We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
The smooth diameter sphere theorem presented in this note shows that it is possible to isolate the standard sphere among all other complete Riemannian manifolds with positive Ricci curvature by using merely curvature and diameter assumptions, and that in fact any violation of smooth rigidity in Cheng\'s maximal diameter theorem must be accompanied by a blow-up of sectional curvatures:
For any given m and C there exists a positive constant