On the groups of c-projective transformations of complete Kähler manifolds

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Abstract

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini–Study metric. This establishes a stronger version of the recently proved Yano–Obata conjecture for complete Kähler manifolds.

Keywords

Kähler manifolds C-projective geometry Automorphism groups of geometric structures Geometric rigidity Integrable systems 

Mathematics Subject Classification

32Q15 32J27 53A20 53C24 22F50 37J35 

Notes

Acknowledgements

We would like to thank the referee for helpful comments and suggestions leading to improvements of our article.

References

  1. 1.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Hamiltonian 2-forms in Kähler geometry I: general theory. J. Differ. Geom. 73, 359–412 (2006)CrossRefMATHGoogle Scholar
  2. 2.
    Bolsinov, A.V., Matveev, V.S., Rosemann, S.: Local Normal Forms for c-projectively Equivalent Metrics and Proof of the Yano-Obata Conjecture in Arbitrary Signature. Proof of the Projective Lichnerowicz Conjecture for Lorentzian metrics. arXiv:1510.00275
  3. 3.
    Calderbank, D.M.J., Matveev, V.S., Rosemann, S.: Curvature and the c-projective mobility of Kähler metrics with hamiltonian 2-forms. Compos. Math. 152, 1555–1575 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Calderbank, D.M.J., Eastwood, M.G., Matveev, V.S., Neusser, K.: C-projective Geometry, to appear in Memoirs of AMS. arXiv:1512.04516
  5. 5.
    Domashev, V.V., Mikeš, J.: On the theory of holomorphically projective mappings of Kählerian spaces. Math. Notes 23, 160–163 (1978). (translation from Mat. Zametki 23 (1978), 297–304)CrossRefMATHGoogle Scholar
  6. 6.
    Fedorova, A., Kiosak, V., Matveev, V.S., Rosemann, S.: The only closed Kähler manifold with degree of mobility \(\ge 3\) is \((CP(n), g_{{\rm Fubini-Study}})\). Proc. Lond. Math. Soc. 105, 153–188 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ferrand, J.: Transformations conformes et quasi-conformes des variétés riemanniennes compactes. Acad. R. Bel. Cl. Sci. Mém. Coll. 39(5), 1–44 (1971)MATHGoogle Scholar
  8. 8.
    Ferrand, J.: The action of conformal transformations on a Riemannian manifold. Math. Ann. 304(2), 277–291 (1996)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Frances, C.: Sur le group d’automorphismes des géométries paraboliques de rang un. Annales scientifiques de l’Ecole Normale Supérieure 40(5), 741–764 (2007)CrossRefMATHGoogle Scholar
  10. 10.
    Ishihara, S., Obata, M.: Affine transformations in a Riemannian manifold. Tohoku Math. J. Secons Ser. 7(3), 146–150 (1955)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kiosak, V., Matveev, V.S.: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Math. Phys. 297(2), 401–426 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kiyohara, K., Topalov, P.J.: On Liouville integrability of h-projectively equivalent Kähler metrics. Proc. Am. Math. Soc. 139, 231–242 (2011)CrossRefMATHGoogle Scholar
  13. 13.
    Lichnerowicz, A.: Geometry of Groups of Transformations, translated from the French and edited by Michael Cole. Noordhoff International Publishing, Leyden (1977)MATHGoogle Scholar
  14. 14.
    Matveev, V.S.: Lichnerowicz–Obata conjecture in dimension two. Comm. Math. Helv. 81(3), 541–570 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Matveev, V.S.: Proof of projective Lichnerowicz–Obata conjecture. J. Differ. Geom. 75, 459–502 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Matveev, V.S.: On the number of nontrivial projective transformations of closed manifolds (in Russian). Fundam. Prikl. Mat. 20, 125–131 (2015). (an English translation was published in Journal of Math. Sciences 223 (2017), 734–738)Google Scholar
  17. 17.
    Matveev, V.S.: Projectively Invariant Objects and the Index of the Group of Affine Transformations in the Group of Projective Transformations, to appear in Bulletin of the Iranian Math. Soc,  https://doi.org/10.1007/s41980-018-0024-y; arXiv:1604.01238
  18. 18.
    Matveev, V.S., Rosemann, S.: Proof of the Yano–Obata conjecture for h-projective transformations. J. Differ. Geom. 92(2), 221–261 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mikes, J.: Holomorphically projective mappings and their generalizations. Geom. 3. J. Math. Sci. (New York) 89(3), 1334–1353 (1998)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Obata, M.: The conjectures on conformal transformations of Riemannian manifolds, J. Differ. Geom. 6, 247–258 (1971/72)Google Scholar
  21. 21.
    Otsuki, T., Tashiro, Y.: On curves in Kählerian spaces. Math. J. Okayama Univ. 4, 57–78 (1954)MathSciNetMATHGoogle Scholar
  22. 22.
    Schoen, R.: On the conformal and CR-automorphism groups. Geom. Funct. Anal. 5(2), 464–481 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Sinjukov, N.S.: Geodesic Mappings of Riemannian Spaces. Nauka, Moscow (1979). (in Russian)MATHGoogle Scholar
  24. 24.
    Topalov, P.J.: Geodesic compatibility and integrability of geodesic flows. J. Math. Phys. 44(2), 913–929 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Tashiro, Y.: On a holomorphically projective correspondence in an almost complex space. Math. J. Okayama Univ. 6, 147–152 (1957)MathSciNetMATHGoogle Scholar
  26. 26.
    Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. International Series of Monographs in Pure and Applied Mathematics. 49 A Pergamon Press Book. The Macmillan Co., New York (1965)Google Scholar
  27. 27.
    Zeghib, A.: On discrete projective transformation groups of Riemannian manifolds. Adv. Math. 297, 26–53 (2016)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakultät für Mathematik und Informatik, Institut für MathematikFriedrich-Schiller-Universität JenaJenaGermany
  2. 2.Faculty of Mathematics and Physics, Mathematical InstituteCharles UniversityPragueCzech Republic

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