×

Symplectic twistor spaces. (English) Zbl 0810.53056

Summary: We introduce some canonical 2-form in the twistor bundles of any Riemannian manifold \(M\). This form is always closed and turns out to be nondegenerate in the following cases: 1. The curvature of \(M\) is pinched. 2. \(M\) is an Einstein four-dimensional manifold of positive or negative curvature. 3. \(M\) is self-dual and the Ricci curvature is pinched.
We prove the existence of a large class of compact symplectic manifolds which do not admit any Kählerian structure. Finally, we introduce a Lagrangian lift of a totally geodesic submanifold of \(M\) and apply the Lagrangian intersection methods to totally geodesic submanifolds.

MSC:

53C56 Other complex differential geometry
53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Atiah, M.F.;Hitchin, N.J.;Singer, I.M.: Self-duality in four-dimensional Riemannian geometry.Proc. Roy. Soc. 362 (1978), 425-461. · Zbl 0389.53011
[2] Berger, M.: Sur quelques varietes riemannian suffisament pincees.Bull. Soc. Math. 88 (1960), 57-71.
[3] Besse, A.:Manifolds, all of whose geodesics are closed. Springer-Verlag, 1978. · Zbl 0387.53010
[4] Besse, A.:Geometrie riemanniene en dimension 4. Cedic-Fernand Nathan, Paris 1981.
[5] Cordero, L.A.;Fernandez, M.;Gray, A.: Symplectic manifolds with no Kähler structure.Topology 25 (1986), 375-380. · Zbl 0596.53030
[6] Cordero, L.A.;Fernandez, M.:Leon, M.: Examples of compact non-Kählerian almost Kählerian manifolds.Proc. Amer. Math. Soc. 95 (1985), 280-286. · Zbl 0575.53015
[7] Donaldson, S.;Friedman, R.: Connected sums of self-dual manifolds and deformations of singular spaces.Nonlinearity 2 (1989), 197-239. · Zbl 0671.53029
[8] Eells, J.;Sampson, J.H.: Harmonic mappings of Riemannian manifolds.Amer. J. Math. 86 (1964), 109-160. · Zbl 0122.40102
[9] Frankel, Th.:Critical submanifolds of the classical groups and Stiefel manifolds. · Zbl 0134.42602
[10] Friedrich, Th.;Kurke, H.: Compact four-dimensional Einstein manifolds with positive scalar curvature.Math. Nachr. 106 (1982), 271-299. · Zbl 0503.53035
[11] Gromov, M.: Almost flat manifolds.J. Differential Geom. 13 (1986), 223-230. · Zbl 0433.53028
[12] Gromov, M.;Thurston, W.P.: Pinching constants for hyperbolic manifolds.Invent. Math. 89 (1987), 1-12. · Zbl 0646.53037
[13] Hitchin, N.J.: Kählerian twistor spaces.Proc. London Math. Soc. (3),43 (1981), 133-150. · Zbl 0474.14024
[14] Le Brun, C.: Explicit self-dual metrics on \(\mathbb{C}\mathbb{P}^2 \) #...# \(\mathbb{C}\mathbb{P}^2 \) IP 2 IP 2.Differential Geom. 34 (1991), 223-253.
[15] McDuff, D.: Examples of simply-connected symplectic non-Kählerian manifolds.J. Differential Geom. 20 (1984), 267-277. · Zbl 0567.53031
[16] Poon, Y.S.: Compact self-dual manifolds with positive scalar curvature. J. Differential Geom.324 (1986), 96-132. · Zbl 0583.53054
[17] Reznikov, A.:Morse theory and combinatorial identities. Preprint.
[18] Salamon, S.M.: Quaternionic Kähler manifolds.Invent. Math. 67 (1984), 143-171. · Zbl 0486.53048
[19] Sampson, J.H.:Harmonic maps in Kähler Geometry, Harmonic Mappings and Minimal Immersions. Lect. Notes in Math.1161 (1985). · Zbl 0573.53040
[20] Simpson, C.T.:Nonabelian Hodge Theory. Preprint.
[21] Thurston, W.P.: Some simple examples of symplectic manifolds.Proc. Amer. Math. Soc. 55 (1976), 467-468. · Zbl 0324.53031
[22] Weinstein, A.: Symplectic manifolds and their Lagrangian submanifolds.Adv. in Math. 6 (1971), 329-346. · Zbl 0213.48203
[23] Weinstein, A.: Fat bundles and symplectic manifolds.Adv. in Math. 37 (1980), 239-250. · Zbl 0449.53035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.