Advertisement

The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 1002–1017 | Cite as

Non-negatively Curved 6-Manifolds with Almost Maximal Symmetry Rank

  • Christine Escher
  • Catherine SearleEmail author
Article
  • 35 Downloads

Abstract

We classify closed, simply connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

Keywords

Almost maximal symmetry rank Equivariant diffeomorphism 6-Manifolds Non-negative curvature 

Mathematics Subject Classification

Primary 53C20 Secondary 57S25 

Notes

Acknowledgements

Christine Escher and Catherine Searle would like to thank Michael Wiemeler and a referee for pointing out an omission in a previous version of the paper. We are also indebted to the same referee for many helpful comments and suggestions. This material is based in part upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2016 semester. Catherine Searle would also like to acknowledge support by Grants from the National Science Foundation (#DMS-1611780), as well as from the Simons Foundation (#355508, C. Searle).

References

  1. 1.
    Bredon, G.: Introduction to Compact Transformation Groups, vol. 48. Academic Press, Amsterdam (1972)zbMATHGoogle Scholar
  2. 2.
    DeVito, J.: The classification of compact simply connected biquotients in dimension \(6\) and \(7\). Math. Ann. (2016).  https://doi.org/10.1007/s00208-016-1460-8 MathSciNetzbMATHGoogle Scholar
  3. 3.
    Escher, C., Searle, C.: Torus actions, maximality and non-negative curvature (2017). arXiv:1506.08685v3 [math.DG]
  4. 4.
    Fang, F., Rong, X.: Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank. Math. Ann. 332(1), 81–101 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Galaz-García, F.: Fixed-point homogeneous nonnegatively curved Riemannian manifolds in low dimensions. PhD Thesis, University of Maryland, College Park (2009)Google Scholar
  6. 6.
    Galaz-García, F., Kerin, M.: Cohomogeneity two torus actions on non-negatively curved manifolds of low dimension. Math. Z. 276(1–2), 133–152 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Galaz-García, F., Searle, C.: Low-dimensional manifolds with non-negative curvature and maximal symmetry rank. Proc. Am. Math. Soc. 139, 2559–2564 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Galaz-García, F., Searle, C.: Nonnegatively curved \(5\)-manifolds with almost maximal symmetry rank. Geom. Topol. 18(3), 1397–1435 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Galaz-García, F., Spindeler, W.: Nonnegatively curved fixed point homogeneous \(5\)-manifolds. Ann. Global Anal. Geom. 41(2), 253–263 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Alg. 91, 137–142 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grove, K., Wilking, B.: A knot characterization and \(1\)-connected nonnegatively curved 4-manifolds with circle symmetry. Geom. Topol. 18(5), 3091–3110 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ishida, H.: Complex manifolds with maximal torus actions. J. Reine Angew. Math. (2016).  https://doi.org/10.1515/crelle-2016-0023
  13. 13.
    Jupp, P.E.: Classification of certain \(6\)-manifolds. Math. Proc. Camb. Philos. Soc. 73(2), 293–300 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kleiner, B.: Riemannian four-manifolds with nonnegative curvature and continuous symmetry. PhD thesis, U.C. Berkeley (1989)Google Scholar
  15. 15.
    Kuroki, S.: An Orlik-Raymond type classification of simply-connected six-dimensional torus manifolds with vanishing odd degree cohomology. Pac. J. Math. 280(1), 89–114 (2016)CrossRefzbMATHGoogle Scholar
  16. 16.
    McGavran, D., Oh, H.S.: Torus actions on \(5\)- and \(6\)-manifolds. Indiana Univ. Math. J. 31(3), 363–376 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mostert, P.S.: On a compact Lie group acting on a manifold. Ann. Math. 65(2), 447–455 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Neumann, W.D.: 3-dimensional \(G\)-manifolds with 2-dimensional orbits. In: Proc. Conf. on Transformation Groups (New Orleans, LA, 1967), pp. 220–222. Springer, New York (1968)Google Scholar
  19. 19.
    Pak, J.: Actions of torus \(T^n\) on \((n+1)\)-manifolds \(M^{n+1}\). Pac. J. Math. 44(2), 671–674 (1973)CrossRefzbMATHGoogle Scholar
  20. 20.
    Parker, J.: 4-dimensional \(G\)-manifolds with 3-dimensional orbit. Pac. J. Math 125(1), 187–204 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Perelman, G.: The entropy formula for the Ricci Flow and its geometric applications (2002). arXiv:math.DG/0211159
  22. 22.
    Perelman, G.: Ricci Flow with surgery on three-manifolds (2003). arXiv:math.DG/0303109
  23. 23.
    Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003). arXiv:math.DG/0307245
  24. 24.
    Rong, X.: Positively curved manifolds with almost maximal symmetry rank. Geom. Ded. 95, 157–182 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Spindeler, W.: \(S^1\)-actions on \(4\)-manifolds and fixed point homogeneous manifolds of nonnegative curvature. PhD Thesis, Westfälische Wilhelms-Universität Münster (2014)Google Scholar
  27. 27.
    Wall, C.T.C.: Classification problems in differential topology—IV. Topology 6, 273–296 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wiemeler, M.: Torus manifolds and non-negative curvature. J. Lond. Math. Soc. 91(3), 667–692 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wilking, B.: Torus actions on manifolds of positive sectional curvature. Acta Math. 191(2), 259–297 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhubr, A.V.: A decomposition theorem for simply connected \(6\)-manifolds. LOMI Sem. Notes 36, 40–49 (1973) (Russian)Google Scholar
  31. 31.
    Zhubr, A.V.: Classification of simply connected six-dimensional spinor manifolds. Math. USSR Izv. 9(1975), 793–812 (1976)zbMATHGoogle Scholar
  32. 32.
    Zhubr, A.V.: Closed simply connected six-dimensional manifolds: proofs of classification theorems. Algebra Anal. 12(4), 126–230 (2000)MathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsOregon State UniversityCorvallisUSA
  2. 2.Department of Mathematics, Statistics, and PhysicsWichita State UniversityWichitaUSA

Personalised recommendations