Abstract
A twistor correspondence is given for complex conformal space-times with vanishing Bach and Eastwood-Dighton tensors; when the Weyl curvature is algebraically general, these equations are precisely the conformal version of Einstein's vacuum equations with cosmological constant. This gives a fully curved version of the linearized correspondence of Baston and Mason [B-M].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[B] Bach, R.: Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krummentensorbegriffs. Math. Zeit.9, 110–135 (1921)
[B-M] Baston, R., Mason, L.: Conformal gravity, the Einstein Equations and spaces of complex null geodesics. Class. Quantum. Grav.4, 815–826 (1987)
[Bu] Buchdahl, N.: On the relative deRham sequence. Proc. AMS 87, 363–366 (1983)
[C-L] Chau, L.-L. Lim, C.-S.: Geometrical constraints and equations of motion in extended supergravity. Phys. Rev. Lett.56, 294–297 (1986)
[D] Dighton, K.: An introduction to the theory of local twistors. Int. J. Theor. Phys.11, 31–43 (1974)
[E-L] Eastwood, M., LeBrun, C.: Thickenings and supersymmetric extensions of complex manifolds. Am. J. Math.108, 1177–1192 (1986)
[E-L2] Eastwood, M., LeBrun, C.: Fattening complex manifolds. Preprint (1989)
[G] Graham, R.: Private communication
[I-Y] Isenberg, J., Yasskin, P.: Non-self-dual nonlinear graviatons. Gen. Rel. Grav.14, 621–627 (1982)
[I-Y-G] Isenberg, J., Yasskin, P., Green, P.: Non-self-dual Gauge Fields. Phys. Lett.78B, 462–464 (1978)
[K-N-T] Kozameh, C., Newman, E.T., Tod, K.P.: Conformal Einstein spaces. Gen. Rel. Grav.17, 343–352 (1985)
[L1] LeBrun, C.: The first formal neighborhood of ambitwistor space for curved space-time. Lett. Math. Phys.6, 345–354 (1982)
[L2] LeBrun, C.: Spaces of complex null geodesics in complex-Riemannian Geometry. Trans. AMS278, 209–231 (1983)
[L3] LeBrun 3. Ambitwistors and Einstein's Equations. Class. Quantum Grav.2, 555–563 (1985)
[L4] LeBrun, C.: Thickenings and gauge fields. Class. Quantum. Grav.3, 1039–1059 (1986)
[M] Manin, Y. I. Gauge Field Theory and Complex Geometry. Berlin, Heidelberg, New York: 1988; translated from the Russian. Kalibrovochnye polya i kompleksnaya geometriya. Moscow: Nauka, 1984 by Koblitz, N., King J
[Ma] Mason, L.: Private communication
[P] Penrose, R.: Non-linear gravitons and curved twistor theory. Gen. Rel. Grav.7, 31–52 (1976)
[P-R] Penrose, R., Rindler, W.: Spinors and Space-Time. Vol. 2. Cambridge, London, New York: Cambridge Univ. Press 1986
[W] Weinstein, A.: The local structure of Poisson manifold. J. Diff. Geom.18 523–557 (1983)
[Wi1] Witten, E.: An interpretation of classical Yang-Mills theory. Phys. Lett.77B, 394–398 (1978)
[Wi2] Witten, E.: Twistor-like transform in ten dimensions. Nucl. Phys. B266 245–264 (1986)
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Research supported in part by NSF grant DMS-8704401
Rights and permissions
About this article
Cite this article
LeBrun, C. Thickenings and conformal gravity. Commun.Math. Phys. 139, 1–43 (1991). https://doi.org/10.1007/BF02102727
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02102727