Product twistor spaces and Weyl geometry

Davidov, Johann

doi:10.1090/proc/14700

Product twistor spaces and Weyl geometry
HTML articles powered by AMS MathViewer

by Johann Davidov PDF

Proc. Amer. Math. Soc. 148 (2020), 3491-3506 Request permission

Abstract:

Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle $\mathcal {Z}\times \mathcal {Z}\to M$, where $\mathcal {Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symmetric torsion. These structures are defined by means of the connection $D$ and four (Kähler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry, and this is used to supply examples satisfying these conditions.

References

M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
L. Bérard-Bergery and T. Ochiai, On some generalizations of the construction of twistor spaces, Global Riemannian geometry (Durham, 1983) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984, pp. 52–59. MR 757205
Arthur L. Besse, Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR 2371700
Guy Bonneau, Einstein-Weyl structures corresponding to diagonal Kähler Bianchi IX metrics, Classical Quantum Gravity 14 (1997), no. 8, 2123–2135. MR 1468571, DOI 10.1088/0264-9381/14/8/012
David M. J. Calderbank and Henrik Pedersen, Einstein-Weyl geometry, Surveys in differential geometry: essays on Einstein manifolds, Surv. Differ. Geom., vol. 6, Int. Press, Boston, MA, 1999, pp. 387–423. MR 1798617, DOI 10.4310/SDG.2001.v6.n1.a14
J. Davidov and O. Muškarov, Existence of holomorphic functions on twistor spaces, Bull. Soc. Math. Belg. Sér. B 40 (1988), no. 2, 131–151. MR 950996
Johann Davidov, Harmonic almost Hermitian structures, Special metrics and group actions in geometry, Springer INdAM Ser., vol. 23, Springer, Cham, 2017, pp. 129–159. MR 3751965
Johann Davidov, Generalized metrics and generalized twistor spaces, Math. Z. 291 (2019), no. 1-2, 17–46. MR 3936058, DOI 10.1007/s00209-018-2071-8
Guillaume Deschamps, Espace de twisteurs des structures complexes généralisées, Math. Z. 279 (2015), no. 3-4, 703–721 (French, with French summary). MR 3318246, DOI 10.1007/s00209-014-1389-0
Michel Dubois-Violette, Structures complexes au-dessus des variétés, applications, Mathematics and physics (Paris, 1979/1982) Progr. Math., vol. 37, Birkhäuser Boston, Boston, MA, 1983, pp. 1–42 (French). MR 728412, DOI 10.1007/BF02591680
J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 589–640 (1986). MR 848842
Ana Cristina Ferreira, Einstein four-manifolds with skew torsion, J. Geom. Phys. 61 (2011), no. 12, 2341–2351. MR 2838511, DOI 10.1016/j.geomphys.2011.07.009
Paul Gauduchon, Structures de Weyl et théorèmes d’annulation sur une variété conforme autoduale, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 4, 563–629 (French). MR 1153706
Paul Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1\times S^3$, J. Reine Angew. Math. 469 (1995), 1–50 (French). MR 1363825, DOI 10.1515/crll.1995.469.1
M. Gualtieri, Generalized complex geometry, Ph.D. thesis, St John’s College, University of Oxford, arXiv:math.DG/0401221, 2003.
Nigel Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), no. 3, 281–308. MR 2013140, DOI 10.1093/qjmath/54.3.281
Nigel Hitchin, Generalized geometry—an introduction, Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., vol. 16, Eur. Math. Soc., Zürich, 2010, pp. 185–208. MR 2681591, DOI 10.4171/079-1/6
Inyoung Kim, Almost-Kahler Anti-Self-Dual Metrics, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 3271886
Jongsu Kim and Massimiliano Pontecorvo, A new method of constructing scalar-flat Kähler surfaces, J. Differential Geom. 41 (1995), no. 2, 449–477. MR 1331974
Takashi Koda, Self-dual and anti-self-dual Hermitian surfaces, Kodai Math. J. 10 (1987), no. 3, 335–342. MR 929993, DOI 10.2996/kmj/1138037464
Claude LeBrun, Poon’s self-dual metrics and Kähler geometry, J. Differential Geom. 28 (1988), no. 2, 341–343. MR 961518
Claude LeBrun, Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann. 289 (1991), no. 3, 383–392. MR 1096177, DOI 10.1007/BF01446578
Claude LeBrun, Anti-self-dual metrics and Kähler geometry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 498–507. MR 1403950
N. R. O’Brian and J. H. Rawnsley, Twistor spaces, Ann. Global Anal. Geom. 3 (1985), no. 1, 29–58. MR 812312, DOI 10.1007/BF00054490
Henrik Pedersen and Andrew Swann, Riemannian submersions, four-manifolds and Einstein-Weyl geometry, Proc. London Math. Soc. (3) 66 (1993), no. 2, 381–399. MR 1199072, DOI 10.1112/plms/s3-66.2.381
Henrik Pedersen and Andrew Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew. Math. 441 (1993), 99–113. MR 1228613, DOI 10.1515/crll.1993.440.99
Yann Rollin and Michael Singer, Non-minimal scalar-flat Kähler surfaces and parabolic stability, Invent. Math. 162 (2005), no. 2, 235–270. MR 2199006, DOI 10.1007/s00222-004-0436-6
I. M. Singer and J. A. Thorpe, The curvature of $4$-dimensional Einstein spaces, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355–365. MR 0256303
Vivian De Smedt and Simon Salamon, Anti-self-dual metrics on Lie groups, Differential geometry and integrable systems (Tokyo, 2000) Contemp. Math., vol. 308, Amer. Math. Soc., Providence, RI, 2002, pp. 63–75. MR 1955629, DOI 10.1090/conm/308/05312
Izu Vaisman, Some curvature properties of complex surfaces, Ann. Mat. Pura Appl. (4) 132 (1982), 1–18 (1983). MR 696036, DOI 10.1007/BF01760974
F. Witt, Special metric structures and closed forms, Ph.D. thesis, Oxvord University, arXiv:math.DG/0502443, 2004.