M\) of \(\mathcal EM\): By definition, the grading element \(E\) acts by multiplication by \(\dim(\mathfrak g_1)\) on the top exterior power of \(\mathfrak g_1\), which represents the top exterior power of \(T^*M\). On the other hand, the construction implies that a generator of \(\mathbb V_0\subset\mathbb V\) will be a lowest weight vector of the complexification of \(\mathbb V\), so \(E\) acts by a negative number on this. The fact that we deal with a fundamental representation implies that \(\dim(\mathfrak g_{-1})\) is an integral multiple of that number. As in the projective case, this can be phrased as the existence of a tautological section, which can then be used together with copies of \(H(\sigma)\) to obtain a section of a line bundle, which can be trivial, a tensor power of \(\mathcal E\) or a tensor power of \(\mathcal E^*\). In any case, a nowhere vanishing section of \(\mathcal E\) determines a canonical section of that bundle (which is the constant \(1\) in the trivial case), so there is an invariant version of the Monge-Ampère equation. In view of part (2) of Proposition 4.5, for these equations there is always an analog of part (1) of Theorem 4.3.

For some of the structures, there are additional natural sections that can be used together with powers of \(H(\sigma)\) to construct other non-linear invariant operators, for example, the conformal metric for conformal structures and partial (density valued) volume forms for Grassmannian structures. Again part (2) of Proposition 4.5 shows that all these equations can be phrased as equations on \(\mbox{\textsf{P}}^\sigma\), so there should be a relation to submanifold geometry of Weyl structures in the style of part (2) of Theorem 4.3. All this will be taken up in detail elsewhere.

References