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Conformal great circle flows on the 3-sphere


Authors: Adam Harris and Gabriel P. Paternain
Journal: Proc. Amer. Math. Soc. 144 (2016), 1725-1734
MSC (2010): Primary 53D25, 58B20; Secondary 32Q65
DOI: https://doi.org/10.1090/proc/12819
Published electronically: August 12, 2015
MathSciNet review: 3451248
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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 $ \lambda $ 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 a work of Gluck and Gu 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 $ \lambda $ given by rotation by $ \pi /2$ according to the orientation of $ M$.


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Additional Information

Adam Harris
Affiliation: School of Science and Technology, University of New England, Armidale, NSW 2351, Australia
Email: adamh@une.edu.au

Gabriel P. Paternain
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom
Email: g.p.paternain@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/proc/12819
Received by editor(s): August 11, 2014
Received by editor(s) in revised form: April 28, 2015
Published electronically: August 12, 2015
Communicated by: Guofang Wei
Article copyright: © Copyright 2015 American Mathematical Society

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