Conformal Great Circle Flows on the 3-Sphere

Abstract

We consider a closed orientable Riemannian 3-manifold (M,g) and a vector field X with unit norm whose integral curves are geodesics of g. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of g. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that X is the Reeb vector field of the 1-form λ obtained by contracting g with X. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in [4] that a volume preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost complex structure in the kernel of λ given by rotation by π/2 according to the orientation of M.