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)



Chains, null-chains, and CR geometry

Author: Lisa K. Koch
Journal: Trans. Amer. Math. Soc. 338 (1993), 245-261
MSC: Primary 32F40; Secondary 53C56
MathSciNet review: 1100695
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A system of distinguished curves distinct from chains is defined on indefinite nondegenerate $ {\text{CR}}$ hypersurfaces; the new curves are called null-chains. The properties of these curves are explored, and it is shown that two sufficiently nearby points of any nondegenerate $ {\text{CR}}$ hypersurface can be connected by either a chain or a null-chain.

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

  • [B] R. Bryant, Holomorphic curves in Lorentzian $ CR$-manifolds, Trans. Amer. Math. Soc. 272 (1982), 203. MR 656486 (83i:32029)
  • [BDS] D. Burns, K. Diederich, and S. Shnider, Distinguished curves in pseudoconvex boundaries, Duke Math. J. 43 (1977), 407-431. MR 0445009 (56:3354)
  • [BS] D. Burns and S. Shnider, Real hypersurfaces in complex manifolds, Several Complex Variables, Proc. Sympos. Pure Math., Vol. 40, Part 2, Amer. Math. Soc., Providence, R.I., 1977, p. 141. MR 0450603 (56:8896)
  • [Ca] É. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes. I, Œuvres Complètes, Gauthier-Villars, Paris, 1955, pp. 1231-1304; II, Œuvres Complètes, Gauthier-Villars, Paris, 1955, pp. 1217-1238.
  • [Ch] J.-H. Cheng, Chain-preserving differomorphisms and $ CR$ equivalence, preprint.
  • [CM] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
  • [Fa] F. Farris, An intrinsic construction of Fefferman's $ CR$ metric, Pacific J. Math. 123 (1986), 33. MR 834136 (87f:53068)
  • [Fe] C. Fefferman, Monge-Ampère equations, the Berman kernel, and the geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395; Erratum, 104 (1976), 393. MR 0407320 (53:11097a)
  • [G] C. R. Graham, On Sparling's characterization of Fefferman metrics, Amer. J. Math. 109 (1897), 853-841. MR 910354 (90f:32021)
  • [H] S. Harris, A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J. 31 (1982), 289. MR 652817 (83j:53064)
  • [J] H. Jacobowitz, Chains in $ CR$ geometry, J. Differential Geometry 21 (1985), 163. MR 816668 (87f:32046)
  • [Ko] L. Koch, Chains in $ CR$ geometry and Lorentz manifolds, Trans. Amer. Math. Soc. 307 (1988), 827. MR 940230 (89k:32038)
  • [Kr] N. Kruzhilin, Local automorphisms and mappings of smooth strictly pseudoconvex hypersurfaces, Math. USSR-Izv. 26 (1986), 531. MR 794956 (86j:32046)
  • [Ku] D. Kupeli, Degenerate submanifolds in semi-Riemannian geometry, Geom. Dedicata 24 (1986), 337. MR 914829 (88m:53117)
  • [L] J. Lee, The Fefferman metric and pseudohermetian invariants, Trans. Amer. Math. Soc. 296 (1986), 411. MR 837820 (87j:32063)
  • [P] H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo (1907), 185.
  • [S] G. Sparling, Twistor theory and the characterization of Fefferman's conformal structures, preprint.
  • [W] S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry 13 (1978), 25. MR 520599 (80e:32015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F40, 53C56

Retrieve articles in all journals with MSC: 32F40, 53C56

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society