Holomorphic curves in Lorentzian CR-manifolds

Bryant, Robert L.

doi:10.1090/S0002-9947-1982-0656486-4

Holomorphic curves in Lorentzian CR-manifolds
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by Robert L. Bryant

Trans. Amer. Math. Soc. 272 (1982), 203-221

DOI: https://doi.org/10.1090/S0002-9947-1982-0656486-4

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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.

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