Let M be a compact n-manifold of RicMββ₯(nβ1)H (H is
a constant). We are concerned with the following space form rigidity: M is
isometric to a space form of constant curvature H under either of the
following conditions:
(i) There is Ο>0 such that for any xβM, the open Ο-ball at
xβ in the (local) Riemannian universal covering space, (UΟββ,xβ)β(BΟβ(x),x), has the maximal volume i.e., the volume of a Ο-ball in the
simply connected n-space form of curvature H.
(ii) For H=β1, the volume entropy of M is maximal i.e. nβ1 ([LW1]).
The main results of this paper are quantitative space form rigidity i.e.,
statements that M is diffeomorphic and close in the Gromov-Hausdorff topology
to a space form of constant curvature H, if M almost satisfies, under some
additional condition, the above maximal volume condition. For H=1, the
quantitative spherical space form rigidity improves and generalizes the
diffeomorphic sphere theorem in [CC2].Comment: The only change from the early version is an improvement on Theorem
A: we replace the non-collapsing condition on M by on M~ (the
Riemannian universal cover), and the corresponding modification is adding
"subsection c" in Section