On some tubes over \(J\)-holomorphic curves in \(S^6\)
Let \(S^{6}\) be the \(6\)-dimensional sphere with its standard almost complex structure. An intrinsic way to understand this structure is to consider \(S^{6}\) as the submanifold of square roots of \(-1\) in the Cayley algebra \(\mathfrak{C}\) of real octonions; the complex structure \(J_{m}\) at \(m\in S^{6}\) is then the Cayley multiplication by \(m\). The automorphism group of \(\mathfrak{C}\) -- the compact group \(G_{2}\) -- acts transitively on \(S^{6}\). The sphere \(S^{6}\) has a canonically defined (up to sign) \(G_{2}\)-invariant complex-valued \(3\)-form \(\omega\): \[ \omega_{m}\left( \xi_{1},\xi_{2},\xi_{3}\right) =\left\langle m\xi_{1} ,\xi_{2}\xi_{3}\right\rangle \qquad\left( m\in S^{6},\;\xi_{j}\in T_{m} S^{6}\right) , \] where \(\left\langle \;,\;\right\rangle \) is the scalar product in \(\mathfrak{C}\). For \(\xi\in G^{3}(T_{m}S^{6})\), the Grassmannian of oriented \(3\)-dimensional subspaces of \(T_{m}S^{6}\), let \(\omega(\xi)=\omega_{m}\left( \xi_{1},\xi _{2},\xi_{3}\right) \), where \(\left( \xi_{1},\xi_{2},\xi_{3}\right) \) is any oriented orthonormal basis of \(\xi\). For \(\kappa\in\mathbb{C}\), \(\left| \kappa\right| \leq1\), define \[ V_{\kappa}=\left\{ \xi\in G^{3}(TS^{6})\mid\omega(\xi)=\kappa\right\} ; \] the \(V_{\kappa}\)'s are the orbits of \(G_{2}\) acting on the Grassmann bundle \(G^{3}(TS^{6})\). A \(3\)-submanifold of \(S^{6}\) is called a \(V_{\kappa} \)-submanifold if its tangent space belongs to \(V_{\kappa}\) at each point. A \(J\)-holomorphic curve (or pseudoholomorphic curve) in \(S^{6}\) is a map \(\phi:M^{2}\rightarrow S^{6}\) whose tangent plane \(\text{d}\phi (T_{m}M)\) is globally \(J\)-invariant for all \(m\in M\). In this paper, the authors consider ``tubes'' over a \(J\)-holomorphic curve \(\phi\), parametrized by the second or the first unit normal bundle of \(\phi\) in \(S^{6}\). They characterize the \(J\)-holomorphic curves \(\phi\) and the values of the parameter (``radius'') of the tube, for which such a tube is a \(V_{\kappa}\)-submanifold.
scientific article; zbMATH DE number 5015947