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)

 
 

 

Lewy's curves and chains on real hypersurfaces


Author: James J. Faran
Journal: Trans. Amer. Math. Soc. 265 (1981), 97-109
MSC: Primary 32C05; Secondary 32F25
DOI: https://doi.org/10.1090/S0002-9947-1981-0607109-0
MathSciNet review: 607109
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Lewy's curves on an analytic real hypersurface $ M = \{ r(z,z) = 0\} $ in $ {{\mathbf{C}}^2}$ are the intersections of $ M$ with any of the Segre hypersurfaces $ {Q_w} = \{ z:r(z,w) = 0\} $. If $ M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy's curves are chains, $ M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in $ {{\mathbf{C}}^n}$, $ n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.


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

  • [1] S. S. Chern, On the projective structure of a real hypersurface in $ {{\mathbf{C}}_{n + 1}}$, Math. Scand. 36 (1975), 74-82. MR 0379910 (52:814)
  • [2] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
  • [3] J. J. Faran, Segre families and real hypersurfaces, Thesis, University of California at Berkeley, 1978.
  • [4] B. Segre, I. Intoruo al problema di Poincaré della rappresentazione pseudo-conforme, Rend Acc. Lincei 13 (1931), 676-683; II. Questioni geometriche legate colla teoria delle funzioni di due variabili complesse, Rend. Semin. Mat. Roma 7 (1931).
  • [5] S. M. Webster, Real hypersurfaces in complex space, University of California at Berkeley, 1975.
  • [6] -, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53-68. MR 0463482 (57:3431)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32C05, 32F25

Retrieve articles in all journals with MSC: 32C05, 32F25


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0607109-0
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society