PINCHING BELOW 1/4, INJECTIVITY RADIUS, AND CONJUGATE RADIUS

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 1/4-pinched. Moreover, some famous examples due to M. Berger show that the statement is even false, unless M(n) is at least 1/9-pinched. It has been a longstanding problem whether the pinching constant can be pushed below 1/4 for odd dimensional manifolds or not. In this paper we prove that this is indeed possible. The pinching constant delta is-an-element-of [1/9, 1/4) 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 1/4 and up to now still depends on the dimension.