Affine connections and defining functions of real hypersurfaces in $\textbf {C}^{n}$
HTML articles powered by AMS MathViewer
- by Hing Sun Luk PDF
- Trans. Amer. Math. Soc. 259 (1980), 579-588 Request permission
Abstract:
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.References
- 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
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 259 (1980), 579-588
- MSC: Primary 53C05; Secondary 32F25, 53B05
- DOI: https://doi.org/10.1090/S0002-9947-1980-0567098-3
- MathSciNet review: 567098