Injectivity radius on complete manifolds with positive and bounded curvature I have two question:
1)  Are there any examples of complete manifold with strictly positive and bounded section curvature which has zero injectivity radius? 
2)  Is there a sequence of non-compact complete manifolds with strictly positive and bounded section curvature with injectivity radius approach to zero?
I think one may construct these examples from Beger's spheres, but I cannot do it rigorously.
 A: If by strictly positive, you mean that there is a $\epsilon>0$ so that the $Sect >\epsilon$, then there are no such examples. The reason for this is that this curvature assumption implies that a manifold is compact. Since the injectivity radius of a point is a positive and continuous function on a manifold, it has a non-zero minimum.
Igor Belegradek pointed out (thanks for this) that strictly positive also has a another meaning, which is that the sectional curvature is everywhere positive, without a positive lower bound. If we allow this as the definition of "strictly positive", there is a simple counterexample, which is just to take a disjoint union of countably many collapsing Berger spheres. Some of the sectional curvature of the Berger spheres converge to zero as the metric collapses (See here), but the curvature is always positive throughout the collapse. This also addresses your second question in terms of this definition of positive curvature.
However, it seems worth asking whether we can create a connected manifold with positive curvature and zero injectivity radius. I was unable to find a simple argument to rule out such examples, but I strongly suspect that none exist. As I mentioned before, the only spaces we need to worry about are non-compact. In this case, one can either use the Soul Theorem or results of Gromoll and Meyer to see that the manifold must be diffeomorphic to $\mathbb{R}^n$. Such a manifold also has no closed geodesics and there is a lower bound on the distance between conjugate points, which is strong evidence that there should be control on the injectivity radius. However, I'm not sure how to derive such an estimate.
Maybe someone more familiar with this can help.
