Does a simply connected complete riemannian manifold with POSITIVE upper curvature bound have positive injectivity radius?

Question

For example: I am thinking that some sort of rauch comparison theorem could be helpful

Answer 1

I am nearly sure that the answer is negative, but do not have specific examples. The reason is that there are sequences $(S^3, g_i)$ of of Riemannian metrics on $S^3$ such that the sectional curvature of $g_i$ is uniformly bounded above, but injectivity radius of $g_i$ converges to zero. These are so called "Berger spheres". In particular, this example show that one cannot get a lower injectivity radius bound by appealing to comparison theorems. Constructing a (noncompact) simple-connected example with a single Riemannian metric would require some dirty work (possibly, gluing Berger spheres by removing foliated solid tori and then gluing results along the boundaries) while preserving an upper curvature bound).

Edit: Here is an easy example. Start with the half-plane $H=\{z\in {\mathbb C}: Im(z)>0\}$ with the standard hyperbolic metric. Consider the cyclic group $Z=\langle u\rangle$ of isometries of $H$ generated by the translation $z\mapsto z+1$. Consider the quotient surface $S=U/Z$, where $U\subset H$ consists of points with $Im(z)\ge 1$. Then injectivity radius of $S$ is not bounded away from zero and its curvature $\equiv -1$. Now, attach to the boundary of $S$ the 2-dimensional disk and extend the Riemannian metric from $S$ smoothly to this disk. The result is a Riemannian metric on a surface $S'$ diffeomorphic to the plane, such that the curvature is bounded above (it is nonconstant only in a compact region) and injectivity radius not bounded below.