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Transactions of the American Mathematical Society

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Chains on CR manifolds and Lorentz geometry


Author: Lisa K. Koch
Journal: Trans. Amer. Math. Soc. 307 (1988), 827-841
MSC: Primary 32F25; Secondary 53C50
DOI: https://doi.org/10.1090/S0002-9947-1988-0940230-2
MathSciNet review: 940230
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Abstract: We show that two nearby points of a strictly pseudoconvex CR manifold are joined by a chain. The proof uses techniques of Lorentzian geometry via a correspondence of Fefferman. The arguments also apply to more general systems of chain-like curves on CR manifolds.


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

  • [1] R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer-Verlag, New York, 1982. MR 658304 (83i:57016)
  • [2] D. Burns and S. Shnider, Real hypersurfaces in complex manifolds, Several Complex Variables, Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R. I., 1977, pp. 141-168. MR 0450603 (56:8896)
  • [3] D. Burns, K. Diederich, and S. Shnider, Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44 (1977), 407-431. MR 0445009 (56:3354)
  • [4] E. Cartan, Sur l'equivalence pseudo-conforme des hypersurfaces de l'espace de deux variables complexes. I, Ann. of Math. (2) 11 (1932), 17; II, Ann. Scuola Norm. Sup. Pisa 1 (1932), 333. MR 1556687
  • [5] S. Chandrasekhar and J. P. Wright, The geodesies in Godel's universe, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 341. MR 0129972 (23:B3007)
  • [6] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
  • [7] F. Farris, An intrinsic construction of Fefferman's $ CR$ metric, Pacific J. Math. 123 (1986), 33-45. MR 834136 (87f:53068)
  • [8] C. Fefferman, Monge-Ampere equations, the Bergman kernel, and the geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), 395; Erratum, 104 (1976), 393. MR 0407320 (53:11097a)
  • [9] K. Godel, An example of a new type of cosmological solution of Einstein's field equations of gravitation, Rev. Modern Phys. 21 (1949), 447-450. MR 0031841 (11:216g)
  • [10] C. R. Graham, On Sparling's characterization of Fefferman metrics, preprint.
  • [11] S. Harris, A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J. 31 (1982), 289-308. MR 652817 (83j:53064)
  • [12] S. W. Hawking and G. F. R. Ellis, The large-scale structure of space-time, Cambridge Univ. Press, Cambridge, 1973. MR 0424186 (54:12154)
  • [13] M. W. Hirsch, Differential topology, Springer-Verlag, New York, 1976. MR 0448362 (56:6669)
  • [14] H. Jacobowitz, Chains in $ CR$ geometry, J. Differential Geom. 21 (1985), 163. MR 816668 (87f:32046)
  • [15] J. Lee, The Fefferman metric and pseudohermitian invariants, preprint.
  • [16] D. Kramer, H. Stephani, E. Herlt, M. MacCallum, and E. Schmutzer (eds.), Exact solutions of Einstein's field equations, Cambridge Univ. Press, Cambridge, 1980. MR 614593 (82h:83002)
  • [17] S. Kobayashi and K. Nomizu, Foundations of differential geometry, I & II, Interscience, New York, 1969.
  • [18] L. Nirenberg, Lectures on partial differential equations, CBMS Regional Conf. Ser. Math., no. 17, Amer. Math. Soc., Providence, R. I., 1973. MR 0450755 (56:9048)
  • [19] -, On a question of Hans Lewy, Russian Math. Surveys 29 (1974), 241 251. MR 0492752 (58:11823)
  • [20] R. Penrose, Techniques of differential topology in relativity, CBMS Regional Conf. Ser. in Math., no. 7, Amer. Math. Soc., Providence, R. I., 1972. MR 0469146 (57:8942)
  • [21] H. Poincaré, Les fonctions analytiques de deux variables et la représentation conforme, Rend. Circ. Mat. Palermo (1907), 185.
  • [22] A. Sparling, Twistor theory and the characterization of Fefferman's conformal structures, preprint.
  • [23] S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. MR 520599 (80e:32015)
  • [24] E. Spanier, Algebraic topology, Springer-Verlag, New York, 1966. MR 666554 (83i:55001)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0940230-2
Article copyright: © Copyright 1988 American Mathematical Society

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