Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 090, 26 pages      arXiv:2411.18961      https://doi.org/10.3842/SIGMA.2025.090

The Fefferman Metric for Twistor CR Manifolds and Conformal Geodesics in Dimension Three

Taiji Marugame
Department of Mathematics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan

Received June 03, 2025, in final form October 16, 2025; Published online October 24, 2025

Abstract
We give an explicit description of the Fefferman metric for twistor CR manifolds in terms of Riemannian structures on the base conformal 3-manifolds. As an application, we prove that chains and null chains on twistor CR manifolds project to conformal geodesics, and that any conformal geodesic has lifts both to a chain and a null chain. By using this correspondence, we give a variational characterization of conformal geodesics in dimension three which involves the total torsion functional.

Key words: twistor CR manifold; Fefferman metric; conformal geodesic.

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References

  1. Bailey T.N., Eastwood M.G., Conformal circles and parametrizations of curves in conformal manifolds, Proc. Amer. Math. Soc. 108 (1990), 215-221.
  2. Bailey T.N., Eastwood M.G., Gover A.R., Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191-1217.
  3. Barros M., Ferrández A., A conformal variational approach for helices in nature, J. Math. Phys. 50 (2009), 103529, 20 pages.
  4. Barros M., Ferrández A., On the energy density of helical proteins, J. Math. Biol. 69 (2014), 1801-1813.
  5. Burns Jr. D., Diederich K., Shnider S., Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44 (1977), 407-431.
  6. Čap A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. 79 (2006), 11-37, arXiv:math.DG/0504389.
  7. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Math. Surveys Monogr., Vol. 154, American Mathematical Society, Providence, RI, 2009.
  8. Čap A., Slovák J., Zádník V., On distinguished curves in parabolic geometries, Transform. Groups 9 (2004), 143-166, arXiv:math.DG/0308051.
  9. Cheng J.-H., Marugame T., Matveev V.S., Montgomery R., Chains in CR geometry as geodesics of a Kropina metric, Adv. Math. 350 (2019), 973-999, arXiv:1806.01877.
  10. Chern A., Knöppel F., Pedit F., Pinkall U., Commuting Hamiltonian flows of curves in real space forms, in Integrable Systems and Algebraic Geometry. Vol. 1, London Math. Soc. Lecture Note Ser., Vol. 458, Cambridge University Press, Cambridge, 2020, 291-328, arXiv:1809.01394.
  11. Dunajski M., Kryński W., Variational principles for conformal geodesics, Lett. Math. Phys. 111 (2021), 127, 18 pages, arXiv:2104.13105.
  12. Dunajski M., Tod P., Conformal geodesics on gravitational instantons, Math. Proc. Cambridge Philos. Soc. 173 (2022), 123-154, arXiv:1906.08375.
  13. Farris F.A., An intrinsic construction of Fefferman's CR metric, Pacific J. Math. 123 (1986), 33-45.
  14. Fefferman C.L., Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395-416, Errarum, Ann. of Math. 104 (1976), 393-394.
  15. Friedrich H., Schmidt B.G., Conformal geodesics in general relativity, Proc. Roy. Soc. London Ser. A 414 (1987), 171-195.
  16. Holland J., On causal geomtries, Ph.D. Thesis, University of Pittsburgh, 2015, available at https://d-scholarship.pitt.edu/24007/.
  17. Koch L.K., Chains, null-chains, and CR geometry, Trans. Amer. Math. Soc. 338 (1993), 245-261.
  18. Kruglikov B., Conformal geodesics are not variational in higher dimensions, arXiv:2505.06739.
  19. Kruglikov B., Matveev V.S., Steneker W., Variationality of Conformal Geodesics in dimension 3, Anal. Math. Phys. 15 (2025), 123, 25 pages, arXiv:2412.04890.
  20. Kuo T.-M., Conformal circles and local diffeomorphisms, arXiv:2210.05864.
  21. LeBrun C.R., Twistor CR manifolds and three-dimensional conformal geometry, Trans. Amer. Math. Soc. 284 (1984), 601-616.
  22. Lee J.M., The Fefferman metric and pseudo-Hermitian invariants, Trans. Amer. Math. Soc. 296 (1986), 411-429.
  23. Low H.C., Twistor CR manifold by non-Riemannian connections, Ph.D. Thesis, University of Pittsburgh, 2020, available at https://www.proquest.com/docview/2467809346.
  24. Marugame T., Hyperkähler ambient metrics associated with twistor CR manifolds, arXiv:2309.02778.
  25. Sato H., Yamaguchi K., Lie contact manifolds, in Geometry of Manifolds (Matsumoto, 1988), Perspect. Math., Vol. 8, Academic Press, Boston, MA, 1989, 191-238.

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