# 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.

• Here is an idea: start from a paraboloid in $\mathbb R^3$ and introduce a sequence of "pimples" that run off to infinity and each approximate a tiny cone. The easiest way to visualize this is by attaching little cones to a parabola in $\mathbb R^2$ so that the result is a convex curve, then approximate by strictly convex curve, and consider the corresponding surface of revolution. Near each attached "cone" the injectivity radius will be small. The trick will be to pick cone regions so that the curvature stays bounded above and injectivity radius tends to zero. – Igor Belegradek Apr 29 '19 at 18:41
• Thanks for your answer, I tried to construct a exampale as you have suggested ,But when attaching some cones to parabola, I should take this cone long and thin to make sure injectivity radius tends to zero,. this seems not true Under the curvature bounded assumption. are there any point I have omitted? – Yuchen Bi May 4 '19 at 8:14

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.
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.
• It seems likely that by "strictly positive" the OP means "$>0$". If so, then your first paragraph is incorrect, e.g. the paraboloid in $\mathbb R^3$ has positive curvture everywhere. – Igor Belegradek Apr 29 '19 at 15:36