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)



Holomorphic curves in Lorentzian CR-manifolds

Author: Robert L. Bryant
Journal: Trans. Amer. Math. Soc. 272 (1982), 203-221
MSC: Primary 32F25; Secondary 53C40
MathSciNet review: 656486
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A CR-manifold is said to be Lorentzian if its Levi form has one negative eigenvalue and the rest positive. In this case, it is possible that the CR-manifold contains holomorphic curves. In this paper, necessary and sufficient conditions are derived (in terms of the "derivatives" of the CR-structure) in order that holomorphic curves exist. A "flatness" theorem is proven characterizing the real Lorentzian hyperquadric $ {Q_5} \subseteq {\mathbf{C}}{P^3}$, and examples are given showing that nonflat Lorentzian hyperquadrics can have a richer family of holomorphic curves than the flat ones.

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

  • [R] Bryant, S. S. Chern and P. A. Griffiths, Notes on exterior differential systems, Proc. 1980 Peking Symposium on Differential Equations, Peking University, 1980.
  • [E] Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom. 1 (1967), 111-127. MR 0233294 (38:1616)
  • [E] Cartan, "Le calcul de variations et certaines familles de courbes" in Oeuvres, Vol. II, Gauthier-Villars, Paris, 1983, pp. 1011-1034.
  • [S] S. Chern, Complex manifolds without potential theory, 2nd ed., Springer-Verlag, Berlin and New York, 1979. MR 533884 (80f:32001)
  • [S] S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. MR 0425155 (54:13112)
  • [P] A. Griffiths, On Cartan's method of Lie groups and moving frames as applied to existence and uniqueness questions in differential geometry, Duke Math. J. 41 (1974), 775-814. MR 0410607 (53:14355)
  • [H] B. Lawson, Jr., Minimal varieties in constant curvature manifolds, Ph.D. Thesis, Stanford, Calif., 1968.
  • [L] Nirenberg, On a question of Hans Lewy, Uspehi Mat. Nauk 29 (1965), 251-262.
  • [R] Penrose, Twistor algebra, J. Math. Phys. 8 (1967), 345-366. MR 0216828 (35:7657)
  • [F] Sommer, Komplex-analytische Blätterung reeller Hyperflächen im $ {{\mathbf{C}}^n}$, Math. Ann. 137 (1959), S 392-411. MR 0108821 (21:7533)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F25, 53C40

Retrieve articles in all journals with MSC: 32F25, 53C40

Additional Information

Keywords: Prolongation, CR-structure, Levi form, holomorphic curve, Lorentzian hyperquadric
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society