Let Mn be a manifold of sectional curvature, 0 < δ < KMn≤1, let X be an Alexandrov space of curvature ≥ −1. Suppose the Gromov-Hausdorff distance of Mn and X is less than ∈(n,δ) > 0. Our main results are: (A) If X has the lowest possible dimension, n−1/2, then a covering space of Mn of order ≤ n+1/2 is diffeomorphic to a lens space, Sn/ℤq, such that 0 < c(n,δ)[vol(Mn)]−1 ≤ q ≤ vol(Sδn)[vol(Mn)]−1, where Sδn is the sphere of constant curvature δ. (B) If X has nonempty boundary, then a coveringspace of Mn of order ≤ n+1/2 is diffeomorphic to a lens space, provided ∈ depends also on the Hausdorff measure of X.
|Original language||English (US)|
|Number of pages||24|
|Journal||Journal of Differential Geometry|
|State||Published - 1999|
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology