We say that a Riemannian manifold $M$ is $ parabolic$ if the only positive superharmonic functions on $M$ are constants. Then what are some examples/characterizations of parabolic Riemannian manifolds that have infinite injectivity radius at each point on $M$, apart from $\Bbb R^2$ ?
My intuitions: I think parabolicity is connected with non-negative sectional curvature whereas infinite injectivity radius is associated with non-positive sectional curvature. So are such manifolds necessarily flat or are there some interesting examples? Thanks in advance for help!