Tenseness of Riemannian flows

Hiraku Nozawa[1]; José Ignacio Royo Prieto[2]

  • [1] Department of Mathematical Sciences College of Science and Engineering Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577 (Japan)
  • [2] Universidad del País Vasco UPV/EHU Departamento de Matemática Aplicada Alameda de Urquijo s/n 48013 Bilbao (Spain)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 4, page 1419-1439
  • ISSN: 0373-0956

Abstract

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We show that any transversally complete Riemannian foliation ℱ of dimension one on any possibly non-compact manifold M is tense; namely, M admits a Riemannian metric such that the mean curvature form of ℱ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa’s characterization of tautness.

How to cite

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Nozawa, Hiraku, and Royo Prieto, José Ignacio. "Tenseness of Riemannian flows." Annales de l’institut Fourier 64.4 (2014): 1419-1439. <http://eudml.org/doc/275460>.

@article{Nozawa2014,
abstract = {We show that any transversally complete Riemannian foliation $\mathcal\{F\}$ of dimension one on any possibly non-compact manifold $M$ is tense; namely, $M$ admits a Riemannian metric such that the mean curvature form of $\mathcal\{F\}$ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa’s characterization of tautness.},
affiliation = {Department of Mathematical Sciences College of Science and Engineering Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577 (Japan); Universidad del País Vasco UPV/EHU Departamento de Matemática Aplicada Alameda de Urquijo s/n 48013 Bilbao (Spain)},
author = {Nozawa, Hiraku, Royo Prieto, José Ignacio},
journal = {Annales de l’institut Fourier},
keywords = {Riemannian foliation; taut foliation; mean curvature; basic cohomology},
language = {eng},
number = {4},
pages = {1419-1439},
publisher = {Association des Annales de l’institut Fourier},
title = {Tenseness of Riemannian flows},
url = {http://eudml.org/doc/275460},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Nozawa, Hiraku
AU - Royo Prieto, José Ignacio
TI - Tenseness of Riemannian flows
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1419
EP - 1439
AB - We show that any transversally complete Riemannian foliation $\mathcal{F}$ of dimension one on any possibly non-compact manifold $M$ is tense; namely, $M$ admits a Riemannian metric such that the mean curvature form of $\mathcal{F}$ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa’s characterization of tautness.
LA - eng
KW - Riemannian foliation; taut foliation; mean curvature; basic cohomology
UR - http://eudml.org/doc/275460
ER -

References

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  1. Jesús A. Álvarez López, A finiteness theorem for the spectral sequence of a Riemannian foliation, Illinois J. Math. 33 (1989), 79-92 Zbl0644.57014MR974012
  2. Jesús A. Álvarez López, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), 179-194 Zbl0759.57017MR1175918
  3. Jesús A. Álvarez López, On the first secondary invariant of Molino’s central sheaf, Ann. Polon. Math. 64 (1996), 253-265 Zbl0863.53021MR1410344
  4. Grant Cairns, Richard H. Escobales, Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold, J. Austral. Math. Soc. Ser. A 62 (1997), 46-63 Zbl0884.57023MR1427628
  5. Alberto Candel, Lawrence Conlon, Foliations. I, 23 (2000), American Mathematical Society, Providence, RI Zbl0936.57001MR1732868
  6. Patrick Caron, Yves Carrière, Flots transversalement de Lie R n , flots transversalement de Lie minimaux, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), A477-A478 Zbl0453.57019MR595919
  7. Yves Carrière, Flots riemanniens, Astérisque (1984), 31-52 Zbl0548.58033MR755161
  8. Demetrio Domínguez, Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), 1237-1276 Zbl0964.53019MR1657170
  9. A. El Kacimi-Alaoui, G. Hector, Décomposition de Hodge basique pour un feuilletage riemannien, Ann. Inst. Fourier (Grenoble) 36 (1986), 207-227 Zbl0586.57015MR865667
  10. Étienne Ghys, Classification des feuilletages totalement géodésiques de codimension un, Comment. Math. Helv. 58 (1983), 543-572 Zbl0534.57015MR728452
  11. Étienne Ghys, Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier (Grenoble) 34 (1984), 203-223 Zbl0525.57024MR766280
  12. André Haefliger, Some remarks on foliations with minimal leaves, J. Differential Geom. 15 (1980), 269-284 Zbl0444.57016MR614370
  13. André Haefliger, Pseudogroups of local isometries, Differential geometry (Santiago de Compostela, 1984) 131 (1985), 174-197, Pitman, Boston, MA Zbl0656.58042MR864868
  14. André Haefliger, Leaf closures in Riemannian foliations, A fête of topology (1988), 3-32, Academic Press, Boston, MA Zbl0667.57012MR928394
  15. Franz W. Kamber, Philippe Tondeur, Foliated bundles and characteristic classes, (1975), Springer-Verlag, Berlin-New York Zbl0308.57011MR402773
  16. Franz W. Kamber, Philippe Tondeur, Duality for Riemannian foliations, Singularities, Part 1 (Arcata, Calif., 1981) 40 (1983), 609-618, Amer. Math. Soc., Providence, RI Zbl0523.57019MR713097
  17. Franz W. Kamber, Philippe Tondeur, Foliations and metrics, Differential geometry (College Park, Md., 1981/1982) 32 (1983), 103-152, Birkhäuser Boston, Boston, MA Zbl0542.53022MR702530
  18. Franz W. Kamber, Philippe Tondeur, Duality theorems for foliations, Astérisque (1984), 108-116 Zbl0559.58022MR755165
  19. Xosé Masa, Duality and minimality in Riemannian foliations, Comment. Math. Helv. 67 (1992), 17-27 Zbl0778.53029MR1144611
  20. I. Moerdijk, J. Mrčun, Introduction to foliations and Lie groupoids, 91 (2003), Cambridge University Press, Cambridge Zbl1029.58012MR2012261
  21. Pierre Molino, Feuilletages de Lie à feuilles denses, (1982-1983) 
  22. Pierre Molino, Riemannian foliations, 73 (1988), Birkhäuser Boston, Inc., Boston, MA Zbl0633.53001MR932463
  23. Pierre Molino, Vlad Sergiescu, Deux remarques sur les flots riemanniens, Manuscripta Math. 51 (1985), 145-161 Zbl0585.53026MR788676
  24. Hiraku Nozawa, Rigidity of the Álvarez class, Manuscripta Math. 132 (2010), 257-271 Zbl1193.53086MR2609297
  25. Hiraku Nozawa, Haefliger cohomology of Riemannian foliations, (2012) Zbl1269.57011
  26. Bruce L. Reinhart, Foliated manifolds with bundle-like metrics, Ann. of Math. (2) 69 (1959), 119-132 Zbl0122.16604MR107279
  27. José Ignacio Royo Prieto, The Euler class for Riemannian flows, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 45-50 Zbl0987.53009MR1805626
  28. José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak, Tautness for Riemannian foliations on non-compact manifolds, Manuscripta Math. 126 (2008), 177-200 Zbl1155.57026MR2403185
  29. José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak, Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys. 16 (2009), 450-466 Zbl1178.57021MR2551892
  30. Hansklaus Rummler, Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts, Comment. Math. Helv. 54 (1979), 224-239 Zbl0409.57026MR535057
  31. K. S. Sarkaria, A finiteness theorem for foliated manifolds, J. Math. Soc. Japan 30 (1978), 687-696 Zbl0398.57012MR513077
  32. Vlad Sergiescu, Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier (Grenoble) 35 (1985), 137-158 Zbl0563.57012MR810671
  33. Dennis Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979), 218-223 Zbl0409.57025MR535056
  34. Philippe Tondeur, A characterization of Riemannian flows, Proc. Amer. Math. Soc. 125 (1997), 3403-3405 Zbl0899.53023MR1415373
  35. N. Žukova, On the stability of leaves of Riemannian foliations, Ann. Global Anal. Geom. 5 (1987), 261-271 Zbl0658.53026MR962299

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