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Low dimensional polar actions

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Abstract

Polar manifolds are Riemannian \(G\)-manifolds admitting a “section”, i.e., a complete submanifold passing through every orbit and doing so orthogonally. We consider compact simply-connected polar manifolds and achieve an equivariantly diffeomorphic classification in dimensions 5 or less. As an application, we determine which of these polar actions admit an invariant metric with non-negative curvature.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Bergmann, I.: Reducible polar representations. Manuscr. Math. 104(3), 309–324 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boualem, H.: Feuilletages riemanniens singuliers transversalement intégrables. Compos. Math. 95(1), 101–125 (1995)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 2(96), 413–443 (1972)

    Article  MathSciNet  Google Scholar 

  6. Dadok, J.: Polar coordinates induced by actions of compact Lie groups. Trans. Am. Math. Soc. 288(1), 125–137 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Davis, M.W.: Lectures on orbifolds and reflection groups. In: ji, L., Yau, S-T. (eds.) Transformation Groups and Moduli Spaces of Curves, vol. 16 of Adv. Lect. Math. (ALM). International Press, Somerville, MA (2011)

  8. Eschenburg, J.H., Heintze, E.: On the classification of polar representations. Mathematische Zeitschrift 232(3), 391–398 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fang, F., Grove, K., Thorbergsson, G.: Tits geometry and positive curvature. arXiv:1205.6222 [math.DG]

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Galaz-Garcia, F., Searle, C.: Non-negatively curved 5-manifolds with almost maximal symmetry rank. arXiv:1111.3183 [math.DG]

  13. Galaz-Garcia, F., Spindeler, W.: Nonnegatively curved fixed point homogeneous 5-manifolds. Ann. Global Anal. Geom. 41(2), 253–263 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ge, J., Radeschi, M.: Differentiable classification of 4-manifolds with singular Riemannian foliations. arXiv:1312.0667 [math.DG]

  15. Grove, K.: Geometry of and via, symmetries. Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000) vol. 27 of Univ. Lecture Ser. American Mathematical Society, Providence, RI, pp. 31–53 (2002)

  16. Grove, K., Wilking, B.: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. arXiv:1304.4827 [math.DG] (2013)

  17. 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 

  18. Grove, K., Ziller, W.: Polar manifolds and actions. J. Fixed Point Theory Appl. 11(2), 279–313 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. 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 

  20. Hoelscher, C.A.: Classification of cohomogeneity one manifolds in low dimensions. Pac. J. Math. 246(1), 129–185 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. Kleiner, B.: Riemannian Four-Manifolds with Nonnegative Curvature and Continuous Symmetry. Ph.D. thesis, University of California, Berkeley (1990)

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

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Palais, R.S., Terng, C.-L.: A general theory of canonical forms. Trans. Am. Math. Soc. 300(2), 771–789 (1987)

    Article  MATH  MathSciNet  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. Simas, F.: Nonnegatively Curved 5-Manifolds with Non-abelian Symmetry. arXiv:1212.5022 [math.DG]

  29. 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  MATH  MathSciNet  Google Scholar 

  30. Wilking, B.: Nonnegatively and positively curved manifolds. In: Cheeger, J., Grove, K. (eds.) Surveys in Differential Geometry. Vol. XI. International Press, Somerville, MA, pp. 25–62 (2007)

  31. Ziller, W.: Examples of Riemannian manifolds with non-negative sectional curvature. In: Cheeger, J., Grove, K. (eds.) Surveys in Differential Geometry, Vol. XI. International Press, Somerville, MA, pp. 63–102 (2007)

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Authors and Affiliations

  1. IMPA, Rio de Janeiro, Brazil

    Francisco J. Gozzi

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  1. Francisco J. Gozzi
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Correspondence to Francisco J. Gozzi.

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The author was supported by CNPq-BRAZIL.

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Gozzi, F.J. Low dimensional polar actions. Geom Dedicata 175, 219–247 (2015). https://doi.org/10.1007/s10711-014-0037-5

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