Advertisement

Nonnegatively curved quotient spaces with boundary

  • Wolfgang Spindeler
Original Article

Abstract

Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group \(\mathsf {G}\) in a way that the quotient space \(M/\mathsf {G}\) has nonempty boundary. Let \(\pi : M \rightarrow M/\mathsf {G}\) denote the quotient map and B be any boundary stratum of \(M/\mathsf {G}\). Via a specific soul construction for \(M/\mathsf {G}\), we construct a smooth closed submanifold N of M such that \(M {\setminus } \pi ^{-1}(B)\) is diffeomorphic to the normal bundle of N. As an application we show that a simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

Mathematics Subject Classification

53C20 

Notes

Acknowledgements

I am grateful to Burkhard Wilking for his support during the work on my thesis where the techniques developed here originate.

References

  1. 1.
    Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 2(96), 413–443 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Galaz-Garcia, F.: Nonnegatively curved fixed point homogeneous manifolds in low dimensions. Geom. Dedicat. 157, 367–396 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Galaz-Garcia, F., Spindeler, W.: Nonnegatively curved fixed point homogeneous 5-manifolds. Ann. Glob. Anal. Geom. 41(2), 253–263 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Grove, K., Halperin, S.: Dupin hypersurfaces, group actions and the double mapping cylinder. J. Differ. Geom. 26(3), 429–459 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Grove, K., Halperin, S.: Contributions of rational homotopy theory to globalproblems in geometry. Publ. Math. IHES 56(1), 171–177 (1982)CrossRefzbMATHGoogle Scholar
  6. 6.
    Grove, K.: Geometry of, and via, symmetries. In: Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), volume 27 of Univ. Lecture Ser., American Mathematical Society, Providence, RI, pp. 31–53 (2002)Google Scholar
  7. 7.
    Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry-rank. J. Pure Appl. Algebra 91(1–3), 137–142 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grove, K., Searle, C.: Differential topological restrictions curvature and symmetry. J. Differ. Geom. 47(3), 530–559 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Perelman, G.: Alexandrov’s spaces with curvatures bounded from below II. (1991). https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCBFB2+.pdf
  10. 10.
    Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. Surveys Differ Geom 11(1), 137–202 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Spindeler, W.: \({\sf S}^1\)-actions on \(4\)-manifolds and fixed point homogeneous manifolds of nonnegative curvature. Dissertation, WWU Münster (2014). http://miami.uni-muenster.de/Record/272b3efb-9d8d-4ee3-8b15-2ab860f49ed0
  12. 12.
    Spindeler, W.: \({\sf S}^1\)-actions on 4-manifolds and fixed point homogeneous manifolds of nonnegative curvature (2015). arXiv:1510.01548
  13. 13.
    Wörner, A.: Boundary Strata of non-negatively curved Alexandrov spaces and a splitting theorem. Dissertation, WWU Münster (2010)Google Scholar
  14. 14.
    Wiemeler, M.: Torus manifolds and non-negative curvature. J. Lond. Math. Soc. II. Ser. 91(3), 667–692 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wilking, B.: Positively curved manifolds with symmetry. Ann. Math. 163, 607–668 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wilking, B.: A duality theorem for riemannian foliations in nonnegative sectional curvature. Geom. Funct. Anal. 17(4), 1297–1320 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Matemática Mexicana 2019

Authors and Affiliations

  1. 1.DüsseldorfGermany

Personalised recommendations