2 2-Segre structures
2.1 Definitions and examples
Let \(m,n \geq 2\) be integers. A vector \(v \in \mathbb{R}^m\otimes \mathbb{R}^n\) is called decomposable or simple if there exists \(x \in \mathbb{R}^m\) and \(y \in \mathbb{R}^n\) such that \(v=x\otimes y\). Let \(X=(X_\alpha)\) be linear coordinates on \(\mathbb{R}^m\) and \(Y=(Y^k)\) on \(\mathbb{R}^n\). Writing \(Z^k_{\alpha}=X_{\alpha}\otimes Y^k\), the set of simple vectors in \(\mathbb{R}^m\otimes \mathbb{R}^n\) is the zero locus of the the homogeneous quadratic equations \[\tag{2.1} Z^i_{\alpha}Z^k_{\beta}-Z^i_{\beta}Z^k_{\alpha}=0\] and thus is a cone. A subset \(\mathcal{S}\) in a real vector space \(V\) is called an \((m,n)\)-Segre cone, if there exists an isomorphism \(V \simeq \mathbb{R}^m\otimes \mathbb{R}^n\) which yields a bijection between \(\mathcal{S}\) and the cone of simple vectors in \(\mathbb{R}^m\otimes \mathbb{R}^n\).
Let \(\mathcal{S}\subset V\) be a Segre cone. Clearly, the isomorphism \(V \simeq \mathbb{R}^m\otimes\mathbb{R}^n\) is unique up to composition with an element of the group \(G(m,n)\), the subgroup of \(\mathrm{GL}(\mathbb{R}^m\otimes\mathbb{R}^n)\) consisting of maps preserving the cone of simple vectors. Let \(H(m,n)=\mathrm{GL}(m,\mathbb{R})\otimes \mathrm{GL}(n,\mathbb{R})\) and \(\mathbb{Z}_2\subset G(n,n)\) be the subgroup generated by the involution \(x\otimes y \mapsto y \otimes x\) for \(x,y \in \mathbb{R}^n\). Then we have an isomorphism of Lie groups1 \[\tag{2.2} G(m,n)\simeq\left\{\begin{array}{cc} H(m,n) & n \neq m,\\ H(n,n)\rtimes \mathbb{Z}_2 & n=m.\\ \end{array}\right.\]
An \((m,n)\)-Segre structure \(\mathcal{S}\) on a smooth \(mn\)-manifold \(M\) is a choice of an \((m,n)\)-Segre cone \(\mathcal{S}_p\subset T_pM\) in each tangent space of \(M\) which varies smoothly from point to point.
An isomorphism \(f : T_pM \to \mathbb{R}^m\otimes\mathbb{R}^n\) will be called a Segre coframe at \(p\) if it maps \(\mathcal{S}_p\) to the cone of simple vectors in \(\mathbb{R}^m\otimes\mathbb{R}^n\).
The set of Segre coframes at \(p\) will be denoted by \((F_\mathcal{S})_p\) and is the fibre of a principal right \(G(m,n)\)-bundle \(\pi : F_\mathcal{S}\to M\), with right action given by \(R_g(f)=g^{-1}\circ f\) for \(g \in G(m,n)\). The tautological \(1\)-form \(\zeta\) is defined by requiring that \(\zeta_f=f\circ \pi^{\prime}_f : T_fF_\mathcal{S}\to \mathbb{R}^m\otimes\mathbb{R}^n\) for \(f \in F_\mathcal{S}\). It satisfies \(R_g^*\zeta=g^{-1}\circ \zeta\) for \(g \in G(m,n)\).
A linear subspace \(\Pi\subset\mathcal{S}_p\) is called simple. The simple linear subspaces which are of the form \(\Pi \simeq \mathbb{R}^m\otimes y\) for some \(y \in \mathbb{R}^n\) are called \(\alpha\)-planes and the simple linear subspaces which are of the form \(\Pi \simeq x \otimes \mathbb{R}^n\) for some \(x \in \mathbb{R}^m\) are called \(\beta\)-planes.2 Note that \(\alpha\)- and \(\beta\)-planes are not well defined for \(n=m\) unless the Segre structure has been reduced to an \(H(n,n)\)-structure. An immersed connected manifold \(\Sigma \to M\) for which \(T_p\Sigma\) is a \(\beta\)-plane for every point \(p \in \Sigma\) is called a proto \(\beta\)-surface. If, in addition, \(\Sigma \to M\) is maximal in the sense of inclusion, then \(\Sigma\) is called a \(\beta\)-surface. A Segre structure \(\mathcal{S}\) is called \(\beta\)-integrable if every \(\beta\)-plane \(\Pi\) is tangent to a unique \(\beta\)-surface \(\Sigma \to M\). The notion of a (proto) \(\alpha\)-surface and \(\alpha\)-integrability are defined analogously. The necessary and sufficient conditions for a Segre structure of type \((m,n)\) to be \(\alpha\)- or \(\beta\)-integrable were given in [2, 16] (see also [3] for the complex case).
Recall that a pseudo-Riemannian metric \(g\) on a smooth \(4\)-manifold \(M\) with signature \((+,+,-,-)\) is said to have split-signature. Locally \(g\) may be written as \[\tag{2.3} g=\varepsilon^1_1\odot \varepsilon^2_2-\varepsilon^1_2 \odot \varepsilon^2_1\] for some linearly independent \(1\)-forms \(\varepsilon^i_j\). A vector \(v \in TM\) is called null if \(g(v,v)=0\). It follows with (2.3) that the \(g\)-null vectors give rise to a \((2,\! 2)\)-Segre structure on \(M\) and conversely it can be shown that every \((2,\! 2)\)-Segre structure on a \(4\)-manifold \(M\) gives rise to a unique conformal structure of split-signature on \(M\).
Closely related to Segre structures is the notion of an almost Grassmann structure.
A smooth \(mn\)-manifold \(M\) is said to carry an almost Grassmann structure if there exist smooth vector bundles \(E_m\to M\) and \(E_n \to M\) of rank \(m,n\) respectively together with an isomorphism \(TM \simeq E_m\otimes E_n\).
Clearly, an almost Grassmann structure on \(M\) induces a unique Segre structure on \(M\), but the existence of a Segre structure \(\mathcal{S}\) on \(M\) is in general not sufficient for the existence of an almost Grassmann structure inducing \(\mathcal{S}\). The two definitions are however equivalent when \(m+n\) is odd (see the appendix).
The prototypical example of a manifold carrying an almost Grassmann structure is the Grassmannian of \(m\)-planes in \(\mathbb{R}^{m+n}\) (see for instance [16] for details). We will construct the associated Segre structure in ยง4 for the case \(m=2\).
2.2 The structure equations of a 2-Segre structure
We will henceforth restrict our attention to \(H(2,n)\)-structures \(\pi : F_\mathcal{S}\to M\) and simply speak of \(2\)-Segre structures, thus implicitly assuming that \((2,2)\)-Segre structures have been reduced to \(H(2,2)\)-structures. We think of \(H(2,n)\) as a subgroup of \(\mathrm{GL}(2n,\mathbb{R})\) via the Kronecker product and consequently of \(\pi : F_\mathcal{S}\to M\) as a reduction with structure group \(H(2,n)\) of the full coframe bundle \(F\to M\) whose fibre at \(p \in M\) consists of all isomorphisms \(T_pM \to \mathbb{R}^{2n}\).
A linear connection \(\theta\) on \(F \to M\) is said to be adapted to the \(2\)-Segre structure \(\pi : F_\mathcal{S}\to M\) if \(\theta\) pulls-back to \(F_\mathcal{S}\) to become a principal \(H(2,n)\)-connection. A \(2\)-Segre structure is called torsion-free, if it admits an adapted connection \(\theta\) with vanishing torsion \(\tau=\mathrm{d}\zeta + \theta \wedge \zeta\).
Write \(H=H(2,n)\) and \(\mathfrak{h}\subset \mathfrak{gl}(2,\mathbb{R})\otimes \mathfrak{gl}(n,\mathbb{R})\) for the Lie algebra of \(H\). For computational purposes it is convenient to introduce the matrices \[a=\left(\begin{array}{rr} 0&-1\\1 & 0\end{array}\right), \quad b_1=\left(\begin{array}{rr} 0&1\\0 & 0\end{array}\right), \quad b_2=\left(\begin{array}{rr} 0&0\\0 & 1\end{array}\right),\] and write \(e^i_k\) for the \((n\times n)\)-matrix whose entry is \(1\) at the position \((k,i)\) and \(0\) otherwise. Using this notation an \(\mathfrak{h}\)-basis is given by \[a\otimes \mathrm{I}_n, \quad b_1\otimes \mathrm{I}_n, \quad b_2 \otimes \mathrm{I}_n, \quad \mathrm{I}_2 \otimes e^i_k,\] and a principal \(H\)-connection \(\theta\) on \(F_\mathcal{S}\) may be written as \[\tag{2.4} \theta=\chi\otimes\mathrm{I}_n+\mathrm{I}_2\otimes \phi\] with \(\chi=\omega a + 2\xi_1b_1+2\xi_2b_2\) and \(\phi=\phi^i_ke^k_i\) for some linearly independent \(1\)-forms \(\omega,\xi_1,\xi_2,\phi^i_k\) on \(F_\mathcal{S}\). Let \(\xi=\xi_1+\mathrm{i}\xi_2\). Straightforward computations show that we may linearly identify \(\mathbb{R}^{2n}\) with \(\mathbb{C}^n\) in such a way that we can write the first structure equation in complex form:
The connection form \((\omega,\xi,\phi)\) of an \(F_\mathcal{S}\)-adapted connection \(\theta\) satisfies \[\tag{2.5} \mathrm{d}\zeta=-\big(\mathrm{i}\left(\omega-\xi\right)\mathrm{I}_n+\phi\big)\wedge\zeta-\mathrm{i}\,\xi\,\mathrm{I}_n \wedge \bar{\zeta}+\tau.\]
Proof. Omitted.
Here the forms \(\zeta\) and \(\tau\) are thought to be \(\mathbb{C}^n\)-valued and \(\bar{\zeta}\) denotes the \(\mathbb{C}^n\)-valued \(1\)-form on \(F_\mathcal{S}\) which is obtained by complex conjugation of the entries of \(\zeta\).
We have the curvature forms \[\tag{2.6} \begin{align} \Omega&=\mathrm{d}\omega+\omega\wedge\mathrm{i}\,(\xi-\bar{\xi}),\\ \Xi&=\mathrm{d}\xi+\xi\wedge\mathrm{i}\left(\bar{\xi}-2\omega\right),\\ \Phi&=\mathrm{d}\phi+\phi\wedge\phi-\omega\wedge\left(\xi+\bar{\xi}\right)\mathrm{I}_n. \end{align}\] Differentiating the structure equation (2.5) gives the Bianchi-identity \[\tag{2.7} \mathrm{d}\tau=\big(\mathrm{i}\left(\Omega-\Xi\right)\mathrm{I}_n+\Phi\big)\wedge\zeta+\mathrm{i}\,\Xi\,\mathrm{I}_n\wedge\bar{\zeta}.\]
2.3 A quasiholomorphic fibre bundle
Let \(P=S^1\cdot \,\mathrm{GL}(n,\mathbb{R})\subset H(2,n)\) be the closed subgroup consisting of elements of the form \[e^{\mathrm{i}\varphi}\cdot b=\left(\begin{array}{rr}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right)\otimes b\] for some real number \(\varphi\) and \(b\in \mathrm{GL}(n,\mathbb{R})\). Equip the quotient \(X_\mathcal{S}=F_\mathcal{S}/P\) with its canonical smooth structure so that the quotient projection \(\nu : F_\mathcal{S}\to X_\mathcal{S}\) is a smooth surjective submersion. Note that the \(1\)-forms \(\eta^k\) are \(\nu\)-semibasic since they are \(\pi\)-semibasic. Moreover since \(\theta=(\omega,\xi,\phi)\) is a principal \(H\)-connection, it follows with (2.4) that \(\xi\) is \(\nu\)-semibasic as well. Therefore the forms \(\eta^k\) together with \(\xi_1\) and \(\xi_2\) span the \(\nu\)-semibasic \(1\)-forms on \(F_\mathcal{S}\).
Let \(\pi : F_\mathcal{S}\to M^{2n}\) be a \(2\)-Segre structure and \(\theta=(\omega,\xi,\phi)\) an adapted connection. Then there exists a unique almost complex structure \(\mathfrak{J}\) on \(X_\mathcal{S}\) such that a complex-valued \(1\)-form on \(X_\mathcal{S}\) is a \((1,\! 0)\)-form for \(\mathfrak{J}\) if and only if its \(\nu\)-pullback is a linear combination of \(\left\{\zeta^1,\ldots,\zeta^n,\xi\right\}\) with coefficients in \(C^{\infty}(F_\mathcal{S},\mathbb{C})\).
Proof. Denote by \(\frac{\partial}{\partial\eta^l},\; \frac{\partial}{\partial \xi_1}, \; \frac{\partial}{\partial \xi_2}, \; \frac{\partial}{\partial \omega}, \; \frac{\partial}{\partial \phi^i_k},\) the vector fields dual to the coframing \((\eta^l,\xi_1,\xi_2,\omega,\phi^i_k)\). Define the map \(\tilde{\mathfrak{J}} : TF_\mathcal{S}\to TX_\mathcal{S}\) by \[\tilde{\mathfrak{J}}(v)=\nu^{\prime}\left(-\eta^{2k}(v)\frac{\partial}{\partial \eta^{2k-1}}+\eta^{2k-1}(v)\frac{\partial}{\partial \eta^{2k}}-\xi_2(v)\frac{\partial}{\partial \xi_1}+\xi_1(v)\frac{\partial}{\partial \xi_2}\right).\] The \(1\)-forms \(\eta^l,\xi_1,\xi_2\) are \(\nu\)-semibasic and thus we have \(\tilde{\mathfrak{J}}(v+w)=\tilde{\mathfrak{J}}(v)\) for every \(v \in TF_\mathcal{S}\) and \(w \in TF_\mathcal{S}\) tangent to the \(\nu\)-fibres. Since \(\theta\) is a principal \(H\)-connection, the equivariance \((R_h)^*\theta=h^{-1}\theta h\) for \(h \in H\) together with a short computation gives
\[\tag{2.8} \left(R_{e^{\mathrm{i}\varphi} \cdot b}\right)^*\xi=e^{-2\mathrm{i}\varphi} \xi,\] for \(e^{\mathrm{i}\varphi}\cdot b \in S^1\cdot \mathrm{GL}(n,\mathbb{R})\). Moreover we have \[\tag{2.9} (R_h)^*\eta=h^{-1}\eta\] for every \(h \in H\). Identifying \(\mathrm{GL}(n,\mathbb{C})\) with the subgroup of \(\mathrm{GL}(2n,\mathbb{R})\) commuting with \(a \otimes \mathrm{I}_n\) and using the fact that \(S^1\cdot \mathrm{GL}(n,\mathbb{R}) \subset \mathrm{GL}(n,\mathbb{C})\) together with (2.8), (2.9) implies \(\tilde{\mathfrak{J}}\circ\left (R_{e^{\mathrm{i}\alpha} \cdot b}\right)^{\prime}=\tilde{\mathfrak{J}}.\) In other words there exists an almost complex structure \(\mathfrak{J} : TX_\mathcal{S}\to TX_\mathcal{S}\) such that \(\tilde{\mathfrak{J}}= \mathfrak{J} \circ \nu^{\prime}\). Clearly \(\mathfrak{J}\) has the desired properties and these properties uniquely characterise \(\mathfrak{J}\).
It is natural to ask when two \(F_\mathcal{S}\)-adapted connections induce the same almost complex structure. We have:
The \(F_\mathcal{S}\)-adapted connections \(\theta=(\phi,\omega,\xi)\) and \(\theta^{\prime}=(\phi^{\prime},\omega^{\prime},\xi^{\prime})\) induce the same almost complex structure on \(X_\mathcal{S}\) if and only if \(\xi - \xi^{\prime}=\lambda_k\zeta^k\) for some smooth functions \(\lambda_k : F_\mathcal{S}\to \mathbb{C}\). In particular any two \(F_\mathcal{S}\)-adapted connections with the same torsion induce the same almost complex structure.
Proof. Let \(\mathfrak{J}_{\theta}\), \(\mathfrak{J}_{\theta^{\prime}}\) denote the almost complex structures with respect to the connections \(\theta\), \(\theta^{\prime}\) and suppose \(\xi^{\prime}=\xi+\lambda_k\zeta^k\) for some smooth functions \(\lambda_k : F_\mathcal{S}\to \mathbb{C}\). Let \(\alpha\) be a \((1,\!0)\)-form for \(\mathfrak{J}_{\theta}\). Then we may write \[\nu^*\alpha=a_k \zeta^k+a\xi=a_k\zeta^k+a\left(\xi^{\prime}-\lambda_k\zeta^k\right)=\left(a_k-\lambda_k\right)\zeta^k+a\xi^{\prime}\] for some smooth functions \(a,a_k : F_\mathcal{S}\to \mathbb{C}\), thus showing that \(\alpha\) is a \((1,\! 0)\)-form for \(\mathfrak{J}_{\theta^{\prime}}\). Conversely suppose \(\mathfrak{J}_{\theta}=\mathfrak{J}_{\theta^{\prime}}\). Note that \(\xi-\xi^{\prime}\) is \(\pi\)-semibasic and may thus be written as \[\xi-\xi^{\prime}=\lambda_k\zeta^k+\lambda_k^{\prime}\bar{\zeta}^k\] for some smooth functions \(\lambda_k,\lambda_k^{\prime}: F_\mathcal{S}\to \mathbb{C}\). Let \(\alpha\) be a \((1,\!0)\)-form for \(\mathfrak{J}_{\theta}\). Then we may write \[\nu^*\alpha=a_k\zeta^k+a\xi=a_k^{\prime}\zeta^k+a^{\prime}\xi^{\prime}=a_k^{\prime}\zeta^k+a^{\prime}\left(\xi-\lambda_k\zeta^k-\lambda_k^{\prime}\bar{\zeta}^k\right)\] for some smooth functions \(a,a^{\prime},a_k,a_k^{\prime} : F_\mathcal{S}\to \mathbb{C}\). Thus it follows \[(a-a^{\prime})\xi+\left(a_k-a_k^{\prime}+a^{\prime}\lambda_k\right)\zeta^k+a^{\prime}\lambda_k^{\prime}\bar{\zeta}^k=0\] which can hold for an arbitrary \((1,\!0)\)-form \(\alpha\) if and only if \(\lambda_k^{\prime}=0\). Finally it is easy to check that if \((\omega,\xi,\phi)\) and \((\omega^{\prime},\xi^{\prime},\phi^{\prime})\) are two \(F_\mathcal{S}\)-adapted connections with the same torsion, then there exist \(n\) smooth complex-valued functions \(a_k\) on \(F_\mathcal{S}\) such that \[\tag{2.10} \begin{align} \omega^{\prime}-\omega&=\operatorname{Re}(a_k)\operatorname{Im}(\zeta^k),\\ \xi^{\prime}-\xi&=\frac{1}{2\mathrm{i}}\bar{a}_k\zeta^k,\\ \left(\phi^{\prime}\right)^i_k-\phi^i_k&=\operatorname{Re}(a_k \zeta^i )+\delta^i_k\operatorname{Re}(a_l)\operatorname{Re}(\zeta^l).\\ \end{align}\]
Denote by \(\mathcal{A}^{1,0}_{\mathcal{S}}\) and \(\mathcal{A}^{0,1}_{\mathcal{S}}\) the complex-valued \(\pi\)-semibasic \(1\)-forms on \(F_\mathcal{S}\) which can be written as \(a_k \zeta^k\) and \(a_k \bar{\zeta}^k\) respectively. Here \(a_k\) are smooth complex-valued functions on \(F_\mathcal{S}\). Furthermore set \[\mathcal{A}^{p,q}_{\mathcal{S}}=\Lambda^p\left(\mathcal{A}^{1,0}_{\mathcal{S}}\right)\otimes \Lambda^q \left(\mathcal{A}^{0,1}_{\mathcal{S}}\right),\] so that the complex-valued \(\pi\)-semibasic \(k\)-forms \(\mathcal{A}^k_{\mathcal{S}}\) on \(F_\mathcal{S}\) decompose as \[\mathcal{A}^k_{\mathcal{S}}=\bigoplus_{p+q=k}\mathcal{A}^{p,q}_{\mathcal{S}}.\]
The almost complex structure \(\mathfrak{J}\) is integrable if and only if \(\Xi\) and the torsion components \(\tau^i\) lie in \(\mathcal{A}^{2,0}_{\mathcal{S}}\oplus \mathcal{A}^{1,1}_{\mathcal{S}}\). In particular for \(n \geq 3\) every torsion-free \(F_\mathcal{S}\)-adapted connection gives rise to an integrable \(\mathfrak{J}\).
The integrability conditions for the almost complex structure \(\mathfrak{J}\) can also be obtained by applying [18]. We we will instead use Proposition 2.6.
Proof of Proposition 2.9. Using the characterisation of \(\mathfrak{J}\) provided in Lemma 2.7, the first statement is an immediate consequence of the structure equation (2.5), the definition of the curvature form \(\Xi\) in (2.6), and the Newlander-Nirenberg theorem. In order to prove the second statement we need to show that for \(n\geq 3\) the condition \(\tau=0\) implies \(\Xi \in \mathcal{A}^{2,0}_{\mathcal{S}}\oplus \mathcal{A}^{1,1}_{\mathcal{S}}\). Since the curvature form \(\Xi\) is a \(\pi\)-semibasic \(2\)-form we may write \[\tag{2.11} \Xi=x_{kl}\zeta^k\wedge\zeta^l+\tilde{x}_{kl}\bar{\zeta}^k\wedge\zeta^l+\hat{x}_{kl}\bar{\zeta}^k\wedge\bar{\zeta}^l\] for some smooth complex-valued functions \(x_{kl},\tilde{x}_{kl},\hat{x}_{kl}\) on \(F_\mathcal{S}\). Writing out the Bianchi-identity (2.7) in components for \(\tau=0\) gives \[0=(\mathrm{i}(\Omega-\Xi)\delta^i_k+\Phi^i_k)\wedge\zeta^k+\mathrm{i}\,\Xi\,\wedge\bar{\zeta}^i,\] replacing \(\Xi\) with the expansion (2.11) we get \[0=\cdots+\mathrm{i}\hat{x}_{kl}\bar{\zeta}^k\wedge\bar{\zeta}^l\wedge\bar{\zeta}^i\] where the unwritten summands do not contain forms in \(\mathcal{A}^{0,3}_{\mathcal{S}}\). If \(n\geq 3\) there is for every choice of indices \(k,l\) an index \(i \neq k\), \(i \neq l\) so that the Bianchi-identity can hold if and only if \(\hat{x}_{kl}=0\) which is equivalent to \(\Xi\) lying in \(\mathcal{A}^{2,0}_{\mathcal{S}}\oplus \mathcal{A}^{1,1}_{\mathcal{S}}\).
Recall that \(H(2,2)\)-structures \(\pi : F_\mathcal{S}\to M\) correspond to oriented conformal structures of split-signature and thus are always torsion-free. In fact, the logical value of the curvature condition \(\Xi \in \mathcal{A}^{2,0}_{\mathcal{S}}\oplus \mathcal{A}^{1,1}_{\mathcal{S}}\) does not depend on the choice of a particular adapted torsion-free connection, but only on \(F_\mathcal{S}\). We leave it to the reader to check that this curvature condition corresponds to self-duality3 of the associated oriented conformal \(4\)-manifold of split-signature.
In fact, it can be shown that for \(n=2\) the almost complex structure \(\mathfrak{J}\) is integrable if and only if \(\theta\) is torsion-free and the associated split-signature conformal structure is self-dual. For \(n\geq 3\), the almost complex structure \(\mathfrak{J}\) is integrable if and only if \(\theta\) is torsion-free.
Suppose \(\mathfrak{J}\) is integrable, so that the total space of the bundle \(\rho : X_\mathcal{S}\to M\) is a complex \((n+1)\)-manifold. By construction the \(\rho\)-fibres are smoothly embedded submanifolds of \(X_\mathcal{S}\) diffeomorphic to \(\mathrm{GL}(2,\mathbb{R})/\mathrm{GL}(1,\mathbb{C})\). We will argue next, that \((X_\mathcal{S},\mathfrak{J})\) is a quasiholomorphic fibre bundle with fibre \(\mathbb{CP}^1\setminus \mathbb{RP}^1\).
Let \(\pi : B \to M\) be a fibre bundle with fibre \(F\) and \(\mathfrak{J}~\) an almost complex structure on \(B\). Then \((B,\mathfrak{J})\) is called quasiholomorphic if
the almost complex structure \(\mathfrak{J}\) is integrable,
there exists a complex structure \(I\) on \(F\),
each \(\pi\)-fibre \(B_p=\pi^{-1}(p)\) admits a complex structure with respect to which it is biholomorphic to \((F,I)\) and with respect to which the inclusion \(B_p \hookrightarrow B\) is a holomorphic embedding.
Pulling back \(\xi\) with a local section \(s\) of \(\nu : F_\mathcal{S}\to X_\mathcal{S}\) gives a complex-valued \(1\)-form which pulls back to the \(\rho\)-fibres to be non-vanishing and which depends on \(s\) only up to complex multiples. It follows that the fibres of \(\rho : X_\mathcal{S}\to M\) are holomorphically embedded Riemann surfaces with respect to the complex structure induced by \(\xi\). Using the Maurer-Cartan form of \(\mathrm{GL}(2,\mathbb{R})\), it is easy to see that fibres are biholomorphic to \(\mathbb{CP}^1\setminus\mathbb{RP}^1\). In [2, 16] it was shown that for \(n \geq 3\) a \(2\)-Segre structure is \(\beta\)-integrable if and only if it is torsion-free and for \(n=2\) if and only if it is self-dual. Summarising we have:
Let \(\pi : F_\mathcal{S}\to M\) be a \(\beta\)-integrable \(2\)-Segre structure. Then there exists a canonical almost complex structure \(\mathfrak{J}\) on \(X_\mathcal{S}\) so that \((X_\mathcal{S},\mathfrak{J})\) is a quasiholomorphic fibre bundle with fibre \(\mathbb{CP}^1\setminus \mathbb{RP}^1\).
Proof. We pick an \(F_\mathcal{S}\)-adapted connection without torsion and let \(\mathfrak{J}\) be the associated almost complex structure on \(X_\mathcal{S}\) whose existence is guaranteed by Lemma 2.7. Then by Proposition 2.9 and the above remarks, the almost complex structure \(\mathfrak{J}\) is integrable and \((X_\mathcal{S},\mathfrak{J})\) is a quasiholomorphic fibre bundle with fibre \(\mathbb{CP}^1\setminus \mathbb{RP}^1\). Finally, by Lemma 2.8, any other \(F_\mathcal{S}\)-adapted torsion-free connection gives rise to the same almost complex structure \(\mathfrak{J}\).