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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Affine connections and defining functions of real hypersurfaces in $\textbf {C}^{n}$
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by Hing Sun Luk PDF
Trans. Amer. Math. Soc. 259 (1980), 579-588 Request permission


The affine connection and curvature introduced by Tanaka on a strongly pseudoconvex real hypersurface are computed explicitly in terms of its defining function. If Fefferman’s defining function is used, then the Ricci form is shown to be a function multiple of the Levi form. The factor is computable by Fefferman’s algorithm and its positivity implies the vanishing of certain cohomology groups (of the ${\bar \partial _b}$ complex) in the compact case.
  • Charles L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 2, 395–416. MR 407320, DOI 10.2307/1970945
  • S. Kobayashi and K. Nomizu, Foundations of differential geometry, Wiley, New York, Vol. I, 1963; Vol. II, 1969.
  • J. J. Kohn and Hugo Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), 451–472. MR 177135, DOI 10.2307/1970624
  • James Morrow and Kunihiko Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR 0302937
  • Noboru Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 9, Kinokuniya Book Store Co., Ltd., Tokyo, 1975. MR 0399517
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 259 (1980), 579-588
  • MSC: Primary 53C05; Secondary 32F25, 53B05
  • DOI:
  • MathSciNet review: 567098