Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension


Let \(\mathrm{M }^n,\, n \in \{4,5,6\}\), be a compact, simply connected \(n\)-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on \(\mathrm{M }^n\) by a torus \(\mathrm{T }^{n-2}\) is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

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  1. 1.

    Barden, D.: Simply connected five-manifolds. Ann. Math. 82(3), 365–385 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Bérard-Bergery, L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pure et Appl. 55, 47–68 (1976)

    MATH  Google Scholar 

  3. 3.

    Bredon, G.E.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)

    Google Scholar 

  4. 4.

    Cheeger, J.: Some examples of manifolds of non-negative curvature. J. Differ. Geom. 8, 623–628 (1973)

    MATH  MathSciNet  Google Scholar 

  5. 5.

    DeVito. J.: The Classification of Simply Connected Biquotients of Dimension at Most 7 and 3 New Examples of Almost Positively Curved Manifolds, Ph.D. thesis. University of Pennsylvania (2011)

  6. 6.

    Duan, H., Liang, C.: Circle bundles over 4-manifolds. Arch. Math. 85, 278–282 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Eschenburg, J.-H.: Freie isometrische Aktionen auf kompakten Liegruppen mit positiv gekrümmten Orbiträumen. Schriftenreihe Math. Inst. Univ. Münster (2), 32 (1984)

  8. 8.

    Fintushel, R.: Circle actions on simply connected 4-manifolds. Trans. Am. Math. Soc. 230, 147–171 (1977)

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Fintushel, R.: Classification of circle actions on 4-manifolds. Trans. Am. Math. Soc. 242, 377–390 (1978)

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Galaz-Garcia, F.: Nonnegatively curved fixed point homogeneous manifolds in low dimensions. Geom. Dedicata 157, 367–396 (2012)

    Google Scholar 

  11. 11.

    Galaz-Garcia, F., Searle, C.: Low-dimensional manifolds with non-negative curvature and maximal symmetry rank. Proc. Am. Math. Soc. 139, 2559–2564 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Grove, K.: Geometry of, and Via, Symmetries, Conformal, Riemannian and Lagrangian Geometry (Knoxville, TN, 2000), Univ. Lecture Ser., vol. 27, Am. Math. Soc., Providence, RI, pp. 31–53 (2002)

  13. 13.

    Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry-rank. J. Pure Appl. Algebra 91(1–3), 137–142 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Grove, K., Searle, C.: Differential topologicial restrictions curvature and symmetry. J. Differ. Geom. 47, 530–559 (1997)

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Grove, K., Wilking, B.: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry, preprint, arXiv:1304.4827 [math.DG]

  16. 16.

    Grove, K., Wilking, B., Ziller, W.: Positively curved cohomogeneity one manifolds and 3-Sasakian geometry. J. Differ. Geom. 78(1), 33–111 (2008)

    MATH  MathSciNet  Google Scholar 

  17. 17.

    Grove, K., Ziller, W.: Lifting group actions and nonnegative curvature. Trans. Am. Math. Soc. 363(6), 2865–2890 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Hsiang, W.Y., Kleiner, B.: On the topology of positively curved 4-manifolds with symmetry. J. Differ. Geom. 29(3), 615–621 (1989)

    MATH  MathSciNet  Google Scholar 

  19. 19.

    Kerin, M.: On the curvature of biquotients. Math. Ann. 352(1), 155–178 (2012)

    Google Scholar 

  20. 20.

    Kleiner, B.: Riemannian Four-manifolds with Non-negative Curvature and Continuous Symmetry. Ph.D. thesis, University of California, Berkeley (1990)

  21. 21.

    Kim, S., McGavran, D., Pak, J.: Torus group actions in simply connected manifolds. Pac. J. Math. 53(2), 435–444 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Oh, H.S.: 6-dimensional manifolds with effective \({\rm T}^4\) actions. Topol. Appl. 13(2), 137–154 (1982)

    Article  MATH  Google Scholar 

  23. 23.

    Oh, H.S.: Toral actions on 5-manifolds. Trans. Am. Math. Soc. 278(1), 233–252 (1983)

    MATH  Google Scholar 

  24. 24.

    Orlik, P., Raymond, F.: Actions of the torus on 4-manifolds. I. Trans. Am. Math. Soc. 152, 531–559 (1970)

    MATH  MathSciNet  Google Scholar 

  25. 25.

    Pavlov, A.V.: Five-dimensional biquotients of Lie groups, (Russian) Sibirsk. Math. Zh. 45(6), 1323–1328 (2004); translation in. Sib. Math. J. 45(6), 1080–1083 (2004)

  26. 26.

    Pao, P.S.: Nonlinear circle actions on the 4-sphere and twisting spun knots. Topology 17(3), 291–296 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. preprint 2002, arXiv:math/0211159v1 [math.DG]

  28. 28.

    Perelman, G.: Ricci flow with surgery on three-manifolds, preprint 2003, arXiv:math/0303109v1 [math.DG]

  29. 29.

    Searle, C., Yang, D.: On the topology of non-negatively curved simply connected 4-manifolds with continuous symmetry. Duke Math. J. 74(2), 547–556 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. 30.

    Totaro, B.: Cheeger manifolds and the classification of biquotients. J. Differ. Geom. 61, 397–451 (2002)

    MATH  MathSciNet  Google Scholar 

  31. 31.

    Verdiani, L.: Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature. J. Differ. Geom. 68, 31–72 (2004)

    MATH  MathSciNet  Google Scholar 

  32. 32.

    Wallach, N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96, 277–295 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  33. 33.

    Wilking, B.: Torus actions on manifolds of positive sectional curvature. Acta Math. 191(2), 259–297 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Wilking, B.: Positively curved manifolds with symmetry. Ann. Math. 163(2), 607–668 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  35. 35.

    Wilking, B.: Nonnegatively and Positively Curved Manifolds. Surv. Differ. Geom., vol. 11. International Press, Somerville, MA (2007)

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The first named author thanks B. Wilking and K. Grove for useful conversations. The second named author wishes to thank J. DeVito for several interesting and helpful discussions.

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Correspondence to Fernando Galaz-Garcia.

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This research was carried out as part of SFB 878: Groups, Geometry & Actions, at the University of Münster.

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Galaz-Garcia, F., Kerin, M. Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension. Math. Z. 276, 133–152 (2014).

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  • Non-negative curvature
  • Circle action
  • Torus action
  • 4-manifolds
  • 5-manifolds
  • Symmetry rank

Mathematics Subject Classification (2000)

  • 53C20