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Nonnegatively curved five-manifolds with non-abelian symmetry

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Abstract

We classify compact simply-connected 5-dimensional manifolds which admit a metric of nonnegative curvature with a connected non-abelian group acting by isometries. We show that they are diffeomorphic to either \(\hbox {S}^{5}, \hbox {S}^{3} \times \hbox {S}^{2}\), the nontrivial \(\hbox {S}^{3}\)-bundle over \(\hbox {S}^{2}\) or the Wu-manifold, SU(3)/SO(3). This result is a consequence of our equivariant classification of all SO(3) and SU(2)-actions on compact simply-connected five-manifolds. In the case of positive curvature we obtain a partial classification.

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References

  1. Asoh, T.: Smooth \(S^{3}\)-actions on \(n\) manifolds for \(n\le 4\). Hiroshima Math. J. 6(3), 619–634 (1976)

    MathSciNet  MATH  Google Scholar 

  2. Barden, D.: Simply connected five-manifolds. Ann. Math. 2(82), 365–385 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borel, A., Bredon, G., Floyd, E.E., Montgomery, D., Palais, R.: Seminar on transformation groups. In: Borel, A. (ed.) Annals of Mathematics Studies, vol. 46. Princeton University Press, Princeton (1960)

    Google Scholar 

  4. Bredon, G.E.: Introduction to Compact Transformation Groups. Pure and Applied Mathematics, vol. 46. Academic Press, New York (1972)

  5. Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry. AMS Chelsea, Providence (2008) (revised reprint of the 1975 original)

  6. Cheeger, J., Gromoll, D.: The structure of complete manifolds of nonnegative curvature. Bull. Am. Math. Soc. 74, 1147–1150 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coxeter, H.S.M.: Regular Complex Polytopes. Cambridge University Press, London (1974)

    MATH  Google Scholar 

  8. 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. ProQuest LLC, Ann Arbor, MI (2011)

  9. Frankel, T.: Manifolds with positive curvature. Pac. J. Math. 11, 165–174 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Galaz-Garcia, F., Kerin, M.: Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimensions. Math. Z. 276(1–2), 133–152 (2014)

  12. Galaz-Garcia, F., Searle, C.: Non-negatively curved 5-manifolds with almost maximal symmetry rank. Geom. Topol. 18(3), 1397–1435 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grove, K.: Geometry of, and via, symmetries. In: Bona, J. L., Hitchin N. J., Reshetikhin N. (eds.) Conformal, Riemannian and Lagrangian Geometry (Knoxville, TN, 2000), vol. 27 of Univ. Lecture Ser., pp. 31–53. Amer. Math. Soc., Providence (2002)

  14. Grove, K., Kim, C.-W.: Positively curved manifolds with low fixed point cohomogeneity. J. Differ. Geom. 67(1), 1–33 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Grove, K., Searle, C.: Differential topological restrictions curvature and symmetry. J. Differ. Geom. 47(3), 530–559 (1997)

    MathSciNet  MATH  Google Scholar 

  16. Grove, K., Wilking, B.: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. Geom. Topol. 18(5), 3091–3110 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hsiang, W., Hsiang, W.: Differentiable actions of compact connected classical groups. I. Am. J. Math. 89, 705–786 (1967)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Hudson, K.: Classification of \({{\rm SO}}(3)\)-actions on five-manifolds with singular orbits. Mich. Math. J. 26(3), 285–311 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jänich, K.: Differenzierbare Mannigfaltigkeiten mit Rand als Orbiträume differenzierbarer \(G\)-Mannigfaltigkeiten ohne Rand. Topology 5, 301–320 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kleiner, B.: Riemannian 4-Manifolds with Nonnegative Curvature and Continuous Symmetry. Ph.D. thesis, Berkeley (1990)

  22. Molino, P.: Riemannian Foliations. Progress in Mathematics, vol. 73. Birkhäuser Boston, Boston (1988)

    Book  MATH  Google Scholar 

  23. Nakanishi, A.: Classification of \({{\rm SO}}(3)\)-actions on five manifolds. Publ. Res. Inst. Math. Sci. 14(3), 685–712 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Oh, H.S.: Toral actions on \(5\)-manifolds. II. Indiana Univ. Math. J. 32(1), 129–142 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ôike, H.: Simply connected closed smooth 5-actions. Tôhoku Math. J. (2) 33(1), 1–24 (1981)

    Article  MATH  Google Scholar 

  27. Rong, X.: Positively curved manifolds with almost maximal symmetry rank. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), vol. 95, pp. 157–182 (2002)

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Smale, S.: On the structure of 5-manifolds. Ann. Math. 2(75), 38–46 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  30. Steenrod, N.: The Topology of Fibre bundles. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1999)

  31. Thurston, W.P.: In: Levy, S. (ed.) The Geometry and Topology of Three-Manifolds. Electronic version 1.1. MSRI - Mathematical Sciences Research Institute, Berkeley 2002

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Wolf, J.A.: Spaces of Constant Curvature, 6th edn. AMS Chelsea, Providence (2011)

    MATH  Google Scholar 

  34. Ziller, W.: Examples of Riemannian manifolds with non-negative sectional curvature. In: Cheeger, J., Grove, K. (eds.) Surveys in Differential Geometry, vol. 11 of Surv. Differ. Geom., pp. 63–102. Int. Press, Somerville (2007)

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Acknowledgments

This work is part of the author’s Ph.D. thesis at IMPA. He would like to express gratitude to IMPA for its hospitality and to his doctoral supervisors Luis A. Florit and Wolfgang Ziller for long hours of helpful and pleasant conversations during the preparation of this paper.

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  1. UNIRIO-Universidade Federal do Estado do Rio de Janeiro, Rio de Janeiro, Brazil

    Fabio Simas

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  1. Fabio Simas
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Correspondence to Fabio Simas.

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The author was partially supported by CNPq-BRAZIL.

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Simas, F. Nonnegatively curved five-manifolds with non-abelian symmetry. Geom Dedicata 181, 61–82 (2016). https://doi.org/10.1007/s10711-015-0112-6

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