Curvature Properties of Twistor Spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 84-105.

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We review some results on the Riemannian and almost Hermitian geometry of twistor spaces of oriented Riemannian $4$-manifolds with emphasis on their curvature properties.
Keywords: twistor spaces, almost Hermitian structures, Riemannian and Hermitian curvatures of twistor spaces.
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Johann Davidov; Oleg Mushkarov. Curvature Properties of Twistor Spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and mathematical physics, Tome 311 (2020), pp. 84-105. http://geodesic.mathdoc.fr/item/TM_2020_311_a4/

[1] Alexandrov B., Grantcharov G., Ivanov S., “Curvature properties of twistor spaces of quaternionic Kähler manifolds”, J. Geom., 62:1–2 (1998), 1–12 | DOI | MR | Zbl

[2] Ali D., Davidov J., Mushkarov O., “Compatible almost complex structures on twistor spaces and their Gray–Hervella classes”, J. Geom. Phys., 75 (2014), 213–229 | DOI | MR | Zbl

[3] Ali D., Davidov J., Mushkarov O., “Holomorphic curvatures of twistor spaces”, Int. J. Geom. Methods Mod. Phys., 11:3 (2014), 1450022 | DOI | MR | Zbl

[4] Angella D., Otal A., Ugarte L., Villacampa R., On Gauduchon connections with Kähler-like curvature, E-print, 2018, arXiv: 1809.02632v1 [math.DG]

[5] Apostolov V., “Compact $*$-Einstein Hermitian surfaces of negative type”, C. r. Acad. Bulg. sci., 48:11–12 (1995), 19–22 | MR | Zbl

[6] Apostolov V., Armstrong J., Drăghici T., “Local rigidity of certain classes of almost Kähler 4-manifolds”, Ann. Global Anal. Geom., 21:2 (2002), 151–176 | DOI | MR | Zbl

[7] Apostolov V., Armstrong J., Drăghici T., “Local models and integrability of certain almost Kähler 4-manifolds”, Math. Ann., 323:4 (2002), 633–666 | DOI | MR | Zbl

[8] Apostolov V., Davidov J., “Compact Hermitian surfaces and isotropic curvature”, Ill. J. Math., 44:2 (2000), 438–451 | DOI | MR | Zbl

[9] Apostolov V., Davidov J., Muškarov O., “Compact self-dual Hermitian surfaces”, Trans. Amer. Math. Soc., 348:8 (1996), 3051–3063 | DOI | MR | Zbl

[10] Apostolov V., Drăghici T., “Almost Kähler 4-manifolds with $J$-invariant Ricci tensor and special Weyl tensor”, Q. J. Math., 51:3 (2000), 275–294 | DOI | MR | Zbl

[11] Apostolov V., Drăghici T., “The curvature and the integrability of almost-Kähler manifolds: A survey”, Symplectic and contact topology: Interactions and perspectives, Fields Inst. Commun., 35, Amer. Math. Soc., Providence, RI, 2003, 25–53 | MR | Zbl

[12] Apostolov V., Gauduchon P., “The Riemannian Goldberg–Sachs theorem”, Int. J. Math., 8:4 (1997), 421–439 | DOI | MR | Zbl

[13] Apostolov V., Gauduchon P., “Selfdual Einstein Hermitian four-manifolds”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 5, 1:1 (2002), 203–243 | MR | Zbl

[14] Apostolov V., Grantcharov G., Ivanov S., “Hermitian structures on twistor spaces”, Ann. Global Anal. Geom., 16:3 (1998), 291–308 | DOI | MR | Zbl

[15] Armstrong J., “An ansatz for almost-Kähler, Einstein 4-manifolds”, J. reine angew. Math., 542 (2002), 53–84 | MR | Zbl

[16] Atiyah M.F., Hitchin N.J., Singer I.M., “Self-duality in four-dimensional Riemannian geometry”, Proc. R. Soc. London A, 362 (1978), 425–461 | DOI | MR | Zbl

[17] Balas A., “Compact Hermitian manifolds of constant holomorphic sectional curvature”, Math. Z., 189 (1985), 193–210 | DOI | MR | Zbl

[18] Balas A., Gauduchon P., “Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler”, Math. Z., 190 (1985), 39–43 | DOI | MR | Zbl

[19] Besse A.L., Einstein manifolds, Springer, Berlin, 1987 | MR | Zbl

[20] Bishop R.L., Goldberg S.I., “Some implications of the generalized Gauss–Bonnet theorem”, Trans. Amer. Math. Soc., 112 (1964), 508–535 | DOI | MR | Zbl

[21] Bismut J.-M., “A local index theorem for non-Kähler manifolds”, Math. Ann., 284:4 (1989), 681–699 | DOI | MR | Zbl

[22] Blair D.E., Riemannian geometry of contact and symplectic manifolds, Prog. Math., 203, Birkhäuser, Boston, 2002 | MR | Zbl

[23] Blair D.E., Ianus S., “Critical associated metrics on symplectic manifolds”, Nonlinear problems in geometry, Proc. AMS spec. sess., 1985, Contemp. Math., 51, ed. by D.M. DeTurck, Amer. Math. Soc., Providence, RI, 1986, 23–29 | DOI | MR

[24] Blanchard A., “Sur les variétés analytiques complexes”, Ann. sci. Éc. norm. supér. Sér. 3, 73 (1956), 157–202 | MR | Zbl

[25] Boothby W.M., “Hermitian manifolds with zero curvature”, Mich. Math. J., 5 (1958), 229–233 | DOI | MR | Zbl

[26] Calderbank D.M.J., Pedersen H., “Selfdual Einstein metrics with torus symmetry”, J. Diff. Geom., 60:3 (2002), 485–521 | DOI | MR | Zbl

[27] Chen X.X., “On Kähler manifolds with positive orthogonal bisectional curvature”, Adv. Math., 215:2 (2007), 427–445 | DOI | MR | Zbl

[28] Chern S.-s., Complex manifolds without potential theory, 2nd ed., Springer, New York, 1979 | MR | Zbl

[29] Conder M., Maclachlan C., “Compact hyperbolic 4-manifolds of small volume”, Proc. Amer. Math. Soc., 133:8 (2005), 2469–2476 | DOI | MR | Zbl

[30] D'Atri J.E., Nickerson H.K., “Divergence-preserving geodesic symmetries”, J. Diff. Geom., 3 (1969), 467–476 | DOI | MR | Zbl

[31] Davidov J., “Einstein condition and twistor spaces of compatible partially complex structures”, Diff. Geom. Appl., 22:2 (2005), 159–179 | DOI | MR | Zbl

[32] Davidov J., “Harmonic almost Hermitian structures”, Special metrics and group actions in geometry, Proc. INdAM Workshop “New Perspectives in Differential Geometry,” (Rome, 2015), Springer INdAM Ser., 23, Springer, Cham, 2017, 129–159 | DOI | MR | Zbl

[33] Davidov J., Grantcharov G., Muškarov O., “Twistorial examples of $*$-Einstein manifolds”, Ann. Global Anal. Geom., 20:2 (2001), 103–115 | DOI | MR | Zbl

[34] Davidov J., Grantcharov G., Muškarov O., “Curvature properties of the Chern connection of twistor spaces”, Rocky Mt. J. Math., 39:1 (2009), 27–48 | DOI | MR | Zbl

[35] Davidov J., Muškarov O., “Twistor spaces with Hermitian Ricci tensor”, Proc. Amer. Math. Soc., 109:4 (1990), 1115–1120 | DOI | MR | Zbl

[36] Davidov J., Muškarov O., “On the Riemannian curvature of a twistor space”, Acta math. Hung., 58:3–4 (1991), 319–332 | DOI | MR | Zbl

[37] Davidov J., Mushkarov O., “Twistorial examples of almost Hermitian manifolds with Hermitian Ricci tensor”, Acta math. Hung., 156:1 (2018), 194–203 | DOI | MR | Zbl

[38] Davidov J., Muškarov O., Grantcharov G., “Almost complex structures on twistor spaces”, Almost complex structures, Proc. Int. Workshop (Sofia, 1992), World Scientific, Singapore, 1994, 113–149 | MR | Zbl

[39] Davidov J., Muškarov O., Grantcharov G., “Kähler curvature identities for twistor spaces”, Ill. J. Math., 39:4 (1995), 627–634 | DOI | MR | Zbl

[40] Derdziński A., “Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor”, Math. Z., 172 (1980), 273–280 | DOI | MR | Zbl

[41] Derdziński A., “Exemples de métriques de Kähler et d'Einstein autoduales sur le plan complexe”, Géométrie riemannienne en dimension 4: Sémin. Arthur Besse 1978/79, ed. by L. Bérard Bergery, M. Berger, C. Houzel, CEDIC/Fernand Nathan, Paris, 1981, 334–346 | MR

[42] Deschamps G., “Compatible complex structures on twistor space”, Ann. Inst. Fourier, 61:6 (2011), 2219–2248 | DOI | MR | Zbl

[43] Donaldson S.K., “Remarks on gauge theory, complex geometry and 4-manifold topology”, Fields medallists' lectures, ed. by M. Atiyah, D. Iagolnitzer, World Scientific, Singapore, 1997, 384–403 | DOI | MR

[44] Drăghici T.C., “On the almost Kähler manifolds with Hermitian Ricci tensor”, Houston J. Math., 20:2 (1994), 293–298 | MR

[45] Drăghici T.C., “On some 4-dimensional almost Kähler manifolds”, Kodai Math. J., 18:1 (1995), 156–168 | DOI | MR

[46] Drăghici T.C., “Almost Kähler 4-manifolds with $J$-invariant Ricci tensor”, Houston J. Math., 25:1 (1999), 133–145 | MR

[47] Eells J., Salamon S., “Twistorial construction of harmonic maps of surfaces into four-manifolds”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 12 (1985), 589–640 | MR | Zbl

[48] Fine J., Panov D., “Symplectic Calabi–Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold”, J. Diff. Geom., 82:1 (2009), 155–205 | DOI | MR | Zbl

[49] Friedrich T., Grunewald R., “On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold”, Math. Nachr., 123 (1985), 55–60 | DOI | MR | Zbl

[50] Friedrich T., Kurke H., “Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature”, Math. Nachr., 106 (1982), 271–299 | DOI | MR | Zbl

[51] Gauduchon P., “Structures de Weyl et théorèmes d'annulation sur une variété conforme autoduale”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 18:4 (1991), 563–629 | MR | Zbl

[52] Gauduchon P., “Hermitian connections and Dirac operators”, Boll. Unione Mat. Ital. Ser. 7 B, 11:2, suppl (1997), 257–288 | MR | Zbl

[53] Goldberg S.I., “Integrability of almost Kaehler manifolds”, Proc. Amer. Math. Soc., 21 (1969), 96–100 | DOI | MR | Zbl

[54] Goldberg S.I., Kobayashi S., “Holomorphic bisectional curvature”, J. Diff. Geom., 1 (1967), 225–233 | DOI | MR | Zbl

[55] Gray A., “Curvature identities for Hermitian and almost Hermitian manifolds”, Tohoku Math. J., 28 (1976), 601–612 | DOI | MR | Zbl

[56] Gray A., “The structure of nearly Kähler manifolds”, Math. Ann., 223 (1976), 233–248 | DOI | MR | Zbl

[57] Gray A., “Einstein-like manifolds which are not Einstein”, Geom. dedicata, 7 (1978), 259–280 | DOI | MR | Zbl

[58] Gray A., Barros M., Naveira A.M., Vanhecke L., “The Chern numbers of holomorphic vector bundles and formally holomorphic connections of complex vector bundles over almost complex manifolds”, J. reine angew. Math., 314 (1980), 84–98 | MR | Zbl

[59] Gray A., Hervella L., “The sixteen classes of almost Hermitian manifolds and their linear invariants”, Ann. Mat. Pura Appl. Ser. 4, 123 (1980), 35–58 | DOI | MR | Zbl

[60] Gu H., Zhang Z., “An extension of Mok's theorem on the generalized Frankel conjecture”, Sci. China. Math., 53:5 (2010), 1253–1264 | DOI | MR | Zbl

[61] Hitchin N., “Compact four-dimensional Einstein manifolds”, J. Diff. Geom., 9 (1974), 435–441 | DOI | MR | Zbl

[62] Hitchin N.J., “Kählerian twistor spaces”, Proc. London Math. Soc. Ser. 3, 43 (1981), 133–150 | DOI | MR | Zbl

[63] Hitchin N.J., “Twistor spaces, Einstein metrics and isomonodromic deformations”, J. Diff. Geom., 42:1 (1995), 30–112 | DOI | MR | Zbl

[64] Houh C.-S., “On totally real bisectional curvature”, Proc. Amer. Math. Soc., 56 (1976), 261–263 | DOI | MR | Zbl

[65] Jelonek W., “Self-duality and $\mathcal A$-manifolds”, J. London Math. Soc. Ser. 2, 58:3 (1998), 697–708 | DOI | MR | Zbl

[66] Kalafat M., Koca C., “On the curvature of Einstein–Hermitian surfaces”, Ill. J. Math., 62:1–4 (2018), 25–39 | DOI | MR | Zbl

[67] Ki U-H., Nakagawa H., “A characterization of the Cartan hypersurface in a sphere”, Tohoku Math. J. Ser. 2, 39 (1987), 27–40 | MR | Zbl

[68] Ki U-H., Suh Y.J., “On semi-Kaehler manifolds whose totally real bisectional curvature is bounded from below”, J. Korean Math. Soc., 33:4 (1996), 1009–1038 | MR | Zbl

[69] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, v. 2, J. Wiley Sons, Hoboken, NJ, 2009 | MR

[70] LeBrun C., “On complete quaternionic-Kähler manifolds”, Duke Math. J., 63:3 (1991), 723–743 | DOI | MR | Zbl

[71] LeBrun C., “Ricci curvature, minimal volumes, and Seiberg–Witten theory”, Invent. math., 145:2 (2001), 279–316 | DOI | MR | Zbl

[72] Ledger A.J., Harmonic Riemannian spaces, PhD thesis, Univ. Durham, Durham, 1954

[73] Lichnerowicz A., Théorie globale des connexions et des groupes d'holonomie, Ed. Cremonese, Roma, 1955 | MR

[74] Matsushita Y., “Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds”, Math. Z., 207:2 (1991), 281–291 | DOI | MR | Zbl

[75] Mok N., “The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature”, J. Diff. Geom., 27:2 (1988), 179–214 | DOI | MR | Zbl

[76] Mori S., “Projective manifolds with ample tangent bundles”, Ann. Math. Ser. 2, 110 (1979), 593–606 | DOI | MR | Zbl

[77] Muškarov O., “Structures presque hermitiennes sur des espaces twistoriels et leurs types”, C. r. Acad. sci. Paris. Sér. 1, 305 (1987), 307–309 | MR

[78] Nurowski P., Einstein equations and Cauchy–Riemann geometry, PhD thesis, SISSA/ISAS, Trieste, 1993

[79] O'Neill B., “The fundamental equations of a submersion”, Mich. Math. J., 13 (1966), 459–469 | DOI | MR | Zbl

[80] Pedersen H., Tod P., “The Ledger curvature conditions and D'Atri geometry”, Diff. Geom. Appl., 11:2 (1999), 155–162 | DOI | MR | Zbl

[81] Penrose R., “Twistor theory, its aims and achievements”, Quantum gravity: An Oxford symposium, Clarendon Press, Oxford, 1975, 268–407 | MR

[82] Penrose R., “The twistor programme”, Rep. Math. Phys., 12:1 (1977), 65–76 | DOI | MR

[83] Pontecorvo M., “On twistor spaces of anti-self-dual Hermitian surfaces”, Trans. Amer. Math. Soc., 331:2 (1992), 653–661 | DOI | MR | Zbl

[84] Przanowski M., Broda B., “Locally Kähler gravitational instantons”, Acta phys. Polon. B, 14:9 (1983), 637–661 | MR

[85] Sato T., “On some compact almost Kähler manifolds with constant holomorphic sectional curvature”, Geometry of manifolds: Coll. pap. 35th symp. diff. geom. (Matsumoto, 1988), Perspect. Math., 8, ed. by K. Shiohama, Acad. Press, Boston, 1989, 129–139 | MR

[86] Sekigawa K., “On some 4-dimensional compact Einstein almost Kähler manifolds”, Math. Ann., 271 (1985), 333–337 | DOI | MR | Zbl

[87] Singer I.M., Thorpe J.A., “The curvature of 4-dimensional Einstein spaces”, Global analysis: Papers in honor of K. Kodaira, Princeton Math. Ser., 29, Princeton Univ. Press, Princeton, NJ, 1969, 355–365 | MR

[88] Siu Y.-T., Yau S.-T., “Compact Kähler manifolds of positive bisectional curvature”, Invent. math., 59 (1980), 189–204 | DOI | MR | Zbl

[89] Strominger A., “Superstrings with torsion”, Nucl. Phys. B, 274:2 (1986), 253–284 | DOI | MR

[90] Szabo Z.I., “Spectral theory for operator families on Riemannian manifolds”, Differential geometry, Part 3: Riemannian geometry: Proc. Summer Res. Inst. (Los Angeles, 1990), Proc. Symp. Pure Math., 54, Pt. 3, ed. by R. Green, S.-T. Yau, Amer. Math. Soc., Providence, RI, 1993, 615–665 | DOI | MR | Zbl

[91] Tachibana S.-I., “On almost-analytic vectors in certain almost-Hermitian manifolds”, Tohoku Math. J. Ser. 2, 11:3 (1959), 351–363 | DOI | MR | Zbl

[92] To W.-K., Yeung S.-K., “Kähler metrics of negative holomorphic bisectional curvature on Kodaira surfaces”, Bull. London Math. Soc., 43:3 (2011), 507–512 | DOI | MR | Zbl

[93] Tricerri F., Vanhecke L., “Curvature tensors on almost Hermitian manifolds”, Trans. Amer. Math. Soc., 267 (1981), 365–398 | DOI | MR | Zbl

[94] Vaisman I., “Some curvature properties of locally conformally Kähler manifolds”, Trans. Amer. Math. Soc., 259 (1980), 439–447 | MR | Zbl

[95] Vaisman I., “Some curvature properties of complex surfaces”, Ann. Mat. Pura Appl. Ser. 4, 132 (1982), 1–18 | DOI | MR | Zbl

[96] Vezzoni L., “On the Hermitian curvature of symplectic manifolds”, Adv. Geom., 7:2 (2007), 207–214 | DOI | MR | Zbl

[97] Vitter A., “Self-dual Einstein metrics”, Nonlinear problems in geometry, Proc. AMS spec. sess., 1985, Contemp. Math., 51, ed. by D.M. DeTurck, Amer. Math. Soc., Providence, RI, 1986, 113–120 | DOI | MR

[98] Willmore T.J., Riemannian geometry, Clarendon Press, Oxford, 1993 | MR | Zbl

[99] Wolf J.A., Gray A., “Homogeneous spaces defined by Lie group automorphisms. I, II”, J. Diff. Geom., 2 (1968), 77–114, 115–159 | DOI | MR | Zbl

[100] Wood C.M., “Harmonic almost-complex structures”, Compos. math., 99:2 (1995), 183–212 | MR | Zbl

[101] Yang B., Zheng F., “On curvature tensors of Hermitian manifolds”, Commun. Anal. Geom., 26:5 (2018), 1195–1222 ; arXiv: 1602.01189v1 [math.DG] | DOI | MR | Zbl

[102] Yau S.-T., Zhao Q., Zheng F., On Strominger Kähler-like manifolds with degenerate torsion, E-print, 2019, arXiv: 1908.05322v1 [math.DG]