Flat almost complex surfaces in the homogeneous nearly Kähler \(S^3\times S^3\)
A nearly Kähler manifold \((\widetilde M,g,J)\) is an almost Hermitian manifold, with almost complex structure \(J\) and Levi-Civita connection \(\widetilde\nabla\) of the metric \(g\), for which the tensor field \(G=\widetilde\nabla J\) is skew-symmetric. A nearly Kähler manifold is called strictly nearly Kähler if furthermore \(\widetilde\nabla_X J=0\) for any non-vanishing tangent vector \(X\). The most well-known and simple example of a nearly Kähler manifold is the 6-dimensional sphere, whose almost complex structure \(J\) can be defined via octonionic multiplication. The only homogeneous 6-dimensional nearly Kähler manifolds are \(S^6\), \(S^3\times S^3\), \(\mathbb CP^3\), and the flag manifold \(\mathrm{SU}(3)/\mathrm{U}(1)\times \mathrm{U}(1)\). These spaces are compact 3-symmetric spaces. A natural class of submanifolds of nearly Kähler manifolds are the almost complex submanifolds, for which \(J\) sends tangent vectors to tangent ones. If \(M\) is an almost complex surface in \(S^3\times S^3\) with metric \(g\) and complex coordinate \(z=u+iv\) on \(M\), then \(\Lambda dz^2=g(P\partial_z,\partial_z)dz^2\) defines a holomorphic quadratic differential on the surface, where \(P\) is an almost product structure on \(M\). In this paper, by using a special adapted frame, the authors prove a Bonnet-type existence and uniqueness theorem for almost complex surfaces in the homogeneous nearly Kähler manifold \(S^3\times S^3\). It is shown that if \(U\subset\mathbb R^2\) is a simply connected open set endowed with a metric \(g=e^\omega dzd\bar z\), where \(z\) is a complex coordinate on \(U\), \(\omega\) is a smooth nonzero function defined on \(U\), and there exists a function \(\mu:U\to\mathbb C\) satisfying \(\omega_{z\bar z}\sinh\omega-\frac{e^{-\omega}}2|\omega_z|^2+\frac43\sinh^2\omega(1-|\mu|^2)=0\) and \(\mu_{\bar z}+\frac{\omega_{\bar z}e^\omega\mu-\omega_z\bar\mu}{2\sinh\omega}=0\), then there is an almost complex immersion \(\varphi:U\to S^3\times S^3\) with \(\Lambda=1\), unit normal \(N\) and with an adapted frame \(\partial_z,\partial_{\bar z},N,\bar N,G(\partial_z,P\partial_{\bar z}),G(\partial_{\bar z},P\partial_{z})\) such that its second fundamental form \(h\) is \(h(\partial_z,\partial_z)=\frac{-\omega_z}{e^\omega-e^{-\omega}}\bar N+\mu G(\partial_{\bar z},P\partial_{z})\). Moreover, two such almost complex immersions are the same up to an isometry of \(S^3\times S^3\). Also, they show that if \(U\subset\mathbb R^2\) is a simply connected open set endowed with a metric \(g\) and two functions \(\omega:U\to\mathbb R\), \(\mu:U\to\mathbb C\) such that \(g(\partial_z,\partial_{\bar z})=e^\omega\), \(\omega_{z\bar z}+\frac23e^\omega(1-|\mu|^2)=0\), and \(\mu_{\bar z}+\omega_{\bar z}\mu=0\), then there is an almost complex immersion \(\varphi:U\to S^3\times S^3\) with vanising differential \(\Lambda dz^2\) and an adapted frame \(\partial_z,\partial_{\bar z},P\partial_z,P\partial_{\bar z},G(\partial_z,P\partial_{\bar z}),G(\partial_{\bar z},P\partial_{z})\) such that its second fundamental form \(h\) is \(h(\partial_z,\partial_z)=\mu G(\partial_{\bar z},P\partial_{z})\). Moreover, two such almost complex immersions are the same up to an isometry of \(S^3\times S^3\). Finally, the authors give a complete classification of flat almost complex surfaces in the homogeneous nearly Kähler \(S^3\times S^3\) by proving that any flat almost complex surface in \(S^3\times S^3\) must be an open part of the following 2-parameter family \(f(u,v)=(p(u,v),q(u,v))\) whose components are homogeneous tori in \(S^3\) given by \[ p(u,v)=\left(\sqrt{\frac{1+a}{2}}e^{i(c_1u+c_2v)},\sqrt{\frac{1-a}{2}}e^{i(c_3u+c_4v)}\right) \] \[ q(u,v)=\left(\sqrt{\frac{1+b}{2}}e^{i(b_1u+b_2v)},\sqrt{\frac{1-b}{2}}e^{i(b_3u+b_4v)}\right)* i, \] where \(*\) denotes the quaternion multiplication, \((a,b)\in(-1,1)\times(-1,1)\) and \(c_i\), \(b_i\) are constants determined by \(a\), \(b\).
scientific article; zbMATH DE number 6859351
scientific article