CR-hypersurfaces in a space with a pseudoconformal connection

Markowitz, Michael J.

doi:10.1090/S0002-9947-1983-0684496-0

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
DOI: https://doi.org/10.1090/S0002-9947-1983-0684496-0
MathSciNet review: 684496
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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.

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
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, DOI https://doi.org/10.1007/BF02417822
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 425155, DOI https://doi.org/10.1007/BF02392146
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 0355886
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
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 589931, DOI https://doi.org/10.4099/math1924.2.131
S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), no. 1, 25–41. MR 520599
S. M. Webster, The rigidity of C-R hypersurfaces in a sphere, Indiana Univ. Math. J. 28 (1979), no. 3, 405–416. MR 529673, DOI https://doi.org/10.1512/iumj.1979.28.28027
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 430296
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