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A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds

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Book cover Geometry of Manifolds with Non-negative Sectional Curvature

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2110))

Abstract

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantly diffeomorphic to a linear action. We show that a compact, simply connected Riemannian 4- or 5-manifold of quasipositive curvature and maximal symmetry rank must be diffeomorphic to the 4-sphere, complex projective plane or the 5-sphere.

2000 Mathematics Subject Classification. 53C20.

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Notes

  1. 1.

    The author “Fernando Galaz-Garcia” is part of SFB 878: Groups, Geometry & Actions, at the University of Münster.

References

  1. S. Aloff, N.R. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–97 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Berard-Bergery, Les variétés riemanniennes homogènes simplement connexes de dimension impaire á courbure strictement positive (French). J. Math. Pures Appl. (9) 55(1), 47–67 (1976)

    Google Scholar 

  3. M. Berger, Les variétés riemanniennes homogènes normales simplement connexes á courbure strictement positive (French). Ann. Scuola Norm. Sup. Pisa (3) 15, 179–246 (1961)

    Google Scholar 

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

    Google Scholar 

  5. D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry. Graduate Studies in Mathematics, vol. 33 (American Mathematical Society, Providence, 2001)

    Google Scholar 

  6. O. Dearricott, A 7-manifold with positive curvature. Duke Math. J. 158(2), 307–346 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. O. Dearricott, Relations between metrics of almost positive curvature on the Gromoll-Meyer sphere. Proc. Am. Math. Soc. 140(6), 2169–2178 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. J. DeVito, 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. http://repository.upenn.edu/edissertations/311.

  9. J.-H. Eschenburg, M. Kerin, Almost positive curvature on the Gromoll-Meyer sphere. Proc. Am. Math. Soc. 136(9), 3263–3270 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Galaz-Garcia, C. Searle, Non-negatively curved 5-manifolds with almost maximal symmetry rank. Geom. Topol. Preprint: arXiv:1111.3183 [math.DG] (2014, in press)

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

    Article  MATH  MathSciNet  Google Scholar 

  15. D. Gromoll, W. Meyer, An exotic sphere with nonnegative sectional curvature. Ann. Math. 100, 401–408 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  16. K. Grove, Geometry of, and via, symmetries, in Conformal, Riemannian and Lagrangian Geometry, Knoxville, 2000. University Lecture Series, vol. 27 (American Mathematical Society, Providence, 2002), pp. 31–53

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  19. K. Grove, W. Ziller, Curvature and symmetry of Milnor spheres. Ann. Math. (2) 152(1), 331–367 (2000)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  21. K. Grove, L. Verdani, W. Ziller, An exotic T 1 S 4) with positive curvature. Geom. Funct. Anal. 21(3), 499–524 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. R.S. Hamilton, Four-manifolds with positive curvature operator. J. Differ. Geom. 24(2), 153–179 (1986)

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  24. M. Kerin, Some new examples with almost positive curvature. Geom. Topol. 15(1), 217–260 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Kleiner, Riemannian four-manifolds with nonnegative curvature and continuous symmetry. Ph.D. Thesis, U.C. Berkeley, 1989

    Google Scholar 

  27. S. Kobayashi, Transformation Groups in Differential Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70 (Springer, New York/Heidelberg, 1972)

    Google Scholar 

  28. J. Kollár, Circle actions on simply connected 5-manifolds. Topology 45(3), 643–671 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. S.B. Myers, N. Steenrod, The group of isometries of a Riemannian manifold. Ann. Math. (2) 40(2), 400–416 (1939)

    Google Scholar 

  30. W.D. Neumann, 3-Dimensional G-manifolds with 2-dimensional orbits. in 1968 Proceedings Conference on Transformation Groups, New Orleans, 1967, pp. 220–222

    Google Scholar 

  31. H.S. Oh, 6-Dimensional manifolds with effective T 4-actions. Topol. Appl. 13(2), 137–154 (1982)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  33. P. Orlik, F. Raymond, Actions of SO(2) on 3-manifolds, in Proceeding Conference on Transformation Groups, New Orleans, 1967 (Springer, New York, 1968) pp. 297–318

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  35. P. Orlik, F. Raymond, Actions of the torus on 4-manifolds. II. Topology 13, 89–112 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  36. J. Pak, Actions of torus T n on (n + 1)-manifolds M n+1. Pac. J. Math. 44, 671–674 (1973)

    Article  MATH  Google Scholar 

  37. G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll. J. Differ. Geom. 40, 209–212 (1994)

    MATH  MathSciNet  Google Scholar 

  38. P. Petersen, F. Wilhelm, Examples of Riemannian manifolds with positive curvature almost everywhere. Geom. Topol. 3, 331–367 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. P. Petersen, F. Wilhelm, An exotic sphere with positive sectional curvature (2008). arXiv:0805.0812v3 [math.DG]

    Google Scholar 

  40. F. Raymond, Classification of the actions of the circle on 3-manifolds. Trans. Am. Math. Soc. 131, 51–78 (1968)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. C. Searle, D. Yang, 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 

  43. K. Tapp, Quasi-positive curvature on homogeneous bundles. J. Differ. Geom. 65(2), 273–287 (2003)

    MATH  MathSciNet  Google Scholar 

  44. L. Verdiani, Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature, I. Math. Z. 241, 329–339 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  45. L. Verdiani, Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature. J. Differ. Geom. 68(1), 31–72 (2004)

    MATH  MathSciNet  Google Scholar 

  46. N.R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. (2) 96, 277–295 (1972)

    Google Scholar 

  47. F. Wilhelm, An exotic sphere with positive curvature almost everywhere. J. Geom. Anal. 11(3), 519–560 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  48. B. Wilking, The normal homogeneous space has positive sectional curvature. Proc. Am. Math. Soc. 127(4), 1191–1194 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  49. B. Wilking, Manifolds with positive sectional curvature almost everywhere. Invent. Math. 148(1), 117–141 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  52. B. Wilking, Nonnegatively and positively curved manifolds, in Surveys in Differential Geometry, vol. XI (International Press, Somerville, 2007), pp. 25–62

    Google Scholar 

  53. W. Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, in Surveys in Differential Geometry, vol. XI (International Press, Somerville, 2007), pp. 63–102

    Google Scholar 

  54. B. Wilking, Torus actions on manifolds with quasipositive curvature. Notes by M. Kerin (2008) Unpublished

    Google Scholar 

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Acknowledgements

This note originated from talks I gave at the Tercer Miniencuentro de Geometría Diferencial in December of 2010, at CIMAT, México. It is a pleasure to thank Rafael Herrera, Luis Hernández-Lamoneda and CIMAT for their hospitality during the workshop. I also wish to thank Martin Kerin, for sharing his notes with the proof of Theorem 1.3, and Xiaoyang Chen, for some conversations on quasipositively curved manifolds.

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  1. Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Münster, Germany

    Fernando Galaz-Garcia

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Galaz-Garcia, F. (2014). A Note on Maximal Symmetry Rank, Quasipositive Curvature, and Low Dimensional Manifolds. In: Geometry of Manifolds with Non-negative Sectional Curvature. Lecture Notes in Mathematics, vol 2110. Springer, Cham. https://doi.org/10.1007/978-3-319-06373-7_3

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