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Communications in Mathematical Physics

, Volume 262, Issue 1, pp 177–208 | Cite as

On Eta-Einstein Sasakian Geometry

  • Charles P. BoyerEmail author
  • Krzysztof Galicki
  • Paola Matzeu
Article

Abstract

A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold Open image in new window . In the case when the transverse space Open image in new window is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
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References

  1. 1.
    Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønneson-Friedman, C.W.: Hamiltonian 2-forms in Kähler geometry, II. Global Classification. J. Diff. Geom 68, 277–345 (2004)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aubin, T.: Nonlinear analysis on manifolds. Monge-Ampere equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 252, New York: Springer-Verlag, 1982Google Scholar
  3. 3.
    Baily, W.L.: On the imbedding of V-manifolds in projective space. Amer. J. Math. 79, 403–430 (1957)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Barden, D.: Simply connected five-manifolds. Ann. of Math. (2) 82, 365–385 (1965)Google Scholar
  5. 5.
    Baum, H.: Twistor and Killing spinors in Lorentzian geometry. In: Global analysis and harmonic analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4, Paris: Soc. Math. France, 2000, pp. 35–52Google Scholar
  6. 6.
    Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, no. 1, 1–40 (2000)Google Scholar
  7. 7.
    Belgun, F.A.: Normal CR structures on compact 3-manifolds. Math. Z. 238, no. 3, 441–460 (2001)Google Scholar
  8. 8.
    Boyer, C.P., Galicki K.: On Sasakian-Einstein geometry. Internat. J. Math. 11, no. 7, 873–909 (2000)Google Scholar
  9. 9.
    Boyer, C.P., Galicki K.: Einstein manifolds and contact geometry. Proc. Amer. Math. Soc. 129, no. 8, 2419–2430 (electronic) (2001)Google Scholar
  10. 10.
    Boyer, C.P., Galicki K.: New Einstein metrics in dimension five. J. Diff. Geom. 57, no. 3, 443–463 (2001)Google Scholar
  11. 11.
    Boyer, C.P., Galicki K.: Einstein Metrics on Rational Homology Spheres. http://arxiv.org/ list/math.DG/0311355, 2003Google Scholar
  12. 12.
    Boyer, C.P., Galicki K., Kollár J.: Einstein Metrics on Spheres. Ann. of Math. 2, no. 162, 557–580 (2005)Google Scholar
  13. 13.
    Boyer, C.P., Galicki K., Kollár J., Thomas E.: Einstein Metrics on Exotic Spheres in Dimensions 7,11, and 15. Experimental Mathematics 14, 59–64 (2005)MathSciNetGoogle Scholar
  14. 14.
    Boyer, C.P., Galicki K., Nakamaye M.: On positive Sasakian geometry. Geom. Dedicata 101, 93–102 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Boyer, C.P., Galicki K., Nakamaye M.: On the geometry of Sasakian-Einstein 5-manifolds. Math. Ann. 325, no. 3, 485–524 (2003)Google Scholar
  16. 16.
    Boyer, C.P., Galicki K., Nakamaye M.: Sasakian geometry, homotopy spheres and positive Ricci curvature. Topology 42, no. 5, 981–1002 (2003)Google Scholar
  17. 17.
    Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, Vol. 203, Boston, MA: Birkhäuser Boston Inc., 2002Google Scholar
  18. 18.
    Bohle, C.: Killing spinors on Lorentzian manifolds. J. Geom. Phys. 45, no. 3–4, 285–308 (2003)Google Scholar
  19. 19.
    Barth, W., Peters, C., Van de Ven, A.: Compact complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 4, Berlin: Springer-Verlag, 1984Google Scholar
  20. 20.
    Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. Math. 2, 1–14 (1966)CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    Candelas, P., Lynker, M., Schimmrigk, R.: Calabi-Yau manifolds in weighted P 4. Nucl. Phys. B 341, no. 2, 383–402 (1990)Google Scholar
  22. 22.
    Calderbank, D.M.J., Pedersen, H.: Einstein-Weyl geometry. In: Surveys in differential geometry: essays on Einstein manifolds. Surv. Differ. Geom., VI, Boston, MA: Int. Press, 1999, pp. 387–423Google Scholar
  23. 23.
    Cheeger, J., Tian, G.: On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay. Invent. Math. 118, no. 3, 493–571 (1994)Google Scholar
  24. 24.
    El Kacimi-Alaoui, A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compositio Math. 73, no. 1, 57–106 (1990)Google Scholar
  25. 25.
    Fintushel, R., Stern, R.J.: Instanton homology of Seifert fibred homology three spheres. Proc. London Math. Soc. (3) 61, no. 1, 109–137 (1990)Google Scholar
  26. 26.
    Gauduchon, P.: Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1× S 3. J. Reine Angew. Math. 469, 1–50 (1995)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Geiges, H.: Normal contact structures on 3-manifolds. Tohoku Math. J. (2) 49, no. 3, 415–422 (1997)Google Scholar
  28. 28.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley-Interscience [John Wiley & Sons], 1978, Pure and Applied MathematicsGoogle Scholar
  29. 29.
    Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, W.: Sasaki-Einstein metrics on S 2× S 3. Adv. Theor. Phys. 8, 711–734 (2004)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, W.: A new infinite class of Sasaki- Einstein manifolds. Adv. Theor. Math. Phys. 8, no. 6Google Scholar
  31. 31.
    Gauduchon, P., Ornea, L.: Locally conformally Kähler metrics on Hopf surfaces. Ann. Inst. Fourier (Grenoble) 48, no. 4, 1107–1127 (1998)Google Scholar
  32. 32.
    Gray, J.W.: Some global properties of contact structures. Ann. of Math. (2) 69, 421–450, (1959)Google Scholar
  33. 33.
    Guilfoyle, B.S.: The local moduli of Sasakian 3-manifolds. Int. J. Math. Math. Sci. 32, no. 2, 117–127 (2002)Google Scholar
  34. 34.
    Higa, T.: Weyl manifolds and Einstein-Weyl manifolds. Comment. Math. Univ. St. Paul. 42, no. 2, 143–160 (1993)Google Scholar
  35. 35.
    Hwang, A.D., Simanca, S.R.: Extremal Kähler metrics on Hirzebruch surfaces which are locally conformally equivalent to Einstein metrics. Math. Ann. 309, no. 1, 97–106 (1997)Google Scholar
  36. 36.
    Iano-Fletcher, A.R.: Working with weighted complete intersections. In: Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., Vol. 281, Cambridge: Cambridge Univ. Press, 2000, pp. 101–173Google Scholar
  37. 37.
    Johnson, J.M., Kollár, J.: Kähler-Einstein metrics on log del Pezzo surfaces in weighted projective 3-spaces. Ann. Inst. Fourier (Grenoble) 51, no. 1, 69–79 (2001)Google Scholar
  38. 38.
    Kollár, J.: Einstein metrics on 5-dimensional Seifert bundles. http://arxiv.org/list/ math.DG/0408184, 2004Google Scholar
  39. 39.
    Kollár, J.: Einstein metrics on connected sums of S 2× S 3. http://arxiv.org/list/math.DG/ 0402141, 2004Google Scholar
  40. 40.
    Klebanov, I.R., Witten, E.: Superconformal field theory on threebranes at a Calabi-Yau singularity. Nucl. Phys. B 536, no. 1–2, 199–218 (1999)Google Scholar
  41. 41.
    Maldacena, J.: The large-N limit of superconformal field theories and supergravity. Internat. J. Theoret. Phys. 38, no. 4, 1113–1133 (1999) Quantum gravity in the southern cone (Bariloche, 1998)Google Scholar
  42. 42.
    Matzeu, P.: Some examples of Einstein-Weyl structures on almost contact manifolds. Class. Quantum Gravity 17, no. 24, 5079–5087 (2000)Google Scholar
  43. 43.
    Matzeu, P.: Almost contact Einstein-Weyl structures. Manus. Math. 108, no. 3, 275–288 (2002)Google Scholar
  44. 44.
    Milnor, J.: On the 3-dimensional Brieskorn manifolds M(p,q,r). In: Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), Ann. of Math. Studies, No. 84, Princeton, NJ: Princeton Univ. Press, 1975, pp. 175–225Google Scholar
  45. 45.
    Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Moroianu, A.: Parallel and Killing spinors on Spin c manifolds. Commun. Math. Phys. 187, no. 2, 417–427 (1997)Google Scholar
  47. 47.
    Martelli, D., Sparks, J.: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of ads/cft duals. Commun. Math. Phys., to appear, http://arxiv.org/list/th/0411238, 2004Google Scholar
  48. 48.
    Martelli, D.D., Sparks, J., Yau, S.-T.: The geometric dual of a-maximisation for toric Sasaki-Einstein manifolds. http://arxiv.org/list/hep-th/0503183, 2005Google Scholar
  49. 49.
    Narita, F.: Riemannian submersion with isometric reflections with respect to the fibers. Kodai Math. J. 16, no. 3, 416–427 (1993)Google Scholar
  50. 50.
    Narita, F.: Riemannian submersions and Riemannian manifolds with Einstein-Weyl structures. Geom. Dedicata 65, no. 1, 103–116 (1997)Google Scholar
  51. 51.
    Narita, F.: Einstein-Weyl structures on almost contact metric manifolds. Tsukuba J. Math. 22, no. 1, 87–98 (1998)Google Scholar
  52. 52.
    Okumura, M.: Some remarks on space with a certain contact structure. Tôhoku Math. J. (2) 14, 135–145 (1962)Google Scholar
  53. 53.
    Orlik, P.: Weighted homogeneous polynomials and fundamental groups.Topology 9, 267–273 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Petersen, P.: Riemannian geometry, Graduate Texts in Mathematics, Vol. 171, New York: Springer-Verlag, 1998Google Scholar
  55. 55.
    Pedersen, H., Swann, A.: Riemannian submersions, four-manifolds and Einstein-Weyl geometry. Proc. London Math. Soc. (3) 66, no. 2, 381–399 (1993)Google Scholar
  56. 56.
    Reid, M.: Canonical 3-folds. Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Alphen aan den Rijn: Sijthoff & Noordhoff, 1980, pp. 273–310Google Scholar
  57. 57.
    Saeki, O.: Knotted homology spheres defined by weighted homogeneous polynomials. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, no. 1, 43–50 (1987)Google Scholar
  58. 58.
    Sasaki, S.: Almost Contact Manifolds, Part 1. Lecture Notes, Mathematical Institute, Tôhoku University, 1965Google Scholar
  59. 59.
    Sasaki, S.: Almost Contact Manifolds, Part 3. Lecture Notes, Mathematical Institute, Tôhoku University, 1968Google Scholar
  60. 60.
    Saveliev, N.: Invariants for homology 3-spheres. Encyclopaedia of Mathematical Sciences, Vol. 140, Low-Dimensional Topology, 1, Berlin: Springer-Verlag, 2002Google Scholar
  61. 61.
    Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, no. 5, 401–487 (1983)Google Scholar
  62. 62.
    Smale, S.: On the structure of 5-manifolds. Ann. of Math. (2) 75, 38–46 (1962)Google Scholar
  63. 63.
    Takahashi, T.: Deformations of Sasakian structures and its application to the Brieskorn manifolds. Tôhoku Math. J. (2) 30, no. 1, 37–43 (1978)Google Scholar
  64. 64.
    Tanno, S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)zbMATHMathSciNetGoogle Scholar
  65. 65.
    Tanno, S.: Sasakian manifolds with constant φ-holomorphic sectional curvature. Tôhoku Math. J. (2) 21, 501–507 (1969)Google Scholar
  66. 66.
    Tanno, S.: Geodesic flows on C L-manifolds and Einstein metrics on S 3× S 2. In: Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), Amsterdam: North-Holland, 1979, pp. 283–292Google Scholar
  67. 67.
    Thurston, W.P.: Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton, NJ: Princeton University Press, 1997, edited by Silvio LevyGoogle Scholar
  68. 68.
    Tondeur, P.: Geometry of foliations. Monographs in Mathematics, Vol. 90, Basel: Birkhäuser Verlag, 1997Google Scholar
  69. 69.
    Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, no. 3, 339–411 (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Charles P. Boyer
    • 1
    Email author
  • Krzysztof Galicki
    • 2
  • Paola Matzeu
    • 3
  1. 1.Department of Mathematics & StatisticsUniversity of New MexicoAlbuquerqueUSA
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Dipartimento di Matematica e InformaticaUniversitá di CagliariCagliariItaly

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