Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics
Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 921-963.

Nous considérons la correspondance de Jones et Tod entre variétés conformes autoduales admettant un champ de vecteurs conforme et les monopoles abéliens sur les variétés de Weyl-Einstein de dimension 3, et nous montrons que les structures complexes invariantes correspondent aux congruences géodésiques sans distorsion. Comme les variétés de Weyl-Einstein tri-dimensionnelles admettent de nombreuses congruences de ce type, cette correspondance offre un mode de construction général de géométries autoduales, qui inclut les constructions bien connues des métriques kählériennes à courbure scalaire nulle et des structures hypercomplexes avec symétrie. Nous montrons également qu’en présence d’une telle congruence l’équation de Weyl-Einstein équivaut à une paire couplée d’équations de monopoles que nous résolvons dans un cas particulier. À partir de ces nouveaux exemples, appelés “espaces de Weyl-Einstein à symétrie géodésique”, nous construisons des structures hypercomplexes admettant deux champs de vecteurs tri-holomorphes commutant entre eux.

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new Einstein-Weyl spaces, which we call Einstein-Weyl “with a geodesic symmetry”, give rise to hypercomplex structures with two commuting triholomorphic vector fields.

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     title = {Selfdual spaces with complex structures, {Einstein-Weyl} geometry and geodesics},
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Calderbank, David M J.; Pedersen, Henrik. Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics. Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 921-963. doi : 10.5802/aif.1779. http://www.numdam.org./articles/10.5802/aif.1779/

[1] V. Apostolov, P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Int. J. Math., 8 (1997), 421-439. | MR | Zbl

[2] A.L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., vol. 10, Springer, Berlin, 1987. | MR | Zbl

[3] C.P. Boyer, J.D. Finley, Killing vectors in self-dual Euclidean Einstein spaces, J. Math. Phys., 23 (1982), 1126-1130. | MR | Zbl

[4] D.M.J. Calderbank, The geometry of the Toda equation, Edinburgh Preprint MS-99-003, 1999, to appear in J. Geom. Phys. | Zbl

[5] D.M.J. Calderbank, H. Pedersen, Einstein-Weyl geometry, in Essays on Einstein Manifolds (eds. C.R. LeBrun and M. Wang), Surveys in Differential Geometry, vol. V, International Press. | Zbl

[6] D.M.J. Calderbank, K.P. Tod, Einstein metrics, hypercomplex structures and the Toda field equation, Edinburgh Preprint MS-98-011, 1998, to appear in Diff. Geom. Appl. | Zbl

[7] T. Chave, K.P. Tod, G. Valent, (4,0) and (4,4) sigma models with a triholomorphic Killing vector, Phys. Lett., B 383 (1996), 262-270.

[8] M. Dunajski, K.P. Tod, Einstein-Weyl structures from hyper-Kähler metrics with conformal Killing vectors, Preprint ESI 739, Vienna, 1999.

[9] P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann., 267 (1984), 495-518. | MR | Zbl

[10] P. Gauduchon, Structures de Weyl et théorèmes d'annulation sur une variété conforme autoduale, Ann. Sci. Norm. Sup. Pisa, 18 (1991), 563-629. | Numdam | MR | Zbl

[11] P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S1 x S3, J. reine angew. Math., 469 (1995), 1-50. | MR | Zbl

[12] P. Gauduchon, K.P. Tod, Hyperhermitian metrics with symmetry, J. Geom. Phys., 25 (1998), 291-304. | MR | Zbl

[13] G.W. Gibbons, S.W. Hawking, Gravitational multi-instantons, Phys. Lett., 78 B (1978), 430-432.

[14] N.J. Hitchin, Complex manifolds and Einstein equations, in Twistor Geometry and Non-linear Systems (eds H.D. Doebner and T.D. Palev), Primorsko 1980, Lecture Notes in Math., vol. 970, Springer, Berlin, 1982, 79-99. | Zbl

[15] S.A. Huggett, K.P. Tod, An Introduction to Twistor Theory, Cambridge University Press, Cambridge, 1985. | MR | Zbl

[16] P.E. Jones, K.P. Tod, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav., 2 (1985), 565-577. | MR | Zbl

[17] D.D. Joyce, Explicit construction of self-dual 4-manifolds, Duke Math. J., 77 (1995), 519-552. | MR | Zbl

[18] C.R. Lebrun, Counterexamples to the generalized positive action conjecture, Comm. Math. Phys., 118 (1988), 591-596. | MR | Zbl

[19] C.R. Lebrun, Explicit self-dual metrics on ℂP2#...#ℂP2, J. Diff. Geom., 34 (1991), 223-253. | MR | Zbl

[20] C.R. Lebrun, Self-dual manifolds and hyperbolic geometry, in Einstein Metrics and Yang-Mills Connections (eds. T. MAbuchi and S. Mukai), Sanda 1990, Lecture Notes in Pure and Appl. Math., vol. 145, Marcel Dekker, New York, 1993, 99-131. | Zbl

[21] H.-C. Lee, A kind of even-dimensional differential geometry and its application to exterior calculus, Amer. J. Math., 65 (1943), 433-438. | MR | Zbl

[22] A.B. Madsen, Einstein-Weyl structures in the conformal classes of LeBrun metrics, Class. Quantum Grav., 14 (1997), 2635-2645. | MR | Zbl

[23] L.J. Mason, N.M.J. Woodhouse, Integrability, Self-duality and Twistor Theory, Clarendon Press, Oxford, 1996. | MR | Zbl

[24] H. Pedersen, Einstein metrics, spinning top motions and monopoles, Math. Ann., 274 (1986), 35-39. | MR | Zbl

[25] H. Pedersen, A. Swann, Riemannian submersions, four-manifolds and Einstein-Weyl geometry, Proc. London Math. Soc., 66 (1993), 381-399. | MR | Zbl

[26] H. Pedersen, K.P. Tod, Three-dimensional Einstein-Weyl geometry, Adv. Math., 97 (1993), 74-109. | MR | Zbl

[27] H. Pedersen, K.P. Tod, Einstein metrics and hyperbolic monopoles, Class. Quantum Grav., 8 (1991), 751-760. | MR | Zbl

[28] K.P. Tod, Compact 3-dimensional Einstein-Weyl structures, J. London Math. Soc., 45 (1992), 341-351. | MR | Zbl

[29] K.P. Tod, Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Class. Quantum Grav., 12 (1995), 1535-1547. | MR | Zbl

[30] K.P. Tod, Cohomogeneity-one metrics with self-dual Weyl tensor, in Twistor Theory (ed. S. Huggett), Lecture Notes in Pure and Appl. Math., vol. 169, Marcel Dekker, Plymouth, 1995, 171-184. | MR | Zbl

[31] K.P. Tod, The SU(∞)-Toda field equation and special four-dimensional metrics, in Geometry and Physics (eds. J.E. Andersen, J. Dupont, H. Pedersen and A. Swann), Lecture Notes in Pure and Appl. Math., vol. 184, Marcel Dekker, Aarhus, 1995, 307-312. | MR | Zbl

[32] I. Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math., 24 (1976), 338-351. | MR | Zbl

[33] R.S. Ward, Einstein-Weyl spaces and SU(∞) Toda fields, Class. Quantum Grav., 7 (1990), L95-L98. | MR | Zbl

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