Abstract
Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense. When they are equipped with an effective isometric action of a compact Lie group with one-dimensional orbit space, they are said to be of cohomogeneity one. Well-known examples include cohomogeneity-one Riemannian manifolds with a uniform lower sectional curvature bound; such spaces are of interest in the context of non-negative and positive sectional curvature. In the present article we classify closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions 5, 6 and 7. This yields, in combination with previous results for manifolds and Alexandrov spaces, a complete classification of closed, simply connected cohomogeneity-one Alexandrov spaces in dimensions at most 7.
Highlights
Alexandrov spaces are complete length spaces with a lower curvature bound in the triangle comparison sense; they generalize Riemannian manifolds with a uniform lower sectional curvature bound
Instances of Alexandrov spaces include Riemannian orbifolds, orbit spaces of isometric actions of compact Lie groups on Riemannian manifolds with sectional curvature bounded below, or Gromov–Hausdorff limits of sequences of n-dimensional Riemannian manifolds with a uniform lower bound on the sectional curvature
In combination with classification results for Alexandrov spaces of cohomogeneity one in dimensions 2, 3 and 4 and the manifold classification results cited above, our result yields a complete equivariant classification of closed, connected cohomogeneityone Alexandrov spaces in dimensions at most 7
Summary
Alexandrov spaces (with curvature bounded from below) are complete length spaces with a lower curvature bound in the triangle comparison sense; they generalize Riemannian manifolds with a uniform lower sectional curvature bound. We point out that the diagrams (G, H, K−, K+) in Tables 1, 2, 3, 4, 5, 6, and 7 contain, as particular cases, the diagrams of non-smoothable cohomogeneity one actions on closed, connected topological manifolds in [16]; in this special situation the positively curved homogeneous spaces K±∕H are either spheres or the Poincaré homology sphere. One considers each group action individually, taking into account the fact that the groups must satisfy restrictions imposed by the fact that the homogeneous spaces K±∕H are positively curved In this way, one obtains all the possible diagrams (G, H, K−, K+) , which determine the equivariant type of the Alexandrov space.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
