Conformal great circle flows on the 3-sphere (Q2790279)

Let \((M, g)\) be a closed \(3\)-dimensional Riemannian manifold, \(X\) a self-parallel unitary vector field (hence, its integral curves are geodesics), and \(X^\perp \subset TM\) the 2-dimensional distribution, which is orthogonal to \(X\). The authors consider the family \(J\) of complex structures on the \(2\)-planes of \(X^\perp\) defined by the following rule: given a unitary \(w \in X^\perp_p\), the vector \(J_{p}(w)\) is the unique unit vector such that \((w, J_p(w), v)\) is positively oriented. They first show that \(\mathcal L_X J = 0\) if and only if the corresponding \(2\)-dimensional distribution \(X_*(X^\perp) \subset TTM\) is stable under an appropriate family \(\mathbb J\) of linear maps on the tangent spaces of \(TM\), defined in terms of \(J\) and the Levi-Civita connection. Second, they focus on the case in which the manifold is the standard unit sphere \(S^3\) and find a characterisation of local unit self-parallel vector fields satisfying the condition \(\mathcal L_X J = 0\). Such characterisation yields the following corollary: a great circle flow of \(S^3\) preserves the above defined complex structures \(J\) if and only if it is a Hopf fibration.