Skip to main content
Log in

Local embeddability of pseudohermitian manifolds into spheres

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper we solve the problem of local pseudohermitian embeddability into spheres. We state necessary and sufficient conditions for the embeddability as a finite number of equations and rank conditions on the curvature and torsion tensors and their derivatives.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Burns, D., Shnider, S.: Real hypersurfaces in complex manifolds. Proc. Symp. Pure math. 30, 141–167 (1976)

    Google Scholar 

  2. Cho, J.S., Han, C.K.: Complete prolongation and the Frobenius integrability for overdetermined systems of partial differential equations. J. Korean Math. Soc. 39, 237–252 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chern, S.S., Moser, J.K.: Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974)

    MathSciNet  Google Scholar 

  4. Ebenfelt, P., Huang, X., Zaitsev, D.: Rigidity of CR-immersions into spheres. Commum. Anal. Geom. 12(3), 631–670 (2004)

    MathSciNet  Google Scholar 

  5. Faran, J.: The nonembeddability of real hypersurfaces in spheres. Proc. Am. Math. Soc. 103, 902–904 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. Forstneric, F.: Embedding strictly pseudoconvex domains into balls. Trans. Am. Math. Soc. 295, 347–368 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Han, C.K.: Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms. Math. Ann. 309, 401–409 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Han, C.K.: Equivalence problem and complete system of finte order. J. Korean Math. Soc. 37(2), 225–243 (2000)

    MathSciNet  Google Scholar 

  9. Han, C.K.: Solvability of overdetermined PDE systems that admit a complete prolongation and some local problems in CR geometry. J. Korean Math. Soc. 40, 695–708 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lee, J.M.: Pseudo-Einstein structures on CR manifolds. Am. J. Math. 110, 157–178 (1988)

    MATH  Google Scholar 

  11. Lempert, L.: Imbedding Cauchy-Riemann manifolds into a sphere. Int. J. Math. 1, 91–108 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Løw, E.: Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls. Math. Z. 190, 401–410 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. Stensønes, B.: Proper maps which are Lipschitz α up to the boundary. J. Geom. Anal. 6(2), 317–339 (1996)

    Google Scholar 

  14. Tanaka, N.: On pseudo-conformal geometry of hypersurfaces of the space of n complex variables. J. Math. Soc. Japan. 14, 397–429 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tanaka, N.: A differential Geometric Study on Strongly Pseudo-convex Manifolds, Kinokuniya Company Ltd., Tokyo, 1975

  16. Webster, S.M.: Pseudohermitian structures on a real hypersurface. J. Differential Geom. 13, 25–41 (1978)

    MATH  MathSciNet  Google Scholar 

  17. Webster, S.M.: The rigidity of C-R hypersurfaces in a sphere. Indiana Univ. Math. J. 28, 405–416 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Webster, S.M.: Segre polar correspondence and double valued reflection for general ellipsoids. Analysis and geometry in several complex variables (Katata, 1997), Trends Math. Birkhäuser, Boston, MA. 1999, pp. 273–288

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sung-Yeon Kim.

Additional information

The second author was supported by BK21-Yonsei University

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, SY., Oh, JW. Local embeddability of pseudohermitian manifolds into spheres. Math. Ann. 334, 783–807 (2006). https://doi.org/10.1007/s00208-005-0710-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-005-0710-y

Mathematics Subject Classification (2000)

Navigation