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

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.

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.

@article{AIF_2000__50_3_921_0,
     author = {Calderbank, David M J. and Pedersen, Henrik},
     title = {Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {3},
     year = {2000},
     pages = {921-963},
     doi = {10.5802/aif.1779},
     zbl = {0970.53027},
     mrnumber = {2001h:53058},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_3_921_0}
}
Calderbank, David M J.; Pedersen, Henrik. Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics. Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 921-963. doi : 10.5802/aif.1779. http://www.numdam.org/item/AIF_2000__50_3_921_0/

[1] V. Apostolov, P. Gauduchon, The Riemannian Goldberg-Sachs theorem, Int. J. Math., 8 (1997), 421-439. | MR 98g:53080 | Zbl 0891.53054

[2] A.L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb., vol. 10, Springer, Berlin, 1987. | MR 88f:53087 | Zbl 0613.53001

[3] C.P. Boyer, J.D. Finley, Killing vectors in self-dual Euclidean Einstein spaces, J. Math. Phys., 23 (1982), 1126-1130. | MR 84f:53064 | Zbl 0484.53051

[4] D.M.J. Calderbank, The geometry of the Toda equation, Edinburgh Preprint MS-99-003, 1999, to appear in J. Geom. Phys. | Zbl 0979.53046

[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 0996.53030

[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 01613413

[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 87a:53101 | Zbl 0536.53066

[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 93d:32046 | Zbl 0763.53034

[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 97d:53048 | Zbl 0858.53039

[12] P. Gauduchon, K.P. Tod, Hyperhermitian metrics with symmetry, J. Geom. Phys., 25 (1998), 291-304. | MR 2000b:53064 | Zbl 0945.53042

[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 0507.53025

[15] S.A. Huggett, K.P. Tod, An Introduction to Twistor Theory, Cambridge University Press, Cambridge, 1985. | MR 87i:32042 | Zbl 0573.53001

[16] P.E. Jones, K.P. Tod, Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav., 2 (1985), 565-577. | MR 87b:53115 | Zbl 0575.53042

[17] D.D. Joyce, Explicit construction of self-dual 4-manifolds, Duke Math. J., 77 (1995), 519-552. | MR 96d:53049 | Zbl 0855.57028

[18] C.R. Lebrun, Counterexamples to the generalized positive action conjecture, Comm. Math. Phys., 118 (1988), 591-596. | MR 89f:53107 | Zbl 0659.53050

[19] C.R. Lebrun, Explicit self-dual metrics on ℂP2#...#ℂP2, J. Diff. Geom., 34 (1991), 223-253. | MR 92g:53040 | Zbl 0725.53067

[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 0802.53010

[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 5,15h | Zbl 0060.38302

[22] A.B. Madsen, Einstein-Weyl structures in the conformal classes of LeBrun metrics, Class. Quantum Grav., 14 (1997), 2635-2645. | MR 99a:53059 | Zbl 0899.53040

[23] L.J. Mason, N.M.J. Woodhouse, Integrability, Self-duality and Twistor Theory, Clarendon Press, Oxford, 1996. | MR 98f:58002 | Zbl 0856.58002

[24] H. Pedersen, Einstein metrics, spinning top motions and monopoles, Math. Ann., 274 (1986), 35-39. | MR 87i:53070 | Zbl 0566.53058

[25] H. Pedersen, A. Swann, Riemannian submersions, four-manifolds and Einstein-Weyl geometry, Proc. London Math. Soc., 66 (1993), 381-399. | MR 94c:53061 | Zbl 0788.53040

[26] H. Pedersen, K.P. Tod, Three-dimensional Einstein-Weyl geometry, Adv. Math., 97 (1993), 74-109. | MR 93m:53042 | Zbl 0778.53041

[27] H. Pedersen, K.P. Tod, Einstein metrics and hyperbolic monopoles, Class. Quantum Grav., 8 (1991), 751-760. | MR 92f:53053 | Zbl 0726.53024

[28] K.P. Tod, Compact 3-dimensional Einstein-Weyl structures, J. London Math. Soc., 45 (1992), 341-351. | MR 93d:53058 | Zbl 0761.53026

[29] K.P. Tod, Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Class. Quantum Grav., 12 (1995), 1535-1547. | MR 96f:53068 | Zbl 0828.53061

[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 95i:53056 | Zbl 0827.53017

[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 98a:53068 | Zbl 0876.53026

[32] I. Vaisman, On locally conformal almost Kähler manifolds, Israel J. Math., 24 (1976), 338-351. | MR 54 #6048 | Zbl 0335.53055

[33] R.S. Ward, Einstein-Weyl spaces and SU(∞) Toda fields, Class. Quantum Grav., 7 (1990), L95-L98. | MR 91g:83019 | Zbl 0687.53044