A splitting theorem for Kähler manifolds whose Ricci tensors have constant eigenvalues

International Journal of Mathematics

Volumn 12, Issue 7, 2001, Pages 769-789

  Apostolov, Vestislav ^a   Drǎghici, Tedi ^b   Moroianu, Andrei ^c  

^a UNIVERSITÉ DU QUÉBEC À MONTRÉAL   (Canada)

^b FLORIDA INTERNATIONAL UNIVERSITY   (United States)

^c CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035457346     PISSN: 0129167X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129167X01001052     Document Type: Article

References (53)

1. 1. V. Apostolov and J. Armstrong, Symplectic 4-manifolds with Hermitian Weyl tensor, Trans. Amer. Math. Soc. 352 (2000), 4501-4513.
2. 2. V. Apostolov, J. Armstrong and T. Draghici, Local rigidity of certain classes of almost Kähler 4-manifolds, to appear in Ann. Glob. An. Geom. available at math.DG/9911197.
3. 3. V. Apostolov and T. Draghici, Hermitian conformal classes and almost Kähler structures on four manifolds, Diff. Geom. Appl. 11 (1999), 179-195.
4. 4. V. Apostolov, T. Draghici and D. Kotschick, An integrability theorem for almost Kähler 4-manifolds, C. R. Acad. Sci., Paris, Ser. I, t. 329 (1999), 413-418.
5. 5. J. Armstrong, An Ansatz for Almost-Kähler, Einstein 4-manifolds, to appear in J. Reine Angew. Math.
6. 6. M. Atiyah, The signature of fiber bundles, in Global Analysis: Papers in Honor of K. Kodaira, Tokyo University Press 1969.
7. 7. P. de Bartolomeis and D. Burns, Stability of vector bundles and extremal metrics, Invent. Math. 92 (1988), 403-407.
8. 8. W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer-Verlag, 1984.
9. 9. A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb.3, Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1987.
10. 10. R. Bielawski, Ricci-flat Kähler metrics on canonical bundles, preprint 2000, available at math.DG/0006144.
11. 11. D. E. Blair and S. Ianus, Critical associated metrics on symplectic manifolds, Contemp. Math. 51 (1986), 23-29.
12. 12. J.-P. Bourguignon, Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d'Einstein, Invent. Math. 63 (1981), 263-286.
13. 13. R. Bryant, private communication.
14. 14. J. Cheeger and S. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature, J. Diff. Geom. 6 (1971), 119-128.
15. 15. J.E. D'Atri, Holomorphic sectional curvatures of bounded homogeneous domains and related questions, Trans. Amer. Math. Soc. 256 (1979), 405-413.
16. 16. J. E. D'Atri and D. Miatello, A characterization of bounded symmetric domains by curvature, Trans. Amer. Math. Soc. 267 (1983), 531-540.
17. 17. J. Davidov and O. Muškarov, Twistor spaces with Hermitian Ricci tensor, Proc. Amer. Math. Soc. 109(4) (1990), 1115-1120.
18. 18. J. Dorfmeister and K. Nakajiama, The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), 23-70.
19. 19. T. Draghici, Almost Kähler 4-manifolds with J-invariant Ricci tensor, Houston J. Math. 25 (1999), 133-145.
20. 20. L. Hernández-Lamoneda, Curvature versus Almost Hermitian Structures, Geom. Dedicata 79 (2000), 205-218.
21. 21. A. Hwang and M. Singer, A momentum construction for circle-invariant Kähler metrics, available at math.DG/9811024.
22. 22. P. Gauduchon, Complex structures on compact conformal manifolds of negative type, in Complex Analysis and Geometry, eds. V. Ancona, E. Ballico and A. Silva, Marcel Dekker, New York; Basel, Hong Kong, 1996, pp. 201-212.
23. 23. S. G. Gindikin, I. I. Piatetskii-Shapiro and E. B. Vinberg, Homogeneous Kähler manifolds, in Geometry of Homogeneous Bounded Domains, C.I.M.E., 1967, pp. 1-88.
24. 24. S. I. Goldberg, Integrability of almost Kähler manifolds, Proc. Amer. Math. Soc. 21 (1969), 96-100.
25. 25. A. Gray, Curvature identities for Hermitian and almost Hermitian manifolds, Tôhoku Math. J. 28 (1976), 601-612.
26. 26. K. Kodaira, A certain type irregular algebraic surfaces, J. Anal. Math. 19 (1967), 207-215.
27. 27. J. L. Koszul, Sur la forme hermitienne canonique des espaces homogènes complexes, Canad. J. Math. 7 (1955), 562-576.
28. 28. D. Kotschick, Orientations and geometrisations of compact complex surfaces, Bull. London Math. Soc. 29(2) (1997), 145-149.
29. 29. O. Kowalski, Generalized Symmetric Spaces, Lecture Notes in Mathematics 805, 1980.
30. 30. C. LeBrun, Polarized 4-manifolds, extremal Kähler metrics and Seiberg-Witten theory, Math. Res. Lett. 2(5) (1995), 653-662.
31. 31. N. C. Leung, Seiberg-Witten invariants and uniformizations, Math. Ann. 306 (1996), 31-46.
32. 32. G. Maschler, Central Kähler metrics, preprint.
33. 33. A. Moroianu, On Kirchberg inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geom. 15 (1997), 235-242.
34. 34. A. Moroianu, Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz, Ann. Inst. Fourier (Grenoble), 49 (1999), 1637-1659.
35. 35. K. Nakajima, Homogeneous Kähler manifolds of non-degenerate Ricci curvature, J. Math. Soc. Japan 42 (1990), 475-494.
36. 36. K. Nakajima, Homogeneous Kähler manifolds of non-positive Ricci curvature, J. Math. Kyoto University 26 (1986), 547-558.
37. 37. T. Oguro and K. Sekigawa, Four-dimensional almost Kähler-Einstein and *-Einstein manifolds, Geom. Dedicata 69 (1998), 91-112.
38. 38. Z. Olszak, A note on almost Kähler manifolds, Bull. Acad. Polon. Sci. XXVI (1978), 139-141.
39. 39. J. Petean, Indefinite Kähler-Einstein metrics on compact complex surfaces, Comm. Math. Phys. 189 (1997), 227-235.
40. 40. I. I. Piatetskii-Shapiro, Automorphic Functions and the Geometry of Classical Domains, English translation, Gordon and Breach, New York, 1969.
41. 41. I. Satake, Algebraic Structures of Symmetric Domains, Iwanami Shoten Publ. and Princeton University Press, 1980.
42. 42. K. Sekigawa, On some compact Einstein almost Kähler manifolds, J. Math. Soc. Japan 39 (1987), 677-684.
43. 43. H. Shima, On homogeneous Kähler manifolds of solvable Lie groups, J. Math. Soc. Japan 25 (1973), 422-445.
44. 44. H. Shima, On homogeneous Kähler manifolds with non-degenerate canonical Hermitian form of signature (2, 2(n - 1)), Osaka J. Math. 10 (1973), 477-493.
45. 45. C. H. Taubes, The Seiberg-Witten Invariants and Symplectic Forms, Math. Res. Lett. 1 (1994), 809-822.
46. 46. C. H. Taubes, SW ⇒ Gr: from the Seiberg-Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9(3) (1996), 845-918.
47. 47. F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398.
48. 48. E. B. Vinberg, S. G. Gindinkin and I. I. Piatetskii-Shapiro, Classification and canonical realization of complex bounded homogeneous domains, Trudy Moskov. Mat. Obsc. 12 (1963), 359-388; [Trans. Moscow Math. Soc. (1963), 404-437].
49. 48. E. B. Vinberg, S. G. Gindinkin and I. I. Piatetskii-Shapiro, Classification and canonical realization of complex bounded homogeneous domains, Trudy Moskov. Mat. Obsc. 12 (1963), 359-388; [Trans. Moscow Math. Soc. (1963), 404-437].
50. 49. C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), 119-153.
51. 50. S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Comm. Pure Appl. Math. 31 (1978) 339-411.
52. 51. S.-T. Yau, On the scalar curvature of compact Hermitian manifolds, Invent. Math. 29 (1974), 213-239.
53. 52. H. C. Wang, Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1-32.