Explicit representation of super-minimal \(J\)-holomorphic curves in a 6-dimensional sphere
Let \(S^6\) be a 6-dimensional unit sphere with the canonical nearly Kähler structure \((J,g)\) which is defined by using the octonians \(O\) (also called the Cayley division algebra). A Riemann surface \(M^2\) equipped with a smooth map \(\phi: M^2\to S^6= (S^6,J,g)\) is called a \(J\)-holomorphic curve in \(S^6\) if each tangent space of \(M^2\) is invariant under the almost complex structure \(J\) via \(\phi_*\). It is well known that the compact exceptional Lie group \(G_2\) can be regarded as the automorphism group of \((S^6, J,g)\) and the homogeneous space \(G_2/U(2)\) coincides with the Grassmannian \(G_2(\text{Im }O)\) of oriented 2-planes in \(\text{Im }O\). Further, the homogeneous space \(G_2/U(2)\) can be regarded as the total space of the fiber bundle \(\pi: G_2/U(2)\to G_2/SU(3)= S^6\) which is associated to the principal \(SU(3)\)-bundle \(\mu: G_2\to G_2/SU(3)= S^6\). We may define the horizontal distribution \(H\) on the associated bundle \((G_2(\text{Im }O),\pi, S^6)\) which is determined by the \(SU(3)\)-connection on \(G_2\) with respect to the Riemannian structure of \((S^6, g)\) and a subbundle \(L_+= \bigcup_{q\in G_2(\text{Im }O)} L_+(q)\) of \(H\) of rank 4 over \(G_2(\text{Im }O)\) such that \(\pi_*\) is complex linear with respect to the canonical complex structure of \(G_2(\text{Im }O)\) and the nearly Kähler structure \((J,g)\) on \(S^6\). Let \(\psi: M^2\to G_2(\text{Im }O)\) be a holomorphic map from a Riemann surface \(M^2\) to \(G_2(\text{Im }O)\) with the canonical complex structure such that \(\psi_*(T_p M^2)\subset L_+(\psi(p))\) for any point \(p\in M^2\). Then, we may observe that \(\phi= \pi\circ\psi: M^2\to S^6\) is a superminimal \(J\)-holomorphic curve (sometimes called a \(J\)-holomorphic curve with zero torsion or with \(\text{III}=0\), III being the third fundamental form). In 1982, \textit{R. L. Bryant} proved that any compact Riemann surface can be considered as a super-minimal \(J\)-holomorphic curve \((S^6, J,g)\) by making use of the twistor method [J. Differ. Geom. 17, 185-232 (1982; Zbl 0526.53055)]. More precisely, he shows that a super-minimal \(J\)-holomorphic curve \(M^2\) is locally given by \(\phi= \pi\circ\psi: M^2\to S^6\), where \(\psi: M^2\to G_2(\text{Im }O)= Q^5(\mathbb{C})\) (5-dimensional complex quadric) is a holomorphic curve. In the present paper, the author gives another algebraic proof of the above result of Bryant whose idea is based on \textit{S.-S. Chern's} method [Selected papers, Vol. 1, Springer-Verlag, New York, 421-434 (1978; Zbl 0403.01012)]. As an application, the author writes down the Boruvka surface \(S^2(1/6)\) in \(S^6\) as a superminimal \(J\)-holomorphic curve explicitly.
scientific article; zbMATH DE number 1180766