A bridge between complex geometry and Riemannian geometry

Cassa, Antonio

doi:10.1090/S0002-9939-1993-1152272-2

A bridge between complex geometry and Riemannian geometry

Author: Antonio Cassa
Journal: Proc. Amer. Math. Soc. 119 (1993), 621-628
MSC: Primary 53C56; Secondary 32C10
DOI: https://doi.org/10.1090/S0002-9939-1993-1152272-2
MathSciNet review: 1152272
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Abstract: Every complex manifold ${M^n}$ with holomorphic metric can be obtained (at least locally) from a complex manifold ${E^{2 \bullet n - 2}}$ and one of its $\mathbb{C}$ principal bundles . The manifold is made of all (signed) complex geodesies of and is the bundle on of all choices of "times" evolving along the geodesies.

[BM] R. J. Baston and L. J. Mason, Conformal gravity, the Einstein equations and spaces of complex null geodesics, Classical Quantum Gravity 4 (1987), no. 4, 815–826. MR 895903
[EL1] Michael Eastwood and Claude LeBrun, Thickening and supersymmetric extensions of complex manifolds, Amer. J. Math. 108 (1986), no. 5, 1177–1192. MR 859775, https://doi.org/10.2307/2374601
[EL2] Michael Eastwood and Claude LeBrun, Fattening complex manifolds: curvature and Kodaira-Spencer maps, J. Geom. Phys. 8 (1992), no. 1-4, 123–146. MR 1165875, https://doi.org/10.1016/0393-0440(92)90045-3
[IYG] J. Isenberg, P. Yassin, and P. S. Green, Non-self-dual gauge fields, Phys. Lett. B 78 (1978), 462-464.
[L1] Claude LeBrun, Spaces of complex null geodesics in complex-Riemannian geometry, Trans. Amer. Math. Soc. 278 (1983), no. 1, 209–231. MR 697071, https://doi.org/10.1090/S0002-9947-1983-0697071-9
[L2] Claude LeBrun, The first formal neighbourhood of ambitwistor space for curved space-time, Lett. Math. Phys. 6 (1982), no. 5, 345–354. MR 677436, https://doi.org/10.1007/BF00419314
[L3] -, Ambitwistors and Einstein equations, Classical Quantum Gravity 2 (1985), 555-563.
[L4] Claude LeBrun, Thickenings and gauge fields, Classical Quantum Gravity 3 (1986), no. 6, 1039–1059. MR 868717
[L5] Claude LeBrun, Thickenings and conformal gravity, Comm. Math. Phys. 139 (1991), no. 1, 1–43. MR 1116408
[M] Yuri I. Manin, Gauge field theory and complex geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 289, Springer-Verlag, Berlin, 1988. Translated from the Russian by N. Koblitz and J. R. King. MR 954833
[P] Roger Penrose, Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), no. 1, 31–52. MR 439004, https://doi.org/10.1007/bf00762011
[PW] R. Penrose and R. S. Ward, Twistors for flat and curved space-time, General relativity and gravitation, Vol. 2, Plenum, New York-London, 1980, pp. 283–328. MR 617923
[W] E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. B 77 (1978), 394-397.