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 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 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 \(E\).
Consider a Lie group \(G\) with simple Lie algebra \(\mathfrak g\) that is endowed with a \(|1|\)-grading with grading element \(E\), and let \(G_0\subset P\subset G\) be subgroups associated to this grading. Let \(\mathbb V\) be a representation of \(G\) which is irreducible as a representation of \(\mathfrak g\). Then there is a \(G_0\)-invariant decomposition \(\mathbb V=\mathbb V_0\oplus\dots\oplus\mathbb V_N\) such that
Each \(\mathbb V_j\) is an eigenspace of \(E\).
For \(i\in\{-1,0,1\}\) and each \(j\), we have \(\mathfrak g_i\cdot\mathbb V_j\subset\mathbb V_{i+j}\).
For each \(j>0\), restriction of the representation defines a surjection \(\mathfrak g_1\otimes\mathbb V_{j-1}\to\mathbb V_j\).
The canonical \(P\)-invariant filtration on \(\mathbb V\) is given by \(\mathbb V^j=\oplus_{\ell\geq j}\mathbb V_\ell\).
Proof. Note first that the \(|1|\)-grading of \(\mathfrak g\) induces a \(|1|\)-grading on the complexification \(\mathfrak g^{\mathbb C}\) of \(\mathfrak g\), which has the same grading element \(E\) as \(\mathfrak g\). It is also well known that there is a Cartan subalgebra of \(\mathfrak g^{\mathbb C}\) that contains \(E\). Complexifying \(\mathbb V\) and \(\mathfrak g\) if necessary and then passing back to the \(E\)-invariant subspace \(\mathbb V\), we conclude that \(E\) acts diagonalizably on \(\mathbb V\). Denoting the \(\lambda\)-eigenspace for \(E\) in \(\mathbb V\) by \(\mathbb V_\lambda\), it follows readily that \(\mathfrak g_i\cdot\mathbb V_\lambda\subset\mathbb V_{\lambda+i}\) for \(i\in\{-1,0,1\}\). Now take an eigenvalue \(\lambda_0\) with minimal real part, let \(N\) be the smallest positive integer such that \(\lambda_0+N+1\) is not an eigenvalue of \(E\) and put \(\mathbb V_j:=\mathbb V_{\lambda_0+j}\) for \(j=0,\dots,N\). Then, by construction, \(\mathfrak g_{-1}\) acts trivially on \(\mathbb V_0\), \(\mathfrak g_1\) acts trivially on \(\mathbb V_N\), and each \(\mathbb V_j\) is \(\mathfrak g_0\)-invariant. This shows that \(\mathbb V_0\oplus\dots\oplus\mathbb V_N\) is \(\mathfrak g\)-invariant and hence has to coincide with \(\mathbb V\) by irreducibility.
By definition, the adjoint action of each element \(g_0\in G_0\) preserves the grading of \(\mathfrak g\), which easily implies that \(\operatorname{Ad}(g_0)(E)\) acts on \(\mathfrak g_i\) by multiplication by \(i\) for \(i\in\{-1,0,1\}\). This means that \(\operatorname{Ad}(g_0)(E)-E\) lies in the center of \(\mathfrak g\), so \(\operatorname{Ad}(g_0)(E)=E\). But then for \(v\in\mathbb V\), we can compute \(E\cdot g_0\cdot v\) as \(\operatorname{Ad}(g_0)(E)\cdot g_0\cdot v=g_0\cdot E\cdot v\). This shows that each \(\mathbb V_j\) is \(G_0\)-invariant and it only remains to prove the last two claimed properties of the decomposition.
We put \(\tilde{\mathbb V}_0:=\mathbb V_0\) and for \(j>0\), we inductively define \(\tilde{\mathbb V}_j\) as the image of the map \(\mathfrak g_1\otimes\tilde{\mathbb V}_{j-1}\to\mathbb V_j\). Then, by construction, each \(\tilde{\mathbb V}_j\) is a \(\mathfrak g_0\)-invariant subspace of \(\mathbb V_j\), so \(\tilde{\mathbb V}:=\oplus_{j=0}^N\tilde{\mathbb V}_j\subset\mathbb V\) is invariant under the actions of \(\mathfrak g_0\) and \(\mathfrak g_1\). But for \(X\in\mathfrak g_{-1}\) and \(Z\in\mathfrak g_1\), we have \([X,Z]\in\mathfrak g_0\) and for \(v\in\mathbb V\) we get \[X\cdot Z\cdot v=Z\cdot X\cdot v+[Z,X]\cdot v.\] This inductively shows that \(\tilde{\mathbb V}\) is invariant under the action of \(\mathfrak g_{-1}\). Thus it is \(\mathfrak g\)-invariant and hence has to coincide with \(\mathbb V\) by irreducibility, so it remains to verify the claimed description of the canonical \(P\)-invariant filtration.
To do this, we first claim that an element \(v\in\mathbb V\) such that \(Z\cdot v=0\) for all \(Z\in\mathfrak g_1\) has to be contained in \(\mathbb V_N\). It suffices to prove this for the complexification, so we may assume that both \(\mathfrak g\) and \(\mathbb V\) are complex and so there is a highest weight vector \(v_0\in\mathbb V\) which is unique up to scale by irreducibility. It is then well known that \(\mathbb V\) is spanned by vectors obtained from \(v_0\) by the iterated action of elements in negative root spaces of \(\mathfrak g\). Since on such elements \(E\) has non-positive eigenvalues, we conclude that \(v_0\in\mathbb V_N\). Now assume that for some \(j<N\), the space \(W:=\{v\in\mathbb V_j:\mathfrak g_1\cdot v=\{0\}\}\) is non-trivial. Then, by construction, this is a \(\mathfrak g_0\)-invariant subspace of \(\mathbb V_j\) on which the center of \(\mathfrak g_0\) acts by a scalar, so it must contain a vector that is annihilated by all elements in positive root spaces of \(\mathfrak g_0\). But since any positive root space of \(\mathfrak g\) either is a positive root space of \(\mathfrak g_0\) or is contained in \(\mathfrak g_1\), this has to be a highest weight vector for \(\mathfrak g\), which contradicts uniqueness of \(v_0\) up to scale.
Having proved the claim, we can first interpret it as showing that \(\mathbb V^N=\mathbb V_N\). From this the description of the \(P\)-invariant filtration follows by backwards induction: Suppose that that we have shown that \(\mathbb V^{N-j}=\mathbb V_{N-j}\oplus\dots\oplus\mathbb V_N\) and let \(w\in\mathbb V\) be such that for all \(Z\in\mathfrak g_1\), we have \(Z\cdot w\in\mathbb V^{N-j}\). Decomposing \(w=w_0+\dots+w_N\), we conclude that for all \(i<N-j-1\) we must have \(Z\cdot w_i=0\) and hence \(w\in\mathbb V_{N-j-1}\oplus\dots\oplus\mathbb V_N\). Together with the obvious fact that \(\oplus_{\ell\geq N-j-1}\mathbb V_\ell\subset\mathbb V^{N-j-1}\), this implies the description of the \(P\)-invariant filtration.
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].
Suppose that \((G,P)\) corresponds to a \(|1|\)-grading of the Lie algebra \(\mathfrak g\) of \(G\). Let \(\mathbb V\) be a representation of \(G\), which is non-trivial and irreducible as a representation of \(\mathfrak g\), with natural \(P\)-invariant filtration \(\{\mathbb V^j:j=0,\dots,N\}\) such that \(\mathbb V/\mathbb V^1\) has real dimension \(1\). For a parabolic geometry \((p:\mathcal G\to M,\omega)\) of type \((G,P)\) let \(\mathcal VM\) be the tractor bundle induced by \(\mathbb V\), \(\mathcal V^jM\) the subbundle corresponding to \(\mathbb V^j\), and define \(\mathcal EM\) to be the real line bundle \(\mathcal VM/\mathcal V^1M\).
Then the bundle \(A\to M\) of Weyl structures can be naturally identified with the open subbundle in the projectivization \(\mathcal P(\mathcal VM/\mathcal V^2M)\) formed by all lines that are transversal to the subbundle \(\mathcal V^1M/\mathcal V^2M\) of hyperplanes. This in turn leads to an identification of \(A\to M\) with the bundle of all linear connections on the line bundle \(\mathcal EM\to M\).
Proof. By assumption \(\mathbb V^1\subset\mathbb V\) is a \(P\)-invariant hyperplane, so this descends to a \(P\)-invariant hyperplane \(\mathbb V^1/\mathbb V^2\) in \(\mathbb V/\mathbb V^2\). Passing to the projectivization, the complement of this hyperplane is a \(P\)-invariant open subset \(U\subset\mathcal P(\mathbb V/\mathbb V^2)\) and hence defines a natural open subbundle in the associated bundle \(\mathcal P(\mathcal VM/\mathcal V^2M)\).
Now take the decomposition \(\mathbb V=\oplus_{j=0}^N\mathbb V_j\) from Lemma 4.1. Then \(\mathbb V_0\) is a line in \(\mathbb V\) transversal to \(\mathbb V^1\) and hence defines a point \(\ell_0\in U\). We claim that the \(P\)-orbit of \(\ell_0\) is all of \(U\), while its stabilizer subgroup in \(P\) coincides with \(G_0\). This shows that \(U\cong P/G_0\) and thus implies the first claimed description of \(A\to M\). As observed in 2.4, an element \(g\in P\) can be written uniquely as \(\exp(Z)g_0\) for \(g_0\in G_0\) and \(Z\in\mathfrak g_1\). We know that \(\mathbb V_0\) is \(G_0\)-invariant, so for \(w\in\ell_0\) we get \(g_0\cdot w=aw\) for some nonzero element \(a\in\mathbb R\). On the other hand, \(\exp(Z)\cdot w=w+Z\cdot w+\tfrac12Z\cdot Z\cdot w+\dots\), and all but the first two summands lie in \(\mathbb V^2\). This shows that the action of \(\exp(Z)g_0\) sends \(\ell_0\) to the line in \(\mathbb V/\mathbb V^2\) spanned by \(w+Z\cdot w+\mathbb V^2\). But from Lemma 4.1, we know that the action defines a surjection \(\mathfrak g_1\otimes\mathbb V_0\to\mathbb V_1\), which shows that \(P\cdot\ell_0=U\). On the other hand, it is well known that \(\mathfrak g_1\) is an irreducible representation of \(\mathfrak g_0\). Thus also \(\mathfrak g_1\otimes\mathbb V_0\) is irreducible, so \(Z\cdot w=0\) if and only \(Z=0\), which shows that the stabilizer of \(\ell_0\) in \(P\) coincides with \(G_0\).
For the second description, we need some input from the machinery of BGG sequences. There is a natural invariant differential operator \(S:\Gamma(\mathcal EM)\to\Gamma(\mathcal VM)\) which splits the tensorial map \(\Gamma(\mathcal VM)\to\Gamma(\mathcal EM)\) induced by the quotient projection \(\mathcal VM\to\mathcal EM\). It turns out that the operator \(\Gamma(\mathcal EM)\to\Gamma(\mathcal VM/\mathcal V^{i+1}M)\) induced by \(S\) has order \(i\), so there is an induced vector bundle map from the jet prolongation \(J^i\mathcal EM\) to \(\mathcal VM/\mathcal V^{i+1}M\). As proved in [5], the representation \(\mathbb V\) determines an integer \(i_0\) such that this is an isomorphism for all \(i\leq i_0\). For a non-trivial representation \(i_0\geq 1\), so we conclude that \(J^1\mathcal EM\cong \mathcal VM/\mathcal V^2M\). Since \(S\) splits the tensorial projection, we see that, in a point \(x\in M\), the hyperplane \(\mathcal V^1_xM/\mathcal V^2_xM\) corresponds to the jets of sections vanishing in \(x\). Thus lines in \(\mathcal VM/\mathcal V^2M\) that are transversal to \(\mathcal V^1M/\mathcal V^2M\) exactly correspond to lines in \(J^1\mathcal EM\) that are transversal to the kernel of the natural projection to \(\mathcal EM\). Choosing such a line is equivalent to choosing a splitting of this projection and thus of the jet exact sequence for \(J^1\mathcal EM\). It is well known that the choice of such a splitting is equivalent to the choice of a linear connection on \(\mathcal EM\).
(1) Oriented projective structures For \(n\geq 2\), put \(G:=SL(n+1,\mathbb R)\) and let \(P\) be the stabilizer of the ray in \(\mathbb R^{n+1}\) spanned by the first element \(e_0\) in the standard basis. Taking the complementary hyperplane spanned by the remaining basis vectors, one obtains a \(|1|\)-grading on the Lie algebra \(\mathfrak g\) of \(G\) by decomposing into blocks of sizes \(1\) and \(n\) as in \(\left(\begin{smallmatrix}\mathfrak g_0 & \mathfrak g_1\\ \mathfrak g_{-1} & \mathfrak g_0 \end{smallmatrix}\right)\). The subgroup \(G_0\subset P\) is then easily seen to consist of all block diagonal matrices in \(P\).
Now we define \(\mathbb V:=\mathbb R^{(n+1)*}\), the dual of the standard representation of \(G\). In terms of the dual of the standard basis, this decomposes as the sum of \(\mathbb V_0:=\mathbb R\cdot e_0^*\) and \(\mathbb V_1\) spanned by the remaining basis vectors. All properties claimed in Lemma 4.1 are obviously satisfied in this case. The tractor bundle \(\mathcal VM\) corresponding to \(\mathbb V\) is usually called the (standard) cotractor bundle \(\mathcal T^*M\) and the line bundle \(\mathcal EM\) is the bundle \(\mathcal E(1)\) of projective \(1\)-densities. Since \(\mathbb V^2=\{0\}\) in this case, Theorem 4.2 realizes \(A\) as an open subbundle in \(\mathcal P(\mathcal T^*M)\), and this is exactly the construction from [19]. In fact, it is well known that \(\mathcal T^*M\cong J^1\mathcal E(1)\) in this case, this is even used as a definition in [2].
More generally, for \(k\geq 2\), we can take \(\mathbb V\) to be the symmetric power \(S^k\mathbb R^{(n+1)*}\). This visibly decomposes as \(\mathbb V_0\oplus\dots\oplus\mathbb V_k\), where \(\mathbb V_i\) is spanned by the symmetric products of \((e_0^*)^{k-i}\) with \(i\) other basis elements. This corresponds to the tractor bundle \(S^k\mathcal T^*M\), while the quotient \(\mathbb V/\mathbb V^1\) induces the \(k\)th power of \(\mathcal E(1)\), which is usually denoted by \(\mathcal E(k)\) and called the bundle of projective \(k\)-densities. Again, all properties claimed in Lemma 4.1 are obviously satisfied.
(2) Conformal structures For \(p+q=n\geq 3\), we put \(G:=SO(p+1,q+1)\) and we take a basis \(e_0,\dots,e_{n+1}\) for the standard representation \(\mathbb R^{n+2}\) of \(G\) such that the non-trivial inner products are \(\langle e_0,e_{n+1}\rangle=1\), and \(\langle e_i,e_i\rangle=1\) for \(1\leq i\leq p\) and \(\langle e_i,e_i\rangle=-1\) for \(p+1\leq i\leq n\). Splitting matrices into blocks of sizes \(1\), \(n\), and \(1\) defines a \(|1|\)-grading of \(\mathfrak g\) according to \(\left(\begin{smallmatrix} \mathfrak g_0 & \mathfrak g_1 & 0\\ \mathfrak g_{-1} & \mathfrak g_0 & * \\ 0 & * & * \end{smallmatrix}\right)\), where entries marked by \(*\) are determined by other entries of the matrix. Again \(G_0\) turns out to consist of block diagonal matrices and is isomorphic to the conformal group \(CO(p+1,q+1)\) via the adjoint action on \(\mathfrak g_{-1}\). In view of Remark 3.2 we conclude that a parabolic geometry of type \((G,P)\) on a manifold \(M\) is equivalent to a conformal structure.
The standard representation \(\mathbb V\) of \(G\) now decomposes as \(\mathbb V=\mathbb V_0\oplus\mathbb V_1\oplus\mathbb V_2\), with the subspaces spanned by \(e_0\), \(\{e_1,\dots,e_n\}\), and \(e_{n+1}\), respectively. This induces the standard tractor bundle \(\mathcal TM\) on conformal manifolds, and the bundle induced by \(\mathbb V/\mathbb V^1\) is the bundle \(\mathcal E[1]\) of conformal \(1\)-densities. Hence Theorem 4.2 in this case realizes \(A\) as an open subbundle of the projectivization of the quotient of \(\mathcal TM\) by its smallest filtration component. Again, it is well known that this quotient is isomorphic to \(J^1\mathcal E[1]\).
Alternatively, for \(k\geq 2\), we can take \(\mathbb V\) to be the trace-free part \(S^k_0\mathbb R^{n+2}\) in the symmetric power of the standard representation. This leads to the bundle \(S^k_0\mathcal TM\) and the line bundle \(\mathcal E[k]\) of conformal densities of weight \(k\). Here the decomposition from Lemma 4.1 becomes a bit more complicated, since the individual pieces \(\mathbb V_j\) are not irreducible representations of \(G_0\) in general. Still the properties claimed in Lemma 4.1 are obvious via the construction from the decomposition of the standard representation.
(3) Almost Grassmannian structures Here we choose integers \(2\leq p\leq q\), put \(n=p+q\), take \(G:=SL(n,\mathbb R)\) and \(P\subset G\) the stabilizer of the subspace spanned by the first \(p\) vectors of the standard basis of the standard representation \(\mathbb R^n\) of \(G\). Fixing the complementary subspace spanned by the remaining \(q\) vectors in that basis, one obtains a decomposition of the Lie algebra \(\mathfrak g\) of \(G\) into blocks of sizes \(p\) and \(q\), which defines a \(|1|\)-grading as in the projective case. Then \(G_0\) again turns out to consist of block diagonal matrices and hence is isomorphic to \(S(GL(p,\mathbb R)\times GL(q,\mathbb R))\), while \(\mathfrak g_{-1}\) can be identified with the space of \(q\times p\)-matrices endowed with the action of \(G_0\) defined by matrix multiplication from both sides.
Hence the corresponding geometries exist in dimension \(pq\) and they are essentially given by an identification of the tangent bundle with a tensor product of two auxiliary bundles of rank \(p\) and \(q\), respectively, see Section 4.1.3 of [16]. There it is also shown that for these types of structures the Cartan curvature has two fundamental components, but their nature depends on \(p\) and \(q\). For \(p=q=2\), such a structure is equivalent to a split-signature conformal structure, so we will not discuss this case here. If \(p=2\) and \(q>2\), then one of these quantities is the intrinsic torsion of the structure, but the second is a curvature, so this is a case in which there are non-flat, torsion-free examples. For \(p>3\), the intrinsic torsion splits into two components, and torsion-freeness of a geometry implies local flatness.
The basic choice of a representation \(\mathbb V\) that Theorem 4.2 can be applied to is given by \(\Lambda^p\mathbb R^{n*}\), the \(p\)th exterior power of the dual of the standard representation. This decomposes as \(\mathbb V=\mathbb V_0\oplus\dots\oplus\mathbb V_p\), where \(\mathbb V_j\) is spanned by wedge products of elements of the dual of the standard basis that contain \(p-j\) factors from \(\{e_1^*,\dots,e_p^*\}\). The properties claimed in Lemma 4.1 can be easily deduced from the construction from the dual of the standard representation.
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 Example 4.3. 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 Example 4.3, 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 \[\tag{4.1} \det(H(\sigma))=\pm\sigma^{-n-2}\] 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.
Let \(M\) be an oriented smooth manifold of dimension \(n\) which is endowed with a projective structure. Let \(\sigma\in\Gamma(\mathcal E(1))\) be a nowhere-vanishing section and let us denote by \(\nabla^\sigma\) the Weyl connections of the Weyl structure determined by \(\sigma\) and by \(\mbox{\textsf{P}}^{\sigma}\) its Rho tensor. Then we have:
(1) An appropriate constant multiple of \(\sigma\) satisfies (4.1) if and only if \(\nabla^\sigma(\det(\mbox{\textsf{P}}^{\sigma}))=0\) and \(\det(\mbox{\textsf{P}}^\sigma)\) is nowhere vanishing.
(2) The Weyl structure determined by \(\sigma\) is always Lagrangian. If \(M\) is projectively flat, then an appropriate constant multiple of \(\sigma\) satisfies (4.1) if and only if this Weyl structure is non-degenerate and the image of the corresponding section \(s:M\to A\) is a minimal submanifold. This extends to curved projective structures provided that minimality of \(s(M)\) is defined as vanishing of the mean curvature form associated to the canonical connection \(D\) via the definition in [18].
Proof. It is well known how to decompose the curvature of a linear connection on \(TM\) into the projective Weyl curvature and the Rho-tensor, see Section 3.1 of [2] (taking into account the sign conventions mentioned in 2.6). It is also shown there that the Bianchi identity shows that the skew part of the Rho tensor is a non-zero multiple of the trace of the curvature tensor, which describes the action of the curvature on the top exterior power of the tangent bundle and thus on density bundles. This readily implies that \(\mbox{\textsf{P}}^\sigma\) is symmetric, so the Weyl structure defined by \(\sigma\) is Lagrangian by Proposition 3.10.
(1) It is well known that the projective Hessian in terms of a linear connection \(\nabla\) in the projective class and its Rho tensor \(\mbox{\textsf{P}}\) is given by the symmetrization of \(\nabla^2\sigma-\mbox{\textsf{P}}\sigma\), see Section 3.2 of [13]. (The different sign is caused by different sign conventions for the Rho-tensor.) But by definition \(\nabla^\sigma\sigma=0\) and \(\mbox{\textsf{P}}^{\sigma}\) is symmetric, so we conclude that \(H(\sigma)=-\mbox{\textsf{P}}^\sigma\sigma\) and hence \(\det(H(\sigma))=(-1)^n\sigma^n\det(\mbox{\textsf{P}}^{\sigma})\). Now for each \(k\neq 0\), the sections of \(\mathcal E(k)\) that are parallel for \(\nabla^\sigma\) are exactly the constant multiples of \(\sigma^k\), so (1) follows readily.
(2) Since \(\mbox{\textsf{P}}^\sigma\) is symmetric, \(\det(\mbox{\textsf{P}}^\sigma)\) is nowhere vanishing if and only if \(\mbox{\textsf{P}}^\sigma\) is non-degenerate. By Proposition 3.10, this is equivalent to non-degeneracy of the Weyl structure determined by \(\sigma\), and we assume this from now on. Let us write \(\det(\mbox{\textsf{P}}^{\sigma})\) in terms of the tautological section \(\epsilon\) of \(\Lambda^nTM(-n-1)\) as above as \(C(\epsilon\otimes\epsilon\otimes(\mbox{\textsf{P}}^{\sigma})^{\otimes^n})\), where \(C\) denotes the appropriate contraction. Applying \(\nabla^\sigma\) to this, we observe that \(C\) and \(\epsilon\) are projectively invariant bundle maps and thus parallel for any Weyl connection. Thus we conclude that \[\nabla^\sigma\det(\mbox{\textsf{P}}^{\sigma})=(n-1)C\left(\epsilon\otimes\epsilon\otimes(\mbox{\textsf{P}}^{\sigma})^{\otimes^{n-1}} \otimes\nabla^\sigma\mbox{\textsf{P}}^\sigma\right).\] Here we have used that the contraction is symmetric in the bilinear forms we enter and in the right hand side the form index of \(\nabla^{\sigma}\) remains uncontracted. Since we assume that \(\mbox{\textsf{P}}^\sigma\) is invertible, linear algebra tells us that contracting two copies of \(\epsilon\) with \((\mbox{\textsf{P}}^{\sigma})^{\otimes^{n-1}}\) gives \(\det(\mbox{\textsf{P}}^\sigma)\Phi\), where \(\Phi\in\Gamma(S^2TM)\) is the inverse of \(\mbox{\textsf{P}}^\sigma\). Returning to the abstract index notation used in Theorem 3.13 and writing \(\nabla\) and \(\mbox{\textsf{P}}\) instead of \(\nabla^\sigma\) and \(\mbox{\textsf{P}}^\sigma\), we conclude that \(\nabla\det(\mbox{\textsf{P}})=0\) is equivalent to \(0=\mbox{\textsf{P}}^{ab}\nabla_i\mbox{\textsf{P}}_{ab}\).
On the other hand, we know the second fundamental form of \(s(M)\subset A\) from Theorem 3.13. To determine the mean curvature, we have to contract an inverse metric into this expression, and we already know that this inverse metric is just \(\mbox{\textsf{P}}^{ij}\). Vanishing or non-vanishing of the result is independent of the final contraction with \(\mbox{\textsf{P}}^{kc}\). Thus we conclude that \(s(M)\subset A\) is minimal if and only if \[\tag{4.2} 0=\mbox{\textsf{P}}^{ab}(2\nabla_a\mbox{\textsf{P}}_{bi}-\nabla_i\mbox{\textsf{P}}_{ab}).\] In the proof of Theorem 3.13, we have also noted that the Cotton-York tensor is given by the alternation of \(\nabla_i\mbox{\textsf{P}}_{jk}\) in the first two indices. It is well known that this vanishes for projectively flat structures (see [2]) and hence in the projectively flat case, \(\nabla_i\mbox{\textsf{P}}_{jk}\) is completely symmetric. Using this, the claim in the projectively flat case follows immediately.
In the non-flat case, we first have to determine the map \(\varphi\) from Lemma 4 of [18]. In our notation, the map \(P\) used there is given by \(P|_{L^\pm}=\pm\operatorname{id}\). Using this, the beginning of the proof of Theorem 3.12 readily shows that, in the notation used there, for \(\xi\in T_xM\cong L^-_{s(x)}\), we get \(\varphi(\xi^i)=(\xi^i,-\mbox{\textsf{P}}_{ja}\xi^a)\). Now we can combine this with the formula for \(II_D\) from Theorem 3.13, which describes the operator \(\widehat{A}\) used in [18]. This easily shows that, up to a non-zero factor, that the operator \(\widehat{h}\) from [18] is given by \[\widehat{h}(\xi^i,\eta^j,\zeta^k)=\xi^i\eta^j\zeta^k\mbox{\textsf{P}}_{ia}\mbox{\textsf{P}}^{ab}\nabla^s_j\mbox{\textsf{P}}_{kb}.\] By definition, the mean curvature form \(\widehat{H}\) from [18] is the trace over the first and third entry of this. Thus we have to contract \(\widehat{h}_{ijk}\) with \(\mbox{\textsf{P}}^{ik}\), which again leads to \(\mbox{\textsf{P}}^{ab}\nabla^s_j\mbox{\textsf{P}}_{ab}\).
In the special case of a two-dimensional projective structure the minimality condition (4.2) was previously obtained in [28].
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 4.4, we obtain:
Let \((M,[\nabla])\) be a closed oriented locally flat projective manifold. Then \([\nabla]\) is properly convex if and only if \([\nabla]\) arises from a minimal Lagrangian Weyl structure whose Rho tensor is positive definite.
Proof. Suppose that the flat projective structure \([\nabla]\) arises from a minimal Lagrangian Weyl structure \(s\) whose Rho tensor \(\mbox{\textsf{P}}^s\) is positive definite. Since \(s\) is Lagrangian, \(\mbox{\textsf{P}}^s\) is a constant negative multiple of the Ricci tensor of the Weyl connection \(\nabla^s\), see [2], and taking into account the sign issue mentioned in 2.6. By Theorem 3.13, the nowhere vanishing density \(\det(\mbox{\textsf{P}}^s)\) is preserved by \(\nabla^s\), whence an appropriate power defines a volume density that is parallel for \(\nabla^s\). Finally, projective flatness implies that the pair \((\nabla^s,\mbox{\textsf{P}}^s)\) satisfies the hypothesis of Theorem 3.2.1 of [25], which then implies that \([\nabla]\) is properly convex.
Conversely, suppose that \([\nabla]\) is properly convex. By [26] there is a solution \(\sigma\) to the projective Monge-Ampère equation with right hand side \((-1)^{n+2}\sigma^{n+2}\) and such that \(\sigma\) is negative for the natural orientation on \(\mathcal E(1)\). By Theorem 4.4, \([\nabla]\) arises from a minimal Lagrangian Weyl structure. Since the Hessian of \(\sigma\) is positive definite, so is the Rho tensor.
Existence and uniqueness of minimal Lagrangian Weyl structures for a given torsion-free AHS structure is an interesting fully non-linear PDE problem. In the special case of projective surfaces, some partial results regarding uniqueness have been obtained in [27] and [30]. See also [31] for a connection to dynamical systems and [29] for a related variational problem on the space of conformal structures.
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 \(\pi:\otimes^2T^*M\to\circledcirc^2T^*M\).
Suppose that \((G,P)\) corresponds to a \(|1|\)-grading on the simple Lie algebra \(\mathfrak g\) of \(G\). Suppose further, that there is a representation \(\mathbb V\) of \(G\) satisfying the assumptions of Theorem 4.2 whose complexification is a fundamental representation of the complexification of \(\mathfrak g\), and let \(\mathcal EM\) denote the natural line bundle induced by \(\mathbb V/\mathbb V^1\).
(1) There is an invariant differential operator \(H:\Gamma(\mathcal EM)\to\Gamma(\circledcirc^2T^*M\otimes\mathcal EM)\) of second order.
(2) For a nowhere vanishing section \(\sigma\in\Gamma(\mathcal EM)\), let \(\mbox{\textsf{P}}^\sigma\) be the Rho tensor of the Weyl structure determined by \(\sigma\). Then \(H(\sigma)\) is a non-zero multiple of \(\pi(\mbox{\textsf{P}}^\sigma)\sigma\), where \(\pi\) is the projection to the Cartan square.
Proof. (1) The representation \(\mathbb V\) induces a tractor bundle on parabolic geometries of type \((G,P)\) to which the construction of BGG sequences can be applied. The first operator \(H\) in the resulting sequence is defined on \(\Gamma(\mathcal EM)\). Now the condition that \(\dim(\mathbb V/\mathbb V^1)=1\) implies that the complexification of \(\mathbb V\) is the fundamental representation corresponding to the simple root that induces the \(|1|\)-grading that defines \(\mathfrak p\). Since the complexification of \(\mathbb V\) is a fundamental representation, the results of [5] show that the first operator in the BGG sequence has order two and the target space claimed in (1).
(2) It is also known in general (see [14] or [10]) how to write out \(H\) in terms of a Weyl structure with Weyl connection \(\nabla^s\) and Rho tensor \(\mbox{\textsf{P}}^s\): For \(\sigma\in\Gamma(\mathcal EM)\), one then has to form \(\nabla^2\sigma-\mbox{\textsf{P}}^s\sigma\), symmetrize and then project to the Cartan square. But if \(s\) is the Weyl structure determined by \(\sigma\), then by definition \(\nabla^s\sigma=0\) and \(\mbox{\textsf{P}}^s=\mbox{\textsf{P}}^\sigma\), which implies the claim.
Observe that Example 4.3 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 4.8, for these equations there is always an analog of part (1) of Theorem 4.4.
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 4.8 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 4.4. All this will be taken up in detail elsewhere.
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