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, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 141–168. MR 0450603
  • [2] Elie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes, Ann. Mat. Pura Appl. 11 (1933), no. 1, 17–90 (French). MR 1553196, 10.1007/BF02417822
  • [3] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 0425155
  • [4] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [5] Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 0355886
  • [6] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
  • [7] Noboru Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131–190. MR 0589931
  • [8] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25–41. MR 520599
  • [9] S. M. Webster, The rigidity of C-R hypersurfaces in a sphere, Indiana Univ. Math. J. 28 (1979), no. 3, 405–416. MR 529673, 10.1512/iumj.1979.28.28027
  • [10] Keizo 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
  • [11] Kentaro Yano and Yosio Mutô, Sur la théorie des espaces à connexion conforme normale et la géométrie conforme des espaces de Riemann, J. Fac. Sci. Imp. Univ. Tokyo. Sect. I. 4 (1941), 117–169 (French). MR 0005710

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