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 .

The Newman–Penrose formalism for Riemannian 3-manifolds

“…More interesting for us, therefore, is the case of Ric(k, k) > 0. When M is compact, we can recover a result discovered in [7]. Corollary 1.…”

“…The result in [7] appears in a slightly different guise. Specifically, it was shown in [7] that if a closed and orientable Riemannian 3-manifold has a k satisfying the conditions in Corollary 1, then k, · is a contact form and k its Reeb vector field.…”

“…is nondegenerate) if and only if k, [x , y ] | p = 0, which, by Frobenius's theorem, is the case if and only if ω p = 0, it follows that the result in [7] and Corollary 1 are equivalent. (Passing to the smooth orientation covering of M , the result in [7] of course also holds for nonorientable M .)…”

“…Specifically, it was shown in [7] that if a closed and orientable Riemannian 3-manifold has a k satisfying the conditions in Corollary 1, then k, · is a contact form and k its Reeb vector field. But since k, · is a contact form at p ∈ M (in other…”

“…Furthermore, if k is complete, hypersurfaceorthogonal, and satisfies Ric(k, k) ≥ 0, then its divergence must be nonnegative. As an application, we show that if the Riemannian 3-manifold is closed and a unit length k with geodesic flow satisfies Ric(k, k) > 0, then k cannot be hypersurface-orthogonal, thus recovering a result in [7]. Turning next to scalar curvature, we derive an evolution equation for the scalar curvature in terms of unit vector fields k that satisfy the condition R(k, ·, ·, ·) = 0.…”