COVERING RADIUS, VOLUME COMPARISON AND SPHERE RIGIDITY

We prove some relation of volume with the covering radius of Riemannian manifolds and reprove well-known sphere rigidity theorems by using it.

Geometry of Manifolds R. L. Bishop;R. Crittenden
English transl. in russian Math. surveys v.47 Alexandrov spaces with curvature bounded below Y. Burago;M. Gromov;G. Perelman
Uspekhi Math. Nauk v.47 Alexandrov spaces with curvature bounded below Y. Burago;M. Gromov;G. Perelman
Sphere theorems in Alexandrov spaces via new metric invariant M. Cho
will appear in Invent. Math. Convergence and Positive Ricci curvature T. Colding
On the excess of metric spaces and manifolds(preprint) K. Grove;P. Petersen
Acta Math v.169 Volume comparison a la Alexandrov K. Grove;P. Petersen
Invent. Math. v.98 A new version of differentiable sphere theorem Y. Otsu;K. Shiohama;T. Yamaguchi
Proceedings of Symposia in Pure Math. v.54 Recebt developments in sphete theorems K. Shiohama
Perspect. Math. v.8 Positively curved manifolds with restricted diameters, Geometry of manifolds K. Shiohama;T. Yamaguchi
Proceedings of Symposia in Pure Math. v.54 Hausdorff convergence and sphere theorems J-Y. Wu