Abstract
In this paper, a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular, we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group \(\mathbb {H}_{3}\) or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to S U (2,1).
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Notes
Here, 〈⋅,⋅〉 stands for the usual duality between the tangent and the cotangent space.
Here, we are implicitly identifying ν 0 with π ∗ (ν 0 ) via the projection π:Z→M.
Having in mind the usual identification from complex geometry \(\partial _{x}+i\partial _{y}=\partial _{\overline w}\), this becomes the eigenvalue problem \(\partial _{\overline w}f=-c_{12}^{2} f\), whose solutions are of the form \(f=Ae^{-c_{12}^{2}\overline w}\)
The way φ acts on the level sets of P may be nontrivial, i.e., ; this is irrelevant if we restrict to the zero level set of H.
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Acknowledgments
I want to express all my gratitude to my supervisor, prof A. A. Agrachev, for his constant support and many invaluable discussions. I also want to thank him for pointing me out new research directions in several occasions when I was running out of ideas. Without doubt, it is also his merit if this work has been completed. I want to acknowledge also the anonymous referees for their unusual care in pointing out mistakes and for their useful advices to improve the exposition.
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Boarotto, F. Conformal Equivalence of 3D Contact Structures on Lie Groups. J Dyn Control Syst 22, 251–283 (2016). https://doi.org/10.1007/s10883-015-9273-8
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DOI: https://doi.org/10.1007/s10883-015-9273-8