Jung, Seoung Dal; Richardson, Ken

The Mean Curvature of Transverse Kähler Foliations

Doc. Math. 24, 995-1031 (2019)
DOI: 10.25537/dm.2019v24.995-1031
Communicated by Christian Bär

Summary

We study properties of the mean curvature one-form and its holomorphic and antiholomorphic cousins on a transverse Kähler foliation. If the mean curvature of the foliation is automorphic, then there are some restrictions on basic cohomology similar to that on Kähler manifolds, such as the requirement that the odd basic Betti numbers must be even. However, the full Hodge diamond structure does not apply to basic Dolbeault cohomology unless the foliation is taut.

Mathematics Subject Classification

53C12, 53C21, 53C55, 57R30, 58J50

Keywords/Phrases

Riemannian foliation, transverse Kähler foliation, Lefschetz decomposition, mean curvature

References

  • 1. H. Ait Haddou, Foliations and Lichnerowicz basic cohomology, Int. Math. Forum 2 (2007), no. 49-52, 2437-2446. DOI 10.12988/imf.2007.07214; zbl 1141.57303; MR2381831.
  • 2. J. A. Álvarez-López, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), 179-194. DOI 10.1007/BF00130919; zbl 0759.57017; MR1175918.
  • 3. A. Banyaga, Some properties of locally conformal symplectic structures, Comment. Math. Helv. 77 (2002), no. 2, 383-398. DOI 10.1007/s00014-002-8345-z; zbl 1020.53050; MR1915047.
  • 4. C. P. Boyer and K. Galicki, Sasakian geometry. Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. zbl 1155.53002; MR2382957.
  • 5. C. P. Boyer, K. Galicki, and M. Nakamaye, On the geometry of Sasakian-Einstein 5-manifolds, Math. Ann. 325 (2003), no. 3, 485-524. DOI 10.1007/s00208-002-0388-3; zbl 1046.53028; MR1968604; arxiv math/0012047.
  • 6. B. Cappelletti-Montano, A. de Nicola, and I. Yudin, Hard Lefschetz Theorem for Sasakian manifolds, J. Diff. Geom. 101 (2015), 47-66. DOI 10.4310/jdg/1433975483; zbl 1322.58002; MR3356069; arxiv 1306.2896.
  • 7. Y. Carrière, Flots riemanniens, Astérisque 116 (1984), 31-52. zbl 0548.58033; MR0755161.
  • 8. D. Chinea, M. de León, and J. C. Marrero, Topology of cosymplectic manifolds, J. Math. Pures Appl. (9) 72 (1993), no. 6, 567-591. zbl 0845.53025; MR1249410.
  • 9. L. A. Cordero and R. A. Wolak, Properties of the basic cohomology of transversely Kähler foliations, Rendiconti del Circolo Matematico di Palermo, Serie II, 40 (1991), 177-188. DOI 10.1007/BF02844686; zbl 0763.57018; MR1151583.
  • 10. D. Domínguez, Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), no. 6, 1237-1276. DOI 10.1353/ajm.1998.0048; zbl 0964.53019; MR1657170.
  • 11. A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Compositio Math. 73 (1990), no. 1, 57-106. zbl 0697.57014; MR1042454.
  • 12. A. El Kacimi-Alaoui and G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier 36 (1986), no. 3, 207-227. DOI 10.5802/aif.1066; zbl 0586.57015; MR0865667.
  • 13. A. El Kacimi-Alaoui and M. Nicolau, On the topological invariance of the basic cohomology, Math. Ann. 295 (1993), no. 4, 627-634. DOI 10.1007/BF01444906; zbl 0793.57016; MR1214951.
  • 14. O. Goertsches, H. Nozawa, and D. Töben, Rigidity and vanishing of basic Dolbeault cohomology of Sasakian manifolds, J. Symplectic Geom. 14 (2016), no. 1, 31-70. DOI 10.4310/JSG.2016.v14.n1.a2; zbl 1346.53049; MR3523249; arxiv 1206.2803.
  • 15. G. Habib and K. Richardson, Modified differentials and basic cohomology for Riemannian foliations, J. Geom. Anal. 23 (2013), no. 3, 1314-1342. DOI 10.1007/s12220-011-9289-6; zbl 1276.53031; MR3078356; arxiv 1007.2955.
  • 16. G. Habib and L. Vezzoni, Some remarks on Calabi-Yau and hyper-Kähler foliations, Differential Geom. Appl. 41 (2015), 12-32. DOI 10.1016/j.difgeo.2015.03.006; zbl 1348.53032; MR3353736; arxiv 1306.5159.
  • 17. J. Hebda, Curvature and focal points in Riemannian foliations, Indiana Univ. Math. J. 35 (1986), no. 2, 321-331. DOI 10.1512/iumj.1986.35.35019; zbl 0596.53027; MR0833397.
  • 18. S. D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys. 39 (2001), 253-264. DOI 10.1016/S0393-0440(01)00014-6; zbl 1024.53019; MR1848103.
  • 19. S. D. Jung and M. J. Jung, Transversally holomorphic maps between Kähler foliations, J. Math. Anal. Appl. 416 (2014), 683-697. DOI 10.1016/j.jmaa.2014.03.022; zbl 1311.53024; MR3188732.
  • 20. S. D. Jung and H. Liu, Transversal infinitesimal automorphisms on Kähler foliations, Bull. Aust. Math. Soc. 86 (2012), 405-415. DOI 10.1017/S0004972711003431; zbl 1268.53030; MR2995892; arxiv 1106.0358.
  • 21. S. D. Jung and J. S. Pak, Some vanishing theorems on Kähler foliation, Comm. Korean Math. Soc. 11 (1996), 767-781. zbl 0943.58022; MR1432280.
  • 22. S. D. Jung and J. S. Pak, Some vanishing theorems on complete Kähler foliation, Acta Math. Hungar. 77 (1997), no. 1-2, 15-28. DOI 10.1023/A:1006523404114; zbl 0905.53045; MR1485582.
  • 23. S. D. Jung and K. Richardson, Transverse conformal Killing forms and a Gallot-Meyer theorem for foliations, Math. Z. 270 (2012), 337-350. DOI 10.1007/s00209-010-0800-8; zbl 1247.53029; MR2875837; arxiv 0805.4187.
  • 24. S. D. Jung and K. Richardson, Basic Dolbeault cohomology and Weitzenb\``ock formulas on transversely K\''ahler foliations, preprint. arxiv 1709.05469.
  • 25. F. W. Kamber and Ph. Tondeur, Duality for Riemannian foliations, Proc. Sympos. Pure Math., Amer. Math. Soc. 40 (1983), Part I, 609-618. zbl 0523.57019; MR0713097.
  • 26. F. W. Kamber and Ph. Tondeur, Duality theorems for foliations, Asterisque 116 (1984), 108-116. zbl 0559.58022; MR0755165.
  • 27. F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann. 277 (1987), 415-431. DOI 10.1007/BF01458323; zbl 0637.53043; MR0891583.
  • 28. Y. Kordyukov, M. Lejmi, and P. Weber, Seiberg-Witten invariants on manifolds with Riemannian foliations of codimension 4, J. Geom. Phys. 107 (2016), 114-135. DOI 10.1016/j.geomphys.2016.05.012; zbl 1345.53029; MR3521471; arxiv 1506.08088.
  • 29. P. March, M. Min-Oo, and E. A. Ruh, Mean curvature of Riemannian foliations, Canad. Math. Bull. 39 (1996), 95-105. DOI 10.4153/CMB-1996-012-4; zbl 0853.53023; MR1382495.
  • 30. A. Mason, An application of stochastic flows to Riemannian foliations, Houston J. Math. 26 (2000), 481-515. zbl 1159.58311; MR1811936; arxiv math/9812117.
  • 31. M. Min-Oo, E. Ruh, and Ph. Tondeur, Vanishing theorems for the basic cohomology of Riemannian foliations, J. Reine Angew. Math. 415 (1991), 167-174. zbl 0716.53032; MR1096904.
  • 32. A. Moroianu and L. Ornea, Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length, C. R. Math. Acad. Sci. Paris 338 (2004), 561-564. DOI 10.1016/j.crma.2004.01.030; zbl 1049.58029; MR2057030; arxiv 1004.4807.
  • 33. S. Nisikawa and Ph. Tondeur, Transversal infinitesimal automorphisms for harmonic Kähler foliations, Tohoku Math. J. 40 (1988), 599-611. DOI 10.2748/tmj/1178227924; zbl 0664.53014; MR0972248.
  • 34. E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), 1249-1275. DOI 10.1353/ajm.1996.0053; zbl 0865.58047; MR1420923.
  • 35. H. Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comm. Math. Helv. 54 (1979), 224-239. DOI 10.1007/BF02566270; zbl 0409.57026; MR0535057.
  • 36. Ph. Tondeur, Foliations on Riemannian manifolds, Springer-Verlag, New York, 1988. zbl 0643.53024; MR0934020.
  • 37. Ph. Tondeur, Geometry of foliations, Birkhäuser Verlag, 1997. zbl 0905.53002; MR1456994.
  • 38. I. Vaisman, Remarkable operators and commutation formulas on locally conformal Kähler manifolds, Compositio Math. 40 (1980), no. 3, 287-299. zbl 0401.53019; MR0571051.
  • 39. R. Wolak, Sasakian structures: a foliated approach. arxiv 1605.04163.
  • 40. S. Yorozu and T. Tanemura, Green's theorem on a foliated Riemannian manifold and its applications, Acta Math. Hungarica 56 (1990), 239-245. DOI 10.1007/BF01903838; zbl 0734.53028; MR1111308.

Affiliation

Jung, Seoung Dal
Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea
Richardson, Ken
Department of Mathematics, Texas Christian University (TCU), Box 298900, Fort Worth, Texas 76129, USA

Downloads