4 Submanifold theory of Lagrangian connections

We now restrict attention to torsion-free connections on \(T\Sigma\) having symmetric Ricci tensor, so that the zero-section \(o : \Sigma \to T^*\Sigma\) is a Lagrangian submanifold. If furthermore \(\nabla\) is timelike/spacelike, we obtain an induced metric \(g=\mp o^*h_{\nabla}=\pm \mathrm{Ric}(\nabla)\) and the immersion \(o : \Sigma \to T^*\Sigma\) has a well-defined normal bundle and second fundamental form. In particular, we want to compute when \(o : \Sigma \to T^*\Sigma\) is minimal, that is, the trace with respect to \(g\) of its second fundamental form vanishes identically.

4.1 Algebraic preliminaries

Before we delve into the computations, we briefly review the relevant algebraic structure of the theory of oriented surfaces in an oriented Riemannian – and oriented split-signature Riemannian four manifold \((M,g)\). We refer the reader to [4] and [11] for additional details.

First, let \(X :\Sigma \to (M,g)\) be an immersion of an oriented surface \(\Sigma\) into an oriented Riemannian \(4\)-manifold. The bundle of orientation compatible \(g\)-orthonormal coframes of \((M,g)\) is an \(\mathrm{SO}(4)\)-bundle \(\pi : F^+_g \to M\). The Grassmannian \(G^+_2(\mathbb{R}^4)\) of oriented \(2\)-planes in \(\mathbb{R}^4\) is a homogeneous space for the natural action of \(\mathrm{SO}(4)\) and the stabiliser subgroup is \(\mathrm{SO}(2)\times \mathrm{SO}(2)\). Consequently, the pullback bundle \(X^*F^+_g \to \Sigma\) admits a reduction \(F_X\subset X^*F^+_g\) with structure group \(\mathrm{SO}(2)\times \mathrm{SO}(2)\), where the fibre of \(F_X\) at \(p \in \Sigma\) consists of those coframes mapping the oriented tangent plane to \(\Sigma\) at \(X(p)\) to some fixed oriented \(2\)-plane in \(\mathbb{R}^4\), while preserving the orientation.

The second fundamental form of \(X\) is a quadratic form on \(T\Sigma\) with values in the rank two normal bundle of \(X\). Therefore, it is represented by a map \(F_X \to S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) which is equivariant with respect to some suitable representation of \(\mathrm{SO}(2)\times \mathrm{SO}(2)\) on \(S^2(\mathbb{R}_2)\times \mathbb{R}^2\). The relevant representation is defined by the rule \[\varrho(r_{\alpha},r_{\beta})(A)(x,y)=r_{-\beta}A(r_{\alpha}x,r_{\alpha}y), \quad x,y \in \mathbb{R}^2,\] where \(A \in S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) is a symmetric bilinear form on \(\mathbb{R}^2\) with values in \(\mathbb{R}^2\) and \(r_{\alpha},r_{\beta}\) denote counter-clockwise rotations in \(\mathbb{R}^2\) by the angle \(\alpha,\beta\), respectively. As usual, we decompose the \(\mathrm{SO}(2)\times \mathrm{SO}(2)\)-module \(S^2(\mathbb{R}_2)\otimes \mathbb{R}^2\) into irreducible pieces. This yields \[\varrho=\varsigma_{0,-1}\oplus\varsigma_{2,1}\oplus\varsigma_{2,-1},\] where for \((n,m) \in \mathbb{Z}^2\) the complex one-dimensional \(\mathrm{SO}(2)\times \mathrm{SO}(2)\)-representation \(\varsigma_{n,m}\) is defined by the rule \[\varsigma_{n,m}(r_{\alpha},r_{\beta})=\mathrm{e}^{\mathrm{i}(n \alpha+m\beta)}.\] Explicitly, the relevant projections \(S^2(\mathbb{R}_2)\otimes \mathbb{R}^2 \to \mathbb{C}\) are \[\tag{4.1} p_{0,-1}(A)=\frac{1}{2}\left(A^1_{11}+A^1_{22}\right)+\frac{\mathrm{i}}{2}\left(A^2_{11}+A^2_{22}\right),\] \[\tag{4.2} p_{2,-1}(A)=\frac{1}{4}\left(A^1_{11}-A^1_{22}+2A^2_{12}\right)+\frac{\mathrm{i}}{4}\left(A^2_{11}-A^2_{22}-2A^1_{12}\right),\] \[\tag{4.3} p_{2,1}(A)=\frac{1}{4}\left(A^1_{22}-A^1_{11}+2A^2_{12}\right)+\frac{\mathrm{i}}{4}\left(A^2_{11}-A^2_{22}+2A^1_{12}\right)\] and where we write \(A(e_i,e_j)=A^k_{ij}e_k\) with respect to the standard basis \((e_1,e_2)\) of \(\mathbb{R}^2\).

The canonical bundle \(K_{\Sigma}\) of \(\Sigma\) with respect to the complex structure induced by the metric \(X^*g\) and orientation is the bundle associated to the representation \(\varsigma_{1,0}\). Moreover, the conormal bundle, thought of as a complex line bundle, is the bundle \(N^*_X\) associated to the representation \(\varsigma_{0,1}\). Consequently, the second fundamental form of \(X\) defines a section \(H\) of the normal bundle which is the mean curvature vector of \(X\), as well as a quadratic differential \(Q_+\)with values in the conormal bundle and a quadratic differential \(Q_-\) with values in the normal bundle. Consequently, we obtain a quartic differential \(Q_+Q_-\) on \(\Sigma\) which turns out to be holomorphic, provided \(X\) is minimal and \(g\) has constant sectional curvature, see [11].

If we instead consider a split-signature oriented Riemannian \(4\)-manifold \((M,g)\), the bundle of orientation compatible \(g\)-orthonormal coframes of \((M,g)\) is an \(\mathrm{SO}(2,2)\)-bundle \(\pi : F^+_g \to M\). Here, as usual, we take \(\mathrm{SO}(2,2)\) to be the subgroup of \(\mathrm{SL}(4,\mathbb{R})\) stabilising the quadratic form \[q(x)=(x_1)^2+(x_2)^2-(y_1)^2-(y_2)^2,\] where \((x,y) \in \mathbb{R}^{2,2}\). Now the action of \(\mathrm{SO}(2,2)\) on the Grassmannian \(G^+_2(\mathbb{R}^{2,2})\) of oriented \(2\)-planes in \(\mathbb{R}^{2,2}\) is not transitive, it is however transitive on the open submanifolds of oriented timelike/spacelike \(2\)-planes. In both cases, the stabiliser subgroup is \(\mathrm{SO}(2)\times\mathrm{SO}(2)\) as well. Therefore, the submanifold theory of a timelike/spacelike oriented surface in an oriented split-signature Riemannian four manifold is entirely analogous to the Riemannian case. In particular, we also encounter the mean curvature vector \(H\) and the quadratic differentials \(Q_{\pm}\).

4.2 The mean curvature form

Knowing what to expect, we now carry out the submanifold theory of timelike/spacelike Lagrangian connections. Note however, that in addition to the split-signature metric \(h_{\nabla}\), we also have a symplectic form \(\Omega_{\nabla}\). The symplectic form allows to identify the conormal bundle to a Lagrangian spacelike/timelike immersion with the cotangent bundle of \(\Sigma\). In particular we may think of the mean curvature vector \(H\) as a \(1\)-form, the conormal-bundle valued quadratic differential \(Q_+\) as a cubic differential and the normal-bundle valued quadratic differential \(Q_-\) as a \((1,\! 0)\)-form.

The product \(P:=F\times \mathbb{R}_2\) is a principal right \(\mathrm{GL}^+(2,\mathbb{R})\)-bundle over \(T^*\Sigma\), where the \(\mathrm{GL}^+(2,\mathbb{R})\)-right action is given by \((f,\xi)\cdot a=(a^{-1}\circ f,\xi a)\) for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\) and \((f,\xi) \in P\). We define two \(\mathbb{R}^2\)-valued \(1\)-forms on \(P\) \[\rho:=\frac{1}{2}\left(\psi^t+\omega\right)\quad \text{and}\quad \zeta:=\frac{1}{2}\left(\psi^t-\omega\right),\] so that the metric \(h_{\nabla}\) satisfies \[\pi^*h_{\nabla}=\rho^t\rho-\zeta^t\zeta=(\rho^1)^2+(\rho^2)^2-(\zeta^1)^2-(\zeta^2)^2.\] From the equivariance properties (2.7) and (2.8) of \(\psi\) and \(\omega\), we compute \[\tag{4.4} R_a^*\begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\frac{1}{2}\begin{pmatrix} a^t+a^{-1} & a^t-a^{-1} \\ a^t-a^{-1} & a^t+a^{-1}\end{pmatrix}\begin{pmatrix} \rho \\ \zeta \end{pmatrix}.\]

Recall that the Lie algebra of the split-orthogonal group \(\mathrm{O}(2,2)\) consists of matrices of the form \[\begin{pmatrix} \mu & \nu \\ \nu^t & \vartheta \end{pmatrix}\] where \(\mu\) and \(\vartheta\) are skew-symmetric. Consequently, there exist unique \(\mathfrak{o}(2)\)-valued \(1\)-forms \(\mu,\vartheta\) on \(F\times \mathbb{R}_2\) and a unique \(\mathfrak{gl}(2,\mathbb{R})\)-valued \(1\)-form \(\nu\) on \(F\times \mathbb{R}_2\) such that \[\tag{4.5} \mathrm{d}\begin{pmatrix} \rho \\ \zeta\end{pmatrix}=-\begin{pmatrix} \mu & \nu \\ \nu^t & \vartheta \end{pmatrix}\wedge \begin{pmatrix} \rho \\ \zeta\end{pmatrix}.\] In order to compute these connection forms we first remark that since the function \(S=(S_{ij})\) represents the (symmetric) Ricci tensor of \(\nabla\), there must exist unique real-valued functions \(S_{ijk}=S_{jik}\) on \(F\) so that \[\mathrm{d}S_{ij}=S_{ijk}\omega^k+S_{ik}\theta^k_j+S_{kj}\theta^k_i.\] Clearly, the function \((S_{ijk}) : F \to S^2(\mathbb{R}_2)\otimes \mathbb{R}_2\) represents \(\nabla\, \mathrm{Ric}(\nabla)\) with \(k\) being the derivative index.

Recall that the components of \(\psi\) and \(\omega\) – and hence equivalently the components of \(\rho\) and \(\zeta\) – span the \(1\)-forms on \(P\) that are semi-basic for the projection \(\pi : P \to T^*\Sigma\). In particular, if \(\epsilon\) is a \(1\)-form on \(T^*\Sigma\), then there exists a unique map \((e_1,e_2) : P \to \mathbb{R}_{2,2}\) so that \(\pi^*\epsilon=e_1\rho+e_2\zeta\). Since \(\pi^*\epsilon\) is invariant under the \(\mathrm{GL}^+(2,\mathbb{R})\) right action, the function \((e_1,e_2)\) satisfies the equivariance property determined by (4.4). Phrased differently, the cotangent bundle of \(T^*\Sigma\) is the bundle associated to \(\pi : P\to T^*\Sigma\) via the representation \(\varrho : \mathrm{GL}^+(2,\mathbb{R}) \to \mathbb{R}_{2,2}\) defined by the rule \[\tag{4.6} \varrho(a)\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}=\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}\frac{1}{2}\begin{pmatrix} a^t+a^{-1} & a^t-a^{-1} \\ a^t-a^{-1} & a^t+a^{-1}\end{pmatrix}\] for all \(a \in \mathrm{GL}^+(2,\mathbb{R})\) and \((\xi_1,\xi_2)\) in \(\mathbb{R}_{2,2}\).

We will next use this fact to exhibit the conormal bundle of the immersion \(o : \Sigma \to T^*\Sigma\) as an associated bundle to a natural reduction of the pullback bundle \(o^*P\). Note that by construction, the pullback bundle \(o^*P \to \Sigma\) is just the frame bundle \(\upsilon : F \to \Sigma\) and that on \(o^*P\simeq F\) we have \(\psi^t=-S\omega\), thus \[\rho=\frac{1}{2}\left(\mathrm{I}_2-S\right)\omega \quad \text{and}\quad \zeta=-\frac{1}{2}\left(\mathrm{I}_2+S\right)\omega.\] If we assume that \(\nabla\) is timelike/spacelike, then the Ricci tensor of \(\nabla\) is positive/negative definite and hence the equations \(S=\pm \mathrm{I}_2\) define a reduction \(F_{\nabla} \to \Sigma\) with structure group \(\mathrm{SO}(2)\) whose basepoint projection we continue to denote by \(\upsilon\). Note that by construction, \(F_{\nabla} \to \Sigma\) is the bundle of orientation preserving orthonormal coframes of the induced metric \(g=\pm \mathrm{Ric}(\nabla)\). In particular, from (4.6) we see that the pullback bundle \(o^*(T^*(T^*\Sigma))\) is the bundle associated to \(\upsilon : F_{\nabla} \to \Sigma\) via the \(\mathrm{SO}(2)\)-representation on \(\mathbb{R}_{2,2}\) defined by the rule \[\tag{4.7} \varrho(a)\begin{pmatrix} \xi_1 & \xi_2\end{pmatrix}=\begin{pmatrix} \xi_1 a^t & \xi_2 a^t\end{pmatrix}\] for all \(a \in \mathrm{SO}(2)\) and \((\xi_1,\xi_2)\) in \(\mathbb{R}_{2,2}\). Furthermore, on \(F_{\nabla}\) we obtain \(\psi=\mp \omega^t\) and hence \[\tag{4.8} \begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\begin{pmatrix} 0\\ -\omega \end{pmatrix}\] in the timelike case and \[\tag{4.9} \begin{pmatrix} \rho \\ \zeta \end{pmatrix}=\begin{pmatrix} \omega\\ 0 \end{pmatrix}\] in the spacelike case. Recall that if \(\alpha\) is a \(1\)-form on \(\Sigma\) then there exists a unique \(\mathbb{R}_2\)-valued function \(a\) on \(F\) – and hence on \(F_{\nabla}\) as well – so that \(\upsilon^*\alpha=a\omega\). It follows as before that \(T^*\Sigma\) is the bundle associated to \(F_{\nabla}\) via the \(\mathrm{SO}(2)\)-representation defined by the rule \[\tag{4.10} \varrho(a)(\xi)=\xi a^t\] for all \(a \in \mathrm{SO}(2)\) and \(\xi \in \mathbb{R}_2\). Using (4.7), (4.8) and (4.9), we see that the conormal bundle \[N^*_o:=o^*(T^*(T^*\Sigma))/T^*\Sigma\] of \(o\) is (isomorphic to) the bundle associated to \(F_{\nabla}\) via the representation (4.10) as well. We thus have an isomorphism \(N^*_o\simeq T^*\Sigma\) between the conormal bundle of the immersion \(o : \Sigma \to T^*\Sigma\) and the cotangent bundle \(T^*\Sigma\). Of course, the metric \(g\) on \(\Sigma\) provides an isomorphism \(T^*\Sigma\simeq T\Sigma\) and hence \(N_o\simeq T^*\Sigma\), where \(N_o\) denotes the normal bundle of \(o\). The second fundamental form of \(o\) is a quadratic form on \(T\Sigma\) with values in the normal bundle, thus here naturally a section of \(S^2(T^*\Sigma)\otimes T^*\Sigma\).

Denoting by \(S^{ij}\) the functions on \(F\) representing the inverse of the Ricci curvature of \(\nabla\), so that \(S^{ij}S_{jk}=\delta^i_k\), we thus have:

Recall that a (non-degenerate) submanifold of a (pseudo-)Riemannian manifold is called totally geodesic if its second fundamental form vanishes identically. We call a timelike/spacelike connection \(\nabla\) totally geodesic if \(o : \Sigma \to (T^*\Sigma,h_{\nabla})\) is a totally geodesic submanifold. We also get:

4.3 Minimality and the Liouville curvature

The minimality condition for a timelike/spacelike Lagrangian connection can also be expressed in terms of the Liouville curvature of \(\nabla\). To this end we decompose the structure equation³ \[\tag{4.13} \mathrm{d}S_{ij}=S_{ijk}\omega^k+S_{ik}\theta^k_j+S_{kj}\theta^k_i\] into (we compute modulo \(\theta^i_j\)) \[\begin{aligned} \mathrm{d}S_{ij}&=\left(\frac{1}{3}(S_{ijk}+S_{ikj}+S_{jki})+\frac{2}{3}S_{ijk}-\frac{1}{3}(S_{ikj}+S_{jki})\right)\omega_k\\ &=\left(R_{ijk}+\frac{2}{3}L_{(i}\varepsilon_{j)k}\right)\omega^k,\end{aligned}\] where we define \[R_{ijk}=\frac{1}{3}\left(S_{ijk}+S_{ikj}+S_{jki}\right)\] and \[L_i=\frac{2}{3}\varepsilon^{jk}\left(S_{ijk}-\frac{1}{2}(S_{ikj}+S_{jki})\right).\]

Writing \(g=\pm \mathrm{Ric}(\nabla)\) for the induced metric, we define \(\beta \in \Omega^1(\Sigma)\) by \[\beta=\frac{3}{8}\operatorname{tr}_g \mathrm{Sym}\nabla g,\] where \(\mathrm{Sym} : \Gamma\left(T^*\Sigma\otimes S^2(T^*\Sigma)\right) \to \Gamma\left(S^3(T^*\Sigma)\right)\) denotes the natural projection. We have:

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