Twistor CR manifolds and three-dimensional conformal geometry

LeBrun, Claude R.

doi:10.1090/S0002-9947-1984-0743735-9

Twistor CR manifolds and three-dimensional conformal geometry
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by Claude R. LeBrun PDF

Trans. Amer. Math. Soc. 284 (1984), 601-616 Request permission

Abstract:

A $\text {CR}$ (i.e. partially complex) $5$-manifold is contructed as a sphere bundle over an arbitrary $3$-manifold with conformal metric. This so-called twistor $CR$ manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract $\text {CR}$ manifold to arise via the construction. The above correspondence is then used to prove that a twistor $\text {CR}$ manifold is locally imbeddable as a real hypersurface in ${{\mathbf {C}}^3}$ only if it is real-analytic with respect to a suitable atlas.

References

M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
S. Bochner, Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2) 44 (1943), 652–673. MR 9206, DOI 10.2307/1969103
Adrien Douady, Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 1–95 (French). MR 203082
A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957), 121–138 (French). MR 87176, DOI 10.2307/2372388
K. Kodaira, A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2) 75 (1962), 146–162. MR 133841, DOI 10.2307/1970424
C. R. LeBrun, ${\cal H}$-space with a cosmological constant, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1778, 171–185. MR 652038, DOI 10.1098/rspa.1982.0035
Claude LeBrun, Spaces of complex null geodesics in complex-Riemannian geometry, Trans. Amer. Math. Soc. 278 (1983), no. 1, 209–231. MR 697071, DOI 10.1090/S0002-9947-1983-0697071-9
Hans Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2) 64 (1956), 514–522. MR 81952, DOI 10.2307/1969599
Louis Nirenberg, Lectures on linear partial differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17, 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. MR 0450755
R. Penrose, D. W. Sciama, and C. J. Isham (eds.), Quantum gravity, Clarendon Press, Oxford, 1975. An Oxford Symposium held at the Rutherford Laboratory, Chilton, February 15–16, 1974. MR 0459344
Roger Penrose, Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), no. 1, 31–52. MR 439004, DOI 10.1007/bf00762011
Roger Penrose, Physical space-time and nonrealizable CR-structures, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 3, 427–448. MR 693958, DOI 10.1090/S0273-0979-1983-15109-1
Alan Weinstein, Lectures on symplectic manifolds, Regional Conference Series in Mathematics, No. 29, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976. MR 0464312
R. O. Wells Jr., Holomorphic approximation on real-analytic submanifolds of a complex manifold, Proc. Amer. Math. Soc. 17 (1966), 1272–1275. MR 200946, DOI 10.1090/S0002-9939-1966-0200946-3