Pinching below \(1\over 4\), injectivity radius, and conjugate radius (Q1345153)

The injectivity radius of any simply connected, even dimensional Riemannian manifold \(M^ n\) with positive sectional curvature equals its conjugate radius. So far the corresponding result in odd-dimensions has only been known under the additional hypothesis that \(M^ n\) is weakly \({1\over 4}\)-pinched. Moreover, some famous examples due to M. Berger show that the statement is even false, unless \(M^ n\) is at least \({1\over 9}\)-pinched. It has been a long standing problem whether the pinching constant can be pushed below \({1\over 4}\) for odd dimensional manifolds or not. In this paper we prove that this is indeed possible. The pinching constant \(\delta\in [{1\over 9},{1\over 4})\) that is needed in our main theorem does not depend on the dimension. As an application we obtain a sphere theorem for simply connected, odd-dimensional, \(\delta_ n\)-pinched manifolds where the pinching constant \(\delta_ n\) is strictly less than \({1\over 4}\) and up to now still depends on the dimension.