Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension

Abstract

Let \(\mathrm{M }^n,\, n \in \{4,5,6\}\), be a compact, simply connected \(n\)-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on \(\mathrm{M }^n\) by a torus \(\mathrm{T }^{n-2}\) is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.

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Acknowledgments

The first named author thanks B. Wilking and K. Grove for useful conversations. The second named author wishes to thank J. DeVito for several interesting and helpful discussions.

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Correspondence to Fernando Galaz-Garcia.

Additional information

This research was carried out as part of SFB 878: Groups, Geometry & Actions, at the University of Münster.

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Galaz-Garcia, F., Kerin, M. Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension. Math. Z. 276, 133–152 (2014). https://doi.org/10.1007/s00209-013-1190-5

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Keywords

  • Non-negative curvature
  • Circle action
  • Torus action
  • 4-manifolds
  • 5-manifolds
  • Symmetry rank

Mathematics Subject Classification (2000)

  • 53C20