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 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 \(d\Omega=0\).
Let \((p:\mathcal G\to M,\omega)\) be a parabolic geometry of type \((G,P)\) and \(\pi:A\to M\) its associated bundle of Weyl structures. Then the natural 2-form \(\Omega\in\Omega^2(A)\) is closed if and only if \((G,P)\) corresponds to a \(|1|\)-grading and the Cartan geometry \((p:\mathcal G\to M,\omega)\) is torsion-free.
Proof. Let \(D\) be the canonical connection on \(TA\) from 2.3. Since \(\Omega\) is induced by a \(G_0\)-invariant pairing on \(\mathfrak g_-\oplus\mathfrak g_+\) it satisfies \(D\Omega=0\). If \(D\) were torsion-free, then \(d\Omega\) would coincide with the complete alternation of \(D\Omega\) and thus would vanish, too. In the presence of torsion, there still is a relation as follows. Expanding \(D\Omega=0\) by inserting vector fields \(\xi,\eta,\zeta\in\mathfrak X(A)\), we obtain \[0=\xi\cdot\Omega(\eta,\zeta)-\Omega(D_\xi\eta,\zeta)-\Omega(\eta,D_\xi\zeta).\] Now one takes the sum of the right hand side over all cyclic permutations of the arguments and uses skew symmetry of \(\Omega\) to bring all derivatives of vector fields into the first component. Then one may use the definition of the torsion \(\tau\) of \(D\) to rewrite \(D_\xi\eta-D_\eta\xi\) as \([\xi,\eta]+\tau(\xi,\eta)\) and similarly for other combinations of the fields. Then the terms in which one field differentiates the value of \(\Omega\) together with the terms involving a Lie bracket add up to the exterior derivative. One concludes that \(D\Omega=0\) implies \[d\Omega(\xi,\eta,\zeta)=\textstyle\sum_{\text{cycl}}\Omega(\tau(\xi,\eta),\zeta),\] where in the right hand side we have the sum over all cyclic permutations of the arguments. Now let us assume that \((G,P)\) corresponds to a \(|k|\)-grading with \(k>1\). Then \(L^-\) and \(L^+\) decompose into direct sums of subbundles according to the grading of \(\mathfrak g_-\) and \(\mathfrak p_+\), respectively. Now we take \(\xi\in L^-\) of degree \(-1\), \(\eta\in L^+\) of degree \(i>1\) and \(\zeta\in L^+\) of degree \(i-1\). Then by Theorem 2.12, \(\tau\) coincides with \(\{\ ,\ \}\) on any two of these three fields. The restriction of \(d\Omega\) to the subbundles corresponding to these three degrees is induced by the trilinear map \(\mathfrak g_{-1}\times\mathfrak g_i\times\mathfrak g_{i-1}\to\mathbb R\) given by \((X,Y,Z)\mapsto \sum_{\text{cycl}}B([X,Y],Z)\), where \(B\) denotes the Killing form of \(\mathfrak g\). But \(B([X,Y],Z)\) is already totally skew, so \(d\Omega=0\) would imply that \(B([X,Y],Z)=0\) for all elements of the given homogeneities. But non-degeneracy of \(B\) shows that \(B([X,Y],Z)=0\) for all \(Z\) implies \([X,Y]=0\) while for \(Y\in\mathfrak g_i\), the equation \([X,Y]=0\) for all \(X\in\mathfrak g_{-1}\) implies \(Y=0\), see Proposition 3.1.2 in [16].
Thus we may assume from now on that \((G,P)\) corresponds to a \(|1|\)-grading. In this case, the bracket \(\{\ ,\ \}\) is identically zero, so by Theorem 2.12, \(\tau\) vanishes upon insertion of one element from \(L^+\). Hence we see that \(d\Omega\) vanishes upon insertion of two elements of \(L^+\). Decomposing \(\Lambda^3T^*A\) according to \(TA=L^-\oplus L^+\), the only potentially non-zero components of \(d\Omega\) thus are the ones in \(\Lambda^3(L^-)^*\) and in \(\Lambda^2(L^-)^*\otimes (L^+)^*\).
Now if \(\xi,\eta\in\Gamma(L^-)\) and \(\zeta\in\Gamma(L^+)\), then we simply obtain \(d\Omega(\xi,\eta,\zeta)=\Omega(T(\xi,\eta),\zeta)\), where \(T\) is defined in Definition 2.10. Non-degeneracy of \(\Omega\) shows that this vanishes for all \(\xi,\eta,\zeta\) if and only if \(T=0\). This shows that vanishing of \(T\) is a necessary condition for \(\Omega\) being closed. In the case of a \(|1|\)-grading, the pullback of \(T\) along a Weyl structure as in Proposition 2.11 is independent of the Weyl structure and gives the torsion of the Cartan geometry \((p:\mathcal G\to M,\omega)\).
To complete the proof, we thus have to show that (still in the case of a \(|1|\)-grading) vanishing of \(T\) implies that the component of \(d\Omega\) in \(\Lambda^3(L^-)^*\) vanishes identically. As above, Theorem 2.12 shows that this component is given by the sum of \(\Omega(Y(\xi,\eta),\zeta)\) over all cyclic permutations of its arguments. But by construction, this is simply the complete alternation of \(Y\), viewed as a section of \(\Lambda^2L^+\otimes L^+\) via the identification \((L^-)^*\cong L^+\). In terms of the Cartan geometry \((p:\mathcal G\to M)\), it thus suffices to show that the component \(\kappa_+\) of the Cartan curvature in \(\mathfrak p_+\) always has trivial complete alternation.
To do this, we first observe that for a \(|1|\)-graded Lie algebra \(\mathfrak g\), the subalgebra \(\mathfrak g_0\) always splits into its center \(\mathfrak z(\mathfrak g_0)\), which has dimension one, and a semisimple part \(\mathfrak g_0^{ss}\). For a torsion-free geometry, the component \(\kappa_0\) of \(\kappa\) with values in \(\mathfrak g_0\) is the lowest non-vanishing homogeneous component of \(\kappa\), which implies that its values have to lie in \(\mathfrak g_0^{ss}\), compare with Theorem 4.1.1 in [16]. Thus we conclude that, viewed as a function \(\mathcal G\to \mathfrak g\), \(\kappa\) has values in the subspace \(\mathfrak g_0^{ss}\oplus\mathfrak g_1\).
Now we can apply the Bianchi identity in the form of equation (1.25) in Proposition 1.5.9 of [16]. This contains four terms, three of which are evaluations of the function \(\kappa\) or its derivative along some vector field, so these have values in \(\mathfrak g_0^{ss}\oplus\mathfrak g_1\), too. Formulated in terms of functions, the Bianchi identity thus implies that for \(X_1,X_2,X_3\) the cyclic sum over the arguments of \([X_1,\kappa(\omega^{-1}(X_2),\omega^{-1}(X_3))]\) has trivial component in \(\mathfrak z(\mathfrak g_0)\). Now we can replace the \(X_i\) by their components in \(\mathfrak g_-\) without changing the \(\mathfrak g_0\)-component of \([X_1,\kappa(\omega^{-1}(X_2),\omega^{-1}(X_3))]\), which in addition depends only on the \(\mathfrak g_1\)-component of \(\kappa\). Now it is well known that for \(X\in\mathfrak g_{-1}\) and \(Z\in\mathfrak g_1\), the component of \([X,Z]\) in \(\mathfrak z(\mathfrak g_0)\) is a non-zero multiple of \(B(X,Z)\), where \(B\) denotes the Killing form. But this exactly shows that, up to a non-zero factor, the \(\mathfrak z(\mathfrak g_0)\)-component of \(\sum_{\text{cycl}}[X_1,\kappa(\omega^{-1}(X_2),\omega^{-1}(X_3))]\) represents the action of the complete alternation of \(\kappa_+\) on the three vector fields corresponding to the \(X_i\). Thus this complete alternation vanishes identically.
(1) The failure of closedness of \(\Omega\) for \(|k|\)-gradings with \(k>1\) can be described more precisely. The map \((X,Y,Z)\mapsto B([X,Y],Z)\) that shows up in the proof defines a \(G\)-invariant element in \(\Lambda^3\mathfrak g^*\) and hence a bi-invariant \(3\)-form on \(G\). Restricting this to \(\Lambda^3(\mathfrak g_-\oplus\mathfrak p_+)^*\), one obtains a \(G_0\)-invariant trilinear form, which is non-zero provided that \(k>1\). This in turn induces a natural \(3\)-form on each manifold endowed with a Cartan Geometry of type \((G,G_0)\). On a bundle of Weyl structures, the proof of Theorem 3.1 shows that this form always is a component of \(d\Omega\).
(2) The parabolic geometries corresponding to \(|1|\)-gradings form a very interesting class of structures. For a \(|1|\)-grading, the subalgebras \(\mathfrak g_-\) and \(\mathfrak p_+\) become Abelian, whence the name “Abelian parabolic geometries” is sometimes used for these structures. The classification of \(|1|\)-gradings of simple Lie algebras is well known from the theory of Hermitian symmetric spaces, which motivates the more common name “AHS structures” where AHS is shorthand for “almost Hermitian symmetric”.
Suppose that \((G,P)\) corresponds to a \(|1|\)-grading on \(\mathfrak g\). As noted in 2.1, the underlying structure \(p_0:\mathcal G_0\to M\) of a Cartan geometry \((p:\mathcal G\to M)\) simply becomes a reduction of the linear frame bundle of \(M\) to the structure group \(G_0\subset GL(\mathfrak g_{-1})\). Thus AHS structures are a special class of G-structures, whose relevance is explained by the classification results by S. Kobayashi and T. Nagano in [24]. They prove that these are the only structures for which the group acts irreducibly, and which have the property that any automorphism is determined by a finite jet in a point but not by the one-jet in a point. In fact, automorphisms are always determined by the two-jet in a point and the equivalent canonical Cartan geometry of type \((G,P)\) is the most effective description for these structures.
The torsion-freeness condition that shows up in Theorem 3.1 has a natural interpretation in the language of \(G_0\)-structures. As noted in the proof, the torsion \(T\) associated to a Weyl structure in this case is independent of the Weyl structure. It turns out that this coincides with the intrinsic torsion of the \(G_0\)-structure (i.e. the component of the torsion that is independent of the choice of connection). Thus torsion-freeness of the Cartan geometry corresponds to the usual notion of integrability in the language of \(G_0\)-structures.
(3) For some types of AHS-structures, torsion-freeness implies local flatness. Locally flat structures can be equivalently be characterized as being obtained from local charts with values in the homogeneous model \(G/P\), for which the transition functions are given by restrictions of left actions of elements of \(g\). This case anyway plays a very important role in the results we are going to prove, so our results are also relevant to these types of AHS structures.
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 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.
Consider a torsion-free AHS structure \((p:\mathcal G\to M,\omega)\) and let \(\pi:A\to M\) be the corresponding bundle of Weyl structures. Let \(\xi,\eta\in\mathfrak X(M)\) be vector fields and \(\alpha,\beta\in\Omega^1(M)\) be one-forms on \(M\) and consider the corresponding sections \(\tilde\xi,\tilde\eta\in\Gamma(L^-)\) and \(\tilde\alpha,\tilde\beta\in\Gamma(L^+)\).
Then for the Lie brackets on \(A\), we get \([\tilde\alpha,\tilde\beta]=0\) and \([\tilde\xi,\tilde\alpha]=D_{\tilde\xi}\tilde\alpha\in\Gamma(L^+)\). Finally, the \(L^-\)-component of \([\tilde\xi,\tilde\eta]\) coincides with \(\widetilde{[\xi,\eta]}\), while its \(L^+\)-component coincides with \(-Y(\tilde\xi,\tilde\eta)\), see Definition 2.10.
Proof. By definition of the torsion \[\tag{3.1} \tau(X,Z)=D_XZ-D_ZX-[X,Z]\] for all \(X,Z\in\mathfrak X(A)\). If at least one of the two fields is a section of \(L^+\), then the left hand side of (3.1) vanishes by Theorem 2.12. Moreover, all the sections coming from \(M\) are parallel in \(L^+\)-directions. This immediately shows that \([\tilde\alpha,\tilde\beta]=0\) and \(0=D_{\tilde\xi}\tilde\alpha-[\tilde\xi,\tilde\alpha]\). In view of torsion-freeness, Theorem 2.12 further tells us that \(\tau(\tilde\xi,\tilde\eta)=Y(\tilde\xi,\tilde\eta)\in\Gamma(L^+)\). Inserting \(X=\tilde\xi\) and \(Z=\tilde\eta\) into the right hand side of (3.1), the first two terms are sections of \(L^-\), so the claim about the \(L^+\)-component of \([\tilde\xi,\tilde\eta]\) follows. Finally, since \(\tilde\xi\) and \(\tilde\eta\) are lifts to \(A\) of \(\xi\) and \(\eta\), the bracket \([\tilde\xi,\tilde\eta]\in\mathfrak X(A)\) is a lift of \([\xi,\eta]\). Since \(\widetilde{[\xi,\eta]}\) is the unique section of \(L^-\) that projects onto \([\xi,\eta]\), it has to coincide with the \(L^-\)-component of that lift.
In particular, we see that, while \(L^+\) always defines an involutive distribution, \(L^-\) is only involutive if the curvature component \(Y\) from Definition 2.10 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 2.7.
The Ricci-type contraction of \(\rho\) is a non-zero multiple of \(h\).
Proof. By Theorem 2.12, \(\rho+\{\ ,\ \}_0\) vanishes upon insertion of one section of \(L^+\) and coincides with \(W\) on \(\Lambda^2(L^-)^*\). Decomposing \(\Lambda^2TA^*\) according to \(TA=L^+\oplus L^-\), we conclude that the component of \(\rho\) in \(\Lambda^2(L^+)^*\) vanishes, its component in \((L^-)^*\otimes (L^+)^*\) is induced by \(-\{\ ,\ \}_0\), and the component in \(\Lambda^2(L^-)^*\) is induced by \(W\). On the other hand, \(\operatorname{End}_0(TA)=\mathcal G\times_{G_0}\mathfrak g_0\), so this is a subbundle of \(((L^-)^*\otimes L^-)\oplus ((L^+)^*\otimes L^+)\subset TA^*\otimes TA\). Thus we conclude that the Ricci-type contraction of \(\rho\) vanishes on \(L^+\times L^+\), while its components on \(L^-\times L^+\) and \(L^-\times L^-\) are induced by the Ricci-type contractions of \(-\{\ ,\ \}_0\) and \(W\), respectively. By Proposition 2.11, the pullback of \(W\) along any section \(s:M\to A\) represents the Weyl curvature of the Weyl structure determined by \(s\). In the torsion-free case, this is well known to have values in an irreducible representation of \(G_0\) that occurs with multiplicity one in \(\Lambda^*\mathfrak g_-^*\otimes\mathfrak g\), which implies that any contraction of \(W\) vanishes identically.
Hence we see that the Ricci type contraction of \(\rho\) has values in \((L^-)^*\otimes (L^+)^*\) and is induced by the Ricci type contraction of \(-\{\ ,\ \}_0\), so this is a natural bundle map, and we can compute it on the inducing representations. Take a basis \(\{e_i\}\) of \(\mathfrak g_{-1}\) and let \(\{e^i\}\) be the dual basis of \(\mathfrak g_{-1}^*\cong\mathfrak g_1\), which means that for the Killing form \(B\), we get \(B(e_i,e^j)=\delta_i^j\). Now we have to view the \(\mathfrak g_0\) component of the bracket \([\ ,\ ]\) in \(\mathfrak g\) as a map sending \((\mathfrak g_-\oplus\mathfrak p_+)^2\) to an endomorphism of \(\mathfrak g_-\oplus\mathfrak p_+\) via the adjoint action. Hence for \(X,Y\in\mathfrak g_{-1}\) and \(Z,W\in\mathfrak g_1\), the Ricci type contraction sends \(\binom{X}{Z}\) and \(\binom{Y}{W}\) to \[\textstyle\sum_iB\big(\big[\big[\binom{X}{Z},\binom{e_i}{0}\big],\binom{Y}{W}\big], \binom0{e^i})+\sum_i B\big(\big[\big[\binom{X}{Z},\binom{0}{e^i}\big], \binom{Y}{W}\big],\binom{e_i}{0}\big).\] Expanding the first sum using invariance of the Killing form and the fact that \(\mathfrak g_{-1}\) is abelian, we obtain \[\textstyle\sum_iB([[Z,e_i],Y],e^i)=\sum_iB(Z,[e_i,[Y,e^i]])=\sum_iB(Z,[Y,[e_i,e^i]]),\] and in the same way the second sum gives \(\sum_iB([X,[e_i,e^i]],W)\). But the element \(\sum_i[e_i,e^i]\in\mathfrak g_0\) is obtained from the identity map in \(\mathfrak g_{-1}\otimes\mathfrak g_{-1}^*\) via the isomorphism to \(\mathfrak g_{-1}\otimes\mathfrak g_1\) and the bracket in \(\mathfrak g\). Since these both are \(\mathfrak g_0\)-equivariant, \(\sum_i[e_i,e^i]\) is \(\mathfrak g_0\)-invariant and thus contained in the center of \(\mathfrak g_0\). In the \(|1|\)-graded case, this center is spanned by the grading element \(E\). In addition, \(B(E,\textstyle\sum_i[e_i,e^i])=\sum_iB([E,e_i],e^i)=-\dim(\mathfrak g_{-1})\), so \(\sum_i[e_i,e^i]\) is a non-zero multiple of \(E\). Hence the whole contraction gives a non-zero multiple of \(B(Z,Y)+B(X,W)=h(\binom{X}{Z},\binom{Y}{W})\).
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 \[\tag{3.2} h(D_\xi\eta,\zeta)-\tfrac12h(\tau(\xi,\eta),\zeta)+\tfrac12h(\tau(\xi,\zeta),\eta)+ \tfrac12h(\tau(\eta,\zeta),\xi).\] 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
For any torsion-free AHS structure, the pseudo-Riemannian metric \(h\) induced by the canonical almost bi-Lagrangian structure on the bundle \(A\) of Weyl structures is an Einstein metric with non-zero scalar curvature.
Proof. Theorem 2.12 in the torsion-free case shows that \(\tau\) vanishes upon insertion of one section of \(L^+\) and has values in \(L^+\). Thus equation (3.2) shows that \(h(C(\xi,\eta),\zeta)\) vanishes if one of the three fields is a section of \(L^+\). This shows that the only non-zero component of \(C\) is the one mapping \(L^-\times L^-\) to \(L^+\). Now it is standard how to compute the curvature of \(\nabla\) from \(C\) and the curvature \(\rho\) of \(D\) via differentiating the equation defining \(C\). The result contains terms in which \(C\) is differentiated as well as terms in which values of \(C\) are inserted into \(C\). From the form of \(C\) we have just deduced, it follows that the latter terms vanish identically.
Using this, one computes that for \(\xi,\eta,\zeta\in\mathfrak X(A)\) the difference \(R(\xi,\eta)(\zeta)-\rho(\xi,\eta)(\zeta)\) is given by \[\tag{3.3} D_\xi(C(\eta,\zeta))-D_\eta(C(\xi,\zeta))+C(\xi,D_\eta\zeta)-C(\eta,D_\xi\zeta)-C([\xi,\eta],\zeta).\] (This is just the covariant exterior derivative of \(C\) with respect to \(D\) evaluated on \(\xi\) and \(\eta\) and then applied to \(\zeta\).) In view of Lemma 3.4, it suffices to prove that the Ricci-type contraction of this expression vanishes. To compute this contraction, we leave \(\xi\) and \(\zeta\) as entries, insert the elements of a local frame of \(TA\) for \(\eta\) and hook the result into \(h\) together with the elements of the dual frame. First of all, (3.3) visibly vanishes for \(\zeta\in\Gamma(L^+)\). If we insert for \(\eta\) an element of a frame for \(L^-\), then the element of the dual frame will sit in \(L^+\). Since \(C\) has values in \(L^+\), these summands do not contribute to the contraction. Thus we only have to take into account the case that we insert elements of a frame for \(L^+\) for \(\eta\), and then the first and fourth term of (3.3) visibly vanish. The remaining three terms vanish if \(\xi\) is a section of \(L^+\), so what we have to compute is \[\textstyle\sum_ih\left(-D^+_{e^i}(C(\xi,\zeta))+C(\xi,D^+_{e^i}\zeta)-C([\xi,e^i],\zeta),e_i\right)\] for a smooth local frame \(\{e^i\}\) for \(L^+\) with dual frame \(\{e_i\}\) for \(L^-\) and local sections \(\xi,\zeta\in\Gamma(L^-)\). Now we can take \(\xi\) and \(\zeta\) and the local frames to be obtained from vector fields respectively one-forms on \(M\). Then \(D^+_{e^i}\zeta=0\), while \([\xi,e^i]\in\Gamma(L^+)\) by Proposition 3.3 and thus \(C([\xi,e^i],\zeta)=0\).
Thus we are left with computing \(\sum_ih(D^+_{e^i}(C(\xi,\zeta)),e_i)\) with the frames, \(\xi\) and \(\zeta\) all coming from \(M\). In particular, \(e_i\) is parallel for \(D^+\) so since \(D\) is metric for \(h\), we may rewrite this as \(\sum_ie^i\cdot h(C(\xi,\zeta),e_i)\). We can then insert the formula for \(h(C(\xi,\zeta),e_i)\) resulting from (3.2), taking into account that on entries from \(L^-\) the torsion \(\tau\) is determined by the tensor \(Y\) from Definition 2.10. Viewing \(Y\) as a section of \(\Lambda^2(L^-)^*\otimes (L^-)^*\), this leads to \[\textstyle\sum_ie^i\cdot h(C(\xi,\zeta),e_i)=\tfrac12\sum_ie^i\cdot (-Y(\xi,\zeta,e_i)+Y(\xi,e_i,\zeta)+Y(\zeta,e_i,\xi)).\] From the proof of Theorem 3.1, we know that the complete alternation of \(Y\) vanishes, which allows us to rewrite this as \(\sum_ie^i\cdot Y(\zeta,e_i,\xi)\). Under the standing assumption that all sections come from \(M\), they are parallel for \(D^+\), so we can complete the proof by showing that \(\sum_i(D^+_{e^i}Y)(\zeta,e_i,\xi)=0\).
Now by definition, \(Y\) is a component of the Cartan curvature, which descends to a well defined section of the bundle \(\Lambda^2T^*M\otimes\mathcal AM\), where \(\mathcal AM=\mathcal G\times_P\mathfrak g\). By torsion freeness, the full Cartan curvature has the form \((0,W,Y)\) with respect to the decomposition \(\mathfrak g=\mathfrak g_{-1}\oplus\mathfrak g_0\oplus\mathfrak g_1\). Hence by Theorem 2.5, we get \[D^+_\varphi(0,W,Y)=(0,D^+_\varphi W,D^+_\varphi,Y)=-\varphi\bullet (0,W,Y),\] and the action \(\bullet\) is induced by the Lie bracket on \(\mathfrak g\). Since this bracket vanishes on \(\mathfrak g_1\times\mathfrak g_1\) and defines the action of \(\mathfrak g_0\) on \(\mathfrak g_{\pm 1}\), we conclude that \(D^+W=0\) and \(D^+_\varphi Y=W(\varphi)\), where we view \(W\) as an section of \(\Lambda^2(L^--)^*\otimes\operatorname{End}(L^+)\cong \Lambda^2(L^--)^*\otimes(L^-)^*\otimes (L^+)^*\) in the right hand side. But this implies that \(\sum_i(D^+_{e^i}Y)(\zeta,e_i,\xi)\) is given by evaluating a trace of \(W\) on \(\zeta\) and \(\xi\) and thus vanishes.
In the special case of a projective structure on a surface \(\Sigma\), the resulting Einstein metric on the four-manifold \(A\) is also anti-self-dual, see [19] and also [8].
The automorphisms of a projective structure on a smooth manifold \(M\) lift to become isometric symplectomorphisms of \((A,\Omega,h)\), see [19]. For background about automorphisms of a parabolic geometry, see [1, 16, 20].
A pair \((h,\Omega)\), consisting of a split-signature metric \(h\) and a symplectic form \(\Omega\) that are related by an endomorphism which squares to become the identity map, is also known as an almost para-Kähler structure. Here, following [6], we refer to such a pair, or rather its associated triple \((\Omega,L_+,L_-)\), as an almost bi-Lagrangian structure.
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.
Let \((p:\mathcal G\to M,\omega)\) be a torsion-free AHS structure and let \(s:M\to A\) be a smooth section.
(1) The Weyl structure corresponding to \(s\) is called Lagrangian if and only if \(s^*\Omega=0\) and thus \(s(M)\subset A\) is a Lagrangian submanifold.
(2) The Weyl structure corresponding to \(s\) is called non-degenerate if and only if \(s^*h\in\Gamma(S^2T^*M)\) is non-degenerate and thus defines a pseudo-Riemannian metric on \(M\).
These properties can easily be characterized in terms of the Rho tensor.
A Weyl structure is Lagrangian if and only if its Rho tensor is symmetric and non-degenerate if and only if the symmetric part of its Rho tensor is non-degenerate.
Proof. For a point \(x\in M\) and a tangent vector \(\xi\in T_xM\) consider \(T_xs\cdot\xi\in T_{s(x)}(A)\). Since this is a lift of \(\xi\), its \(L^-\)-component has to coincide with \(\tilde\xi(s(x))\). On the other hand, by Proposition 2.8, the \(L^+\)-component of \(T_xs\cdot\xi\) corresponds to \(\mbox{\textsf{P}}(x)(\xi)\in T^*_xM\). Pulling back the pairing between \(L^-\) and \(L^+\), one thus obtains the map \((\xi,\eta)\mapsto \mbox{\textsf{P}}(x)(\eta)(\xi)\) and thus the result follows from the definitions of \(\Omega\) and \(h\).
(1) For any type of parabolic geometry, there are natural line bundles called bundles of scales, an example in the AHS-case is provided by the bundle \(\mathcal EM\) in Theorem 4.2 below. If \(\mathcal EM\) is any bundle of scales on \(M\), then mapping a Weyl structure to the induced Weyl connection on \(\mathcal EM\) induces a bijective correspondence between Weyl structures and linear connections on \(\mathcal EM\). Fixing \(\mathcal EM\), one calls a Weyl structure closed if the corresponding linear connection on \(\mathcal EM\) is flat and exact if in addition there is a global parallel section of \(\mathcal EM\), see [15] and Section 5.1 of [16]. For general types of geometries there is a larger freedom of choice of bundles of scales, but for AHS-structures all bundles of scales lead to the same subclasses of closed and exact Weyl structures.
Together with the general theory of Weyl structures, Proposition 3.10 implies that, on a torsion-free AHS-structure, a Weyl structure is Lagrangian if and only if it is closed. This follows from the relation between curvature and torsion of a Weyl connection and the Cartan curvature as discussed in Example 5.2.3 of [16] in the setting of AHS structures. By Theorem 5.2.3 of that reference, the curvature \(R\in\Omega^2(M,\mathcal G\times_g\mathfrak g_0)\) of the Weyl connection corresponding to \(s\in\Gamma(A)\) can be computed as \(s^*W+\partial s^*\mbox{\textsf{P}}\) for the quantities from Proposition 2.11 and a certain natural bundle map \(\partial\). For the action on a bundle of scales, the component of \(R\) with values in the center \(\mathfrak z(\mathfrak g_0)\) is relevant. For a torsion free geometry, \(W\) is the lowest non-zero component of the Cartan curvature and hence by general results has values in an irreducible subrepresentation which cannot meet this center. Hence the curvature of the Weyl connection is only induced by \(\partial s^*\mbox{\textsf{P}}\), and the component of this in \(\mathfrak z(\mathfrak g_0)\) is immediately seen to be the skew part of the Rho tensor up to a non-zero factor.
This result nicely corresponds to the fact that for the canonical symplectic structure on any cotangent bundle \(T^*N\), the image of a one-form \(\alpha\in\Omega^1(N)\) in \(T^*N\) is a Lagrangian submanifold if and only if \(d\alpha=0\).
(2) In the case of an AHS-structure, the cotangent bundle \(T^*M\) coincides with the associated graded bundle, so Proposition 2.3 shows that a Weyl structure \(s\) determines a diffeomorphism \(\varphi_s:T^*M\to A\). Now we can use this to pull back the geometric structures on \(A\) to \(T^*M\) and in particular, in the torsion-free case, compare the pullback of the symplectic form \(\Omega\in\Omega^2(A)\) to the canonical symplectic structure on \(T^*M\). Recall that the diffeomorphism \(\varphi_s\) is induced by \(\Phi_s:\mathcal G_0\times\mathfrak p_+\to\mathcal G\), \(\Phi_s(u_0,Z)=\sigma(u_0)\cdot\exp(Z)\), where \(\sigma:\mathcal G_0\to\mathcal G\) is the equivariant section determined by \(s\).
Equivariancy of the Cartan connection \(\omega\in\Omega^1(\mathcal G,\mathfrak g)\) then implies that the pullback \(\Phi_s^*\omega\) can be easily expressed explicitly in terms of \(\sigma^*\omega\). Denoting by \(q:\mathcal G_0\times\mathfrak p_+\to T^*M\) the canonical projection, the definition of \(\Omega\) in 3.1 shows that \(q^*\varphi_s^*\Omega=\Phi_s^*\Omega\) sends tangent vectors \(\xi,\eta\) to the alternation of the pairing between \((\Phi_s^*\omega)_-(\xi)\in\mathfrak g_{-1}\) and \((\Phi_s^*\omega)_+(\eta)\in\mathfrak g_1\). On the other hand, it is easy to explicitly describe \(q^*\alpha\in\Omega^1(\mathcal G_0\times\mathfrak p_+)\), where \(\alpha\in\Omega^1(T^*M)\) is the canonical one-form. From this, one can explicitly compute the pullback \(-q^*d\alpha\) of the canonical symplectic form on \(T^*M\) and show that it equals the sum of \(\Phi_s^*\Omega\) and the pullback of the alternation of the Rho-tensor. In particular, generalizing a result from [28] in the projective case, we conclude that \(\varphi_s:T^*M\to A\) is a symplectomorphism if and only if the Weyl-structure \(s\) is Lagrangian. Indeed, it turns out that also the split-signature metric \(\varphi_s^*h\) on \(T^*M\) can be computed explicitly in terms of the underlying AHS-structure. All this will be taken up in more detail elsewhere.
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.
Let \((p:\mathcal G\to M,\omega)\) be a torsion-free AHS structure and \(\pi:A\to M\) its bundle of Weyl structures. Let \(s:M\to A\) be a smooth section corresponding to a Lagrangian Weyl structure, let \(\nabla^s\) denote the corresponding Weyl connections and \(\mbox{\textsf{P}}^s\) the corresponding Rho-tensor. Then the following conditions are equivalent:
(1) The submanifold \(s(M)\subset A\) is totally geodesic for the canonical connection \(D\).
(2) The submanifold \(s(M)\subset A\) is totally geodesic for the Levi-Civita connection of \(h\).
(3) \(\nabla^s\mbox{\textsf{P}}^s=0\)
Proof. We will use abstract index notation to carry out the computations and denote the Rho tensor of \(s\) just by \(\mbox{\textsf{P}}\), so this has the form \(\mbox{\textsf{P}}_{ij}\) and is symmetric by assumption. Since for each \(y\in A\) and \(x:=\pi(y)\in M\), we can identify \(L^-_y\) with \(T_xM\) and \(L^+_y\) with \(T^*_xM\), we can use the index notation also on \(A\), but here tangent vectors have the form \((\xi^i,\alpha_j)\). In this language the proof of Proposition 3.10 shows that for \(x\in M\) the tangent space \(T_{s(x)}s(M)\) consists of all pairs of the form \((\xi^i,\mbox{\textsf{P}}_{ja}\xi^a)\). The condition that \(s(M)\) is totally geodesic with respect to \(D\) means that for vector field \((\xi,\alpha)\) on \(A\) that is tangent to \(s(M)\) along \(s(M)\), also the covariant derivative in directions tangent to \(s(M)\) is tangent to \(s(M)\).
In particular, for a vector field \(\eta\in\mathfrak X(M)\), we know from above that \((\widetilde{\eta^j},\widetilde{\mbox{\textsf{P}}_{kb}\eta^b})\in\mathfrak X(A)\) is tangent to \(s(M)\) along \(s(M)\). Since these fields are parallel for \(D^+\), we see that \(s(M)\) is totally geodesic for \(D\) if and only if all derivatives with respect to \(D\) of that field are tangent to \(s(M)\) along \(s(M)\). But this can be checked by pulling back the components of \(D(\widetilde{\eta^j},\widetilde{\mbox{\textsf{P}}_{kb}\eta^b})\) along \(s\), which by Theorem 2.7 leads to \(\nabla^s_i\eta^j\) and \[\tag{3.4} \nabla^s_i\mbox{\textsf{P}}_{ka}\eta^a=\eta^a\nabla^s_i\mbox{\textsf{P}}_{ka}+\mbox{\textsf{P}}_{ka}\nabla^s_i\eta^a,\] respectively. So evidently, the result is tangent to \(s(M)\) if and only if \(\eta^a\nabla^s_i\mbox{\textsf{P}}_{ka}=0\) and since this has to hold for each \(\eta\), we conclude that (1) is equivalent to (3).
To deal with (2), we use the information on the contorsion tensor \(C\) from 3.3. As observed in the proof of Theorem 3.5, the only non-zero component of \(C\) maps \(L^-\times L^-\) to \(L^+\). This means that \((\widetilde{\eta^j},\widetilde{\mbox{\textsf{P}}_{kb}\eta^b})\) is also parallel in \(L^+\)-directions for the Levi-Civita connection, so as above, we can use the pull back of the full derivative along \(s\) and the result has to be tangent to \(s(M)\). We write the pullback of \(C\) along \(s\) as \(C_{ijk}\) using the convention that \(C(\xi,\eta)_k=\xi^i\eta^jC_{ijk}\). Now formula (3.2) from 3.3 expresses \(C\) in terms of the torsion \(\tau\) of \(D\) (with \(h\) just playing the role of identifying \(L^+\) with the dual of \(L^-\)) and we know that the torsion corresponds to the Cartan curvature quantity \(Y\), see Theorem 2.12. Writing the pullback of this along \(s\) in abstract index notion as \(Y_{ijk}\), we conclude form formula (3.2) that \(C_{ijk}=\tfrac12(-Y_{ijk}+Y_{ikj}+Y_{jki})\). The pullback of the derivative along \(s\) again has first component \(\nabla^s_i\eta^j\) but for the second component, we have to add \(C_{iak}\eta^a\) to the right hand side of (3.4).
But it is a well known fact (see Theorem 5.2.3 of [16]) that the pullback of \(Y\) along \(s\) is given by the covariant exterior derivative of the Rho-tensor of \(s\) and since \(\nabla^s\) is torsion-free, this is expressed in abstract index notation as \(Y_{ijk}=\nabla^s_i\mbox{\textsf{P}}_{jk}-\nabla^s_j\mbox{\textsf{P}}_{ik}\). Inserting this into the formula for \(C_{ijk}\), we immediately conclude that we have to add \(\eta^a\nabla^s_a\mbox{\textsf{P}}_{ik}-\eta^a\nabla^s_k\mbox{\textsf{P}}_{ia}\) to (3.4). As above, this implies that (2) is equivalent to \[\tag{3.5} \nabla^s_i\mbox{\textsf{P}}_{ka}+\nabla^s_a\mbox{\textsf{P}}_{ik}-\nabla^s_k\mbox{\textsf{P}}_{ia}=0.\] Of course, (3.5) is satisfied if \(\nabla^s\mbox{\textsf{P}}=0\). Conversely, if (3.5) holds, then summing over all cyclic permutations of the indices shows that the total symmetrization of \(\nabla^s\mbox{\textsf{P}}\) has to vanish. But subtracting three times this total symmetrization from the left hand side of (3.5), one obtains \(-2\nabla^s_k\mbox{\textsf{P}}_{ia}\), so this has to vanish, too.
The result of Theorem 3.12 is particularly interesting if \(s\) is non-degenerate. By Proposition 3.10 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 3.12 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 3.12.
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 3.12, 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 \(\mathrm{II}^s_h\).
Let \((p:\mathcal G\to M,\omega)\) be a torsion-free AHS-structure with bundle of Weyl structures \(\pi:A\to M\). Let \(s:M\to A\) be a non-degenerate, Lagrangian Weyl structure with Weyl connection \(\nabla^s\) and Rho-tensor \(\mbox{\textsf{P}}\in\Gamma(S^2T^*M)\), then \[\mathrm{II}^s_{D}=-\tfrac12 \mbox{\textsf{P}}^{ka}\nabla^s_i\mbox{\textsf{P}}_{ja} \qquad\text{and}\qquad \mathrm{II}^s_{h}=-\tfrac12 \mbox{\textsf{P}}^{ka}(\nabla^s_i\mbox{\textsf{P}}_{ja}+\nabla^s_j\mbox{\textsf{P}}_{ia}-\nabla^s_a\mbox{\textsf{P}}_{ij}).\]
Proof. We only have to project the derivatives computed in the proof of Theorem 3.12 to the normal space. From the description of tangent and normal spaces, it follows readily that projecting \((\xi^i,\alpha_j)\) to the normal space and taking the \(L^-\)-component of the result, one obtains \(\tfrac12(\xi^i-\mbox{\textsf{P}}^{ia}\alpha_a)\). Using this, the formulae follow directly from the the proof of Theorem 3.12.