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 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 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 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 2.2 and Remark 3.2 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 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 3.5, 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 3.12 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 3.13, 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 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 4.4, we relate Calabi’s equation to our equation for a minimal Lagrangian Weyl structure and as Corollary 4.6, 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. Žadnı̀k for helpful discussions and the anonymous referee for interesting comments that helped improving the article significantly.

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