Annals of Global Analysis and Geometry

, Volume 35, Issue 1, pp 1–37 | Cite as

Mok-Siu-Yeung type formulas on contact locally sub-symmetric spaces

  • Robert PetitEmail author
Original Paper


We derive Mok-Siu-Yeung type formulas for horizontal maps from compact contact locally sub-symmetric spaces into strictly pseudoconvex CR manifolds and we obtain some rigidity theorems for the horizontal pseudoharmonic maps.


Contact locally sub-symmetric spaces Tanaka-Webster connection Pseudoharmonic maps CR maps Rumin complex Mok-Siu-Yeung type formulas Rigidity results 

Mathematics Subject Classification (2000)

32V05 32V10 53C17 53C21 53C24 53C25 53C30 53C35 53D10 58E20 


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  1. 1.
    Alekseevsky D.V., Spiro A.F.: Compact homogeneous CR manifolds. J. Geom. Anal. 12, 183–201 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barletta E.: On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold. Diff. Geom. Appl. 25, 612–631 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barletta E., Dragomir S.: Differential equations on contact Riemannian manifolds. Ann. Scuola. Norm. Sup. Pisa. 30, 63–95 (2001)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Barletta E., Dragomir S.: Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds. Kodai. Math. J. 29, 406–454 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Barletta E., Dragomir S., Urakawa H.: Pseudoharmonic maps from a nondegenerate CR manifolds into a Riemannian manifold. Indiana. Univ. Math. J. 50, 719–746 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bieliavsky P., Falbel E., Gorodski C.: The classification of simply-connected contact sub-Riemannian symmetric spaces. Pac. Math. J. 188, 65–82 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Blair D.E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Math., vol. 509. Springer-Verlag, New York (1976)Google Scholar
  8. 8.
    Blair D.E., Koufogiorgos T., Papantoniou B.J.: Contact metric manifolds satisfying a nullity condition. Israel. J. Math. 91, 189–214 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Boeckx E., Cho J.T.: η-parallel contact metric spaces. Diff. Geom. Appl. 22, 275–285 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Borel A.: On the curvature tensor of the Hermitian symmetric manifolds. Ann. Math. 71, 508–521 (1960)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bourguignon J.P.: Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein. Invent. Math. 63, 263–286 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Calabi E., Vesentini E.: On compact locally symmetric Kahler manifolds. Ann. Math. 71, 472–507 (1960)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Chern S.S., Moser J.: Real hypersurfaces in complex manifolds. Acta. Math. 133, 48–69 (1974)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Cho J.T.: Geometry of contact strongly pseudo-convex CR manifolds. J. Korean. Math. Soc. 43, 1019–1045 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dragomir S.: On pseudo-Hermitian immersions between strictly pseudoconvex CR manifolds. Am. J. Math. 117, 169–202 (1995)zbMATHCrossRefGoogle Scholar
  16. 16.
    Etayo, F., Santamaria, R.: Connections functorially attached to almost complex product structures. Scholar
  17. 17.
    Falbel E., Gorodski C.: On contact sub-Riemannian symmetric spaces. Ann. Scient. Ec. Norm. Sup. 28, 571–589 (1995)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Falbel E., Gorodski C., Rumin M.: Holonomy of sub-Riemannian manifolds. Intern. J. Math. 8(3), 317–344 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jost J., Yau S.T.: Harmonic maps and superrigidity. Proc. Symp. Pure Math. 54, 245–280 (1993)MathSciNetGoogle Scholar
  20. 20.
    Kaup W., Zaitsev D.: On Symmetric Cauchy-Riemann Manifolds. Adv. Math. 49, 145–181 (2000)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Lee J.M.: Pseudo-Einstein structures on CR manifolds. Am. J. Math. 110, 157–178 (1988)zbMATHCrossRefGoogle Scholar
  22. 22.
    Matsushima Y.: On the first Betti number of compact quotient spaces of higher-dimensional symmetric spaces. Ann. of. Math. 75, 312–330 (1962)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Mok N., Siu Y.T., Yeung S.K.: Geometric superrigidity. Invent. Math. 113, 57–83 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Musso E.: Homogeneous pseudo-Hermitian Riemannian manifolds of Einstein type. Amer. J. of Math. 113, 219–241 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Petit R.: Harmonic maps and strictly pseudoconvex CR manifolds. Comm. Anal. Geom. 10, 575–610 (2002)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Petit R.: Spinc-structures and Dirac operators on contact manifolds. Diff. Geom. Appl. 22, 229–252 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Rumin M.: Formes différentielles sur les variétés de contact. J. Diff. Geom. 39, 281–330 (1994)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Sakamoto K., Takemura Y.: Curvature invariants of CR manifolds. Kodai. Math. J. 4, 251–265 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Strichartz R.S.: Sub-Riemannian geometry. J. Diff. Geom. 24, 221–263 (1986)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Tanaka, N.: A Differential Geometric Study on Strongly Pseudoconvex CR manifolds, Lecture Notes in Math., vol. 9. Kyoto University (1975)Google Scholar
  31. 31.
    Tanno S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Tanno S.: Pseudo-conformal invariants of type (1,3) of CR manifolds. Hokkaido. Math. J. 20, 195–204 (1991)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Webster S.: Pseudo-Hermitian structures on a real hypersurface. J. Diff. Geom. 13, 25–41 (1978)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Weil A.: Introduction à l’étude des variétés Kahlériennes. Hermann, Paris (1958)Google Scholar
  35. 35.
    Yeung S.K.: On vanishing theorems and rigidity of locally symmetric manifolds. Geom. Anal. Func. Appl. 11, 175–198 (2001)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Jean Leray, UMR 6629 CNRSUniversité de NantesNantesFrance

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