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Transactions of the American Mathematical Society

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



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
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.

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  • [1] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
  • [2] D. Burns Jr. and S. Shnider, Real hypersurfaces in complex manifolds, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 141–168. MR 0450603
  • [3] D. Burns Jr., K. Diederich, and S. Shnider, Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44 (1977), no. 2, 407–431. MR 0445009
  • [4] Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333–354 (French). MR 1556687
  • [5] S. Chandrasekhar and James P. Wright, The geodesics in Gödel’s universe, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 341–347. MR 0129972
  • [6] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR 0425155
  • [7] Frank A. Farris, An intrinsic construction of Fefferman’s CR metric, Pacific J. Math. 123 (1986), no. 1, 33–45. MR 834136
  • [8] 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 0407320
  • [9] Kurt Gödel, An example of a new type of cosmological solutions of Einstein’s field equations of gravitation, Rev. Modern Physics 21 (1949), 447–450. MR 0031841
  • [10] C. R. Graham, On Sparling's characterization of Fefferman metrics, preprint.
  • [11] Steven G. Harris, A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J. 31 (1982), no. 3, 289–308. MR 652817, 10.1512/iumj.1982.31.31026
  • [12] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR 0424186
  • [13] Morris W. Hirsch, Differential topology, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR 0448362
  • [14] Howard Jacobowitz, Chains in CR geometry, J. Differential Geom. 21 (1985), no. 2, 163–194. MR 816668
  • [15] J. Lee, The Fefferman metric and pseudohermitian invariants, preprint.
  • [16] D. Kramer, H. Stephani, E. Herlt, and M. MacCallum, Exact solutions of Einstein’s field equations, Cambridge University Press, Cambridge-New York, 1980. Edited by Ernst Schmutzer; Cambridge Monographs on Mathematical Physics. MR 614593
  • [17] S. Kobayashi and K. Nomizu, Foundations of differential geometry, I & II, Interscience, New York, 1969.
  • [18] Louis Nirenberg, Lectures on linear partial differential equations, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17. MR 0450755
  • [19] L. Nirenberg, A certain problem of Hans Lewy, Uspehi Mat. Nauk 29 (1974), no. 2(176), 241–251 (Russian). Translated from the English by Ju. V. Egorov; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR 0492752
  • [20] Roger Penrose, Techniques of differential topology in relativity, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7. MR 0469146
  • [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. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), no. 1, 25–41. MR 520599
  • [24] Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554

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Article copyright: © Copyright 1988 American Mathematical Society