Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



CR-hypersurfaces in a space with a pseudoconformal connection

Author: Michael J. Markowitz
Journal: Trans. Amer. Math. Soc. 276 (1983), 117-132
MSC: Primary 32F25; Secondary 53B25
MathSciNet review: 684496
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study a submanifold in a space with a pseudoconformal connection. We assume that the submanifold $ M$ is so situated that it inherits the structure of a $ {\text{CR}}$-hypersurface from the ambient space. $ M$ then supports two natural Cartan connections, the normal pseudoconformal connection of Cartan-Chern-Tanaka and an induced pseudoconformal connection. Analogues of the Gauss-Codazzi equations are derived and applied to determine necessary and sufficient conditions for the equivalence of these connections.

References [Enhancements On Off] (What's this?)

  • [1] D. Burns, Jr. and S. Shnider, Real hypersurfaces in complex manifolds, Proc. Sympos. Pure Math., vol. 30, Amer. Math. Soc., Providence, R. I., 1977, pp. 141-168. MR 0450603 (56:8896)
  • [2] E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes. I, Ann. Mat. Pura Appl. 11 (1932), 17-90 (or Oeuvres II, 2, 1231-1304); II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1 (1932), 333-354 (or Oeuvres III, 2, 1217-1238). MR 1553196
  • [3] S.-S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
  • [4] S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
  • [5] S. Kobayashi, Transformation groups in differential geometry, Springer-Verlag, Berlin and New York, 1972. MR 0355886 (50:8360)
  • [6] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vols. I, II, Wiley, New York, 1963, 1969. MR 1393940 (97c:53001a)
  • [7] N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math. 2 (1976), 131-190. MR 0589931 (58:28645)
  • [8] S. M. Webster, Pseudo-hermitian structures on a real hypersurface, Trans. Amer. Math. Soc. 231 (1977), 179-190. MR 520599 (80e:32015)
  • [9] -, The rigidity of $ C$-$ R$ hypersurfaces in a sphere, Indiana Univ. Math. J. 28 (1978), 405-416. MR 529673 (80d:32022)
  • [10] K. Yamaguchi, Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I, Nagoya Math. J. 62 (1976), 55-96. MR 0430296 (55:3301)
  • [11] K. Yano and Y. Muto, Sur la théorie des espaces à connexion conforme normale et la géométrie conforme des espaces de Riemann, J. Fac. Sci. Tokyo Imp. Univ. 4 (1941), 117-169. MR 0005710 (3:192d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F25, 53B25

Retrieve articles in all journals with MSC: 32F25, 53B25

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society