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The deformation theory of anti-self-dual conformal structures. (English) Zbl 0765.58005

Riemannian geometry in dimension 4 has some special features which are not present in any other dimension. In dimensions \(n\geq 5\) the group SO\((n)\) is simple, whereas SO\((4)\) is not simple. The bundle of 2-forms on an oriented 4-Riemannian manifold is decomposed into its self dual (SD) and anti-self-dual (ASD) parts under the action of the Hodge operator.
For a real 4-dimensional Kähler manifold, the Bochner tensor is equal to the ASD part of the Weyl tensor.
In the paper under review, the authors begin a systematic study of all solutions of the ASD equation on a given smooth closed 4-manifold. A theory of moduli spaces analogous to that of M. F. Atiyah, N. J. Hitchin and I. M. Singer [Proc. R. Soc. Lond., Ser. A 362, 425- 461 (1978; Zbl 0389.53011)] in the case of connections is developed.
The relationship with scalar curvature is investigated. Deformations of \((M,G)\)-structures are considered and interesting examples are given.
The moduli spaces of conformal structures can be used to obtain information about the differential topology of 4-manifolds, in the spirit of the Donaldson invariants.

MSC:

58D27 Moduli problems for differential geometric structures
53C20 Global Riemannian geometry, including pinching
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)

Citations:

Zbl 0389.53011
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References:

[1] Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond., Ser. A362, 425–461 (1978) · Zbl 0389.53011
[2] Atiyah, M.F., Singer, I.M.: The index of elliptic operators: III. Ann. Math.87, 546–604 (1968) · Zbl 0164.24301
[3] Besse, A.: Einstein manifolds. Berlin Heidelberg New York: Springer 1986 · Zbl 0613.53001
[4] Bourguignon, J.-P.: Une stratification de l’espace des structures Riemanniennes. Compos. Math.30, (Fasc. 1), 1–41 (1975) · Zbl 0301.58015
[5] Donaldson, S.K., Friedman, R.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity2, 197–239 (1989) · Zbl 0671.53029
[6] Ebin, D.G.: The manifold of Riemannian metrics. In: Chern, S.-S., Smale, S. (eds.) Global analysis. (Proc. Symp. Pure Math., vol. XV, pp. 11–40) Providence, RI: Am. Math. Soc. 1970 · Zbl 0205.53702
[7] Fischer, A.E., Marsden, J.E.: The manifold of conformally equivalent metrics. Can. J. Math.XXIX (No. 1), 193–209 (1977) · Zbl 0358.58006
[8] Floer, A.: Self-dual conformal structures on l\(\mathbb{C}\)P 2. J. Differ. Geom.33, 551–573 (1991) · Zbl 0736.53046
[9] Goldman, W.M.: Geometric structures on manifolds and varieties of representations. Contemp. Math.74, 169–198 (1988) · Zbl 0659.57004
[10] Goldman, W.M., Millson, J.J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Publ. Math., Inst. Hautes Étud. Sci.67, 43–96 (1988) · Zbl 0678.53059
[11] D’Ambra, G., Gromov, M.: Lectures on transformation groups: geometry and dynamics. Surv. Differ. Geom.1, 19–111 (1991) · Zbl 0752.57017
[12] Hitchin, N.J.: Linear field equations on self-dual spaces. Proc. R. Soc. Lond. Ser. A370, 173–191 (1980) · Zbl 0436.53058
[13] Hitchin, N.J.: Kählerian twistor spaces. Proc. Lond. Math. Soc., III. Ser.43, 133–150 (1981) · Zbl 0474.14024
[14] Hitchin, N.J.: On compact four-dimensional Einstein manifolds. J. Differ. Geom.9, 435–441 (1974) · Zbl 0281.53039
[15] Johnson, D., Millson, J.: Deformation spaces associated to compact hyperbolic manifolds. In: Howe, R. (ed.) Discrete groups in geometry and analysis. Boston Basel Stuttgart. Birkhäuser 1987 · Zbl 0664.53023
[16] Kuiper, N.H.: On conformally flat spaces in the large. Ann. Math.50, 916–924 (1949) · Zbl 0041.09303
[17] LeBrun, C.: Explicit self-dual metrics on \(\mathbb{C}\)P 2#...#\(\mathbb{C}\)P 2. J. Differ. Geom.34, 223–254 (1991) · Zbl 0725.53067
[18] Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc., New Ser.17 (No. 1), 37–91 (1987) · Zbl 0633.53062
[19] Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des variétés riemanniennes, application à la démonstration d’une conjecture de A. Lichnerowicz. C.R. Acad. Sci., Paris, Sér. A269, 583–586 (1969) · Zbl 0201.09701
[20] Obata, M.: Conformal transformations of compact Riemannian manifolds. Ill. J. Math.6, 292–295 (1962) · Zbl 0107.15703
[21] Omori, H.: Infinite dimensional Lie transformation groups. (Lect. Notes Math., vol. 427) Berlin Heidelberg New York: Springer 1974 · Zbl 0328.58005
[22] Sun Poon, Y.: Compact self-dual manifolds with positive scalar curvature. J. Differ. Geom.24, 97–132 (1986) · Zbl 0583.53054
[23] Sun Poon, Y.: Algebraic dimension of twistor spaces. Math. Ann.282, 621–627 (1988) · Zbl 0665.32014
[24] Salamon, S.: Topics in four-dimensional Riemannian geometry. In: Vesentini, E. (ed.) Geometry Seminar ’Luigi Bianchi’ 1982. (Lect. Notes Math., vol. 1022) Berlin Heidelberg New York: Springer 1983 · Zbl 0532.53035
[25] Salamon, S.: Riemannian geometry and holonomy groups. (Pitman Res. Notes Math. Ser., vol. 201) Harlow Essex: Longman 1989 · Zbl 0685.53001
[26] Schoen, R., Yau, S.-T.: Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math.92, 47–71 (1988) · Zbl 0658.53038
[27] Thurston, W.P.: The geometry and topology of 3-manifolds. Princeton lecture notes (1978/79)
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