5 Minimal Lagrangian connections

Proof. In our frame adaption where \(S_{ij}=\pm\delta_{ij}\) on \(F_g\), we obtain from (4.13) \[0=\mathrm{d}S_{ij}=\left(R_{ijk}+\frac{2}{3}L_{(i}\varepsilon_{j)k}\right)\omega_k\pm\delta_{ik}\theta^k_j\pm\delta_{kj}\theta^k_i.\] Therefore, writing \(\theta_{ij}=\delta_{ik}\theta^k_j\), we have \[\tag{5.4} \theta_{(ij)}=\mp\frac{1}{2}\left(R_{ijk}+\frac{2}{3}L_{(i}\varepsilon_{j)k}\right)\omega_k.\] For later usage we introduce the notation \(c_1=\mp(\frac{1}{8}R_{111}-\frac{3}{8}R_{122})\) and \(c_2=\mp(-\frac{3}{8}R_{112}+\frac{1}{8}R_{222})\), so that \(c=c_1+\mathrm{i}c_2\). Equation (5.4) written out gives \[\begin{aligned} \theta_{11}&=\mp\frac{1}{2}R_{111}\omega_1\mp\left(\frac{3}{4}R_{112}+\frac{1}{4}R_{222}\right)\omega_2,\\ \frac{1}{2}(\theta_{12}+\theta_{21})&=\mp\left(\frac{3}{8}R_{112}-\frac{1}{8}R_{222}\right)\omega_1\mp\left(\frac{3}{8}R_{122}-\frac{1}{8}R_{111}\right)\omega_2,\\ \theta_{22}&=\mp\left(\frac{3}{4}R_{122}+\frac{1}{4}R_{111}\right)\omega_1\mp\frac{1}{2}R_{222}\omega_2.\\\end{aligned}\] Defining \[\varphi=\theta_{21}\mp\frac{1}{2}R_{222}\omega_1\pm\left(\frac{1}{4}R_{111}+\frac{3}{4}R_{122}\right)\omega_2,\] we compute \[\tag{5.5} \theta=\begin{pmatrix} -\beta & \star_g \beta-\varphi \\ \varphi-\star_g\beta & -\beta\end{pmatrix}+\begin{pmatrix} c_1\omega^1-c_2\omega^2 & -c_2\omega^1-c_1\omega^2\\ -c_2\omega^1-c_1\omega^2 & -c_1\omega^1+c_2 \omega^2\end{pmatrix}.\] The motivation for the definition of \(\varphi\) is that we have \[\mathrm{d}\omega_1=-\omega_2\wedge\varphi\quad \text{and}\quad \mathrm{d}\omega_2=-\varphi\wedge\omega_1,\] hence \(\varphi\) is the Levi-Civita connection form of \(g\). In particular, we see that timelike/spacelike minimal Lagrangian connections are twisted Weyl connections. Since \(\mathrm{Ric}(\nabla)=\pm g\), it follows that the curvature \(2\)-form of \(\theta\) must satisfy \[\tag{5.6} \Theta=\mathrm{d}\theta+\theta\wedge\theta=\begin{pmatrix} 0 & \pm\omega_1\wedge\omega_2 \\ \mp\omega_1\wedge\omega_2 & 0\end{pmatrix}.\] In order to evaluate this condition we first recall that we write \(\upsilon^*\beta=b_i\omega_i\) and \[\begin{aligned} \mathrm{d}b_1&=b_{11}\omega_1+b_{12}\omega_2+b_2\varphi,\\ \mathrm{d}b_2&=b_{21}\omega_1+b_{22}\omega_2-b_1\varphi,\end{aligned}\] for unique real-valued functions \(b_{ij}\) on \(F_g\). From (5.5) and (5.6) we obtain \[\mathrm{d}\beta=-\frac{1}{2}\left(\mathrm{d}\theta_{11}+\mathrm{d}\theta_{22}\right)=\frac{1}{2}\left(\theta_{12}\wedge\theta_{21}+\theta_{21}\wedge\theta_{12}\right)=0\] showing that \(\beta\) is closed, hence (5.3) is verified. Likewise, we also obtain \[\begin{aligned} \mathrm{d}\varphi&=\frac{1}{2}(\mathrm{d}\theta_{21}-\mathrm{d}\theta_{12})+\mathrm{d}\! \star_g\!\beta\\ &=(b_{11}+b_{22})\omega_1\wedge\omega_2+\frac{1}{2}\left((\theta_{11}-\theta_{22})\wedge(\theta_{21}+\theta_{12})\right)\mp \omega_1\wedge\omega_2\\ &=-\left(2\left((c_1)^2+(c_2)^2\right)-(b_{11}+b_{22})\pm 1\right)\omega_1\wedge\omega_2.\end{aligned}\] Writing \(K_g\) for the Gauss curvature of \(g\), this last equation is equivalent to \[K_g=\pm 1+2\,|C|_g^2+\delta_g\beta,\] which verifies (5.1).

In order to prove (5.2), we use \[\upsilon^*\left(\beta-\mathrm{i}\star_g\beta\right)=(b_1+\mathrm{i}b_2)(\omega^1-\mathrm{i}\omega^2).\] In light of (2.6) the condition (5.2) is equivalent to the condition \[\tag{5.7} \mathrm{d}c\wedge\omega=\overline{b}c\overline{\omega}\wedge\omega+3\mathrm{i}c\varphi\wedge\omega,\] where we use the complex notation \(b=b_1-\mathrm{i}b_2\), \(c=c_1+\mathrm{i}c_2\) and \(\omega=\omega_1+\mathrm{i}\omega_2\). Again, from (5.5) we compute \[c\omega=\frac{1}{2}\left[(\theta_{11}-\theta_{22})-\mathrm{i}\left(\theta_{12}+\theta_{21}\right)\right],\] hence \[\begin{gathered} \mathrm{d}c\wedge\omega=\mathrm{d}(c\omega)-c\mathrm{d}\omega=-\theta_{12}\wedge\theta_{21}\\ +\frac{\mathrm{i}}{2}\left(\theta_{11}\wedge(\theta_{12}-\theta_{21})+\theta_{22}\wedge(\theta_{21}-\theta_{12})\right)-(c_1+\mathrm{i}c_2)(\mathrm{d}\omega_1+\mathrm{i}\mathrm{d}\omega_2).\end{gathered}\] Using (5.5) and the structure equations (2.5) this gives \[\begin{gathered} \mathrm{d}c\wedge\omega=3c_2\omega_1\wedge\varphi+3c_1\omega_2\wedge\varphi-2(b_1c_2+b_2c_1)\omega_1\wedge\omega_2\\ +\mathrm{i}\left(-3c_1\omega_1\wedge\varphi+3c_2\omega_2\wedge\varphi+2(b_1c_1-b_2c_2)\omega_1\wedge\omega_2\right),\end{gathered}\] which is equivalent to \[\begin{gathered} \mathrm{d}c\wedge\omega=(b_1+\mathrm{i}b_2)(c_1+\mathrm{i}c_2)(\omega_1-\mathrm{i}\omega_2)\wedge(\omega_1+\mathrm{i}\omega_2)\\ +3\mathrm{i}(c_1+\mathrm{i}c_2)\varphi\wedge(\omega_1+\mathrm{i}\omega_2),\end{gathered}\] that is, equation (5.7). This completes the proof.

Proof. Clearly, the map sending a torsion-free minimal Lagrangian connection \(\nabla\) into the set of triples \((g,\beta,C)\) satisfying the above structure equations, is surjective. Now suppose the two triples \((g_1,\beta_1,C_1)\) and \((g_2,\beta_2,C_2)\) on \(\Sigma\) satisfy the above structure equations and define the same torsion-free spacelike minimal Lagrangian connection \(\nabla\) on \(T\Sigma\). Then \(g_1=\pm \mathrm{Ric}(\nabla)=g_2\) and consequently we obtain \(\beta_1=\beta_2\) as well as \(C_1=C_2\), since these quantities are defined in terms of \(\nabla \mathrm{Ric}(\nabla)\) by using the metric \(g_1=g_2\).

Proof. The Lagrangian connection \(\nabla\) is totally geodesic if and only if the second fundamental form vanishes identically or equivalently, if \(\beta\) and \(C\) vanish identically. In this case (5.5) implies that \(\theta\) is the Levi-Civita connection of \(g\) and the structure equation (5.1) gives that \(g\) has Gauss curvature \(\pm 1\).