Manifold without conjugate points and positive curvature

Question

I'm looking for an example of a complete riemannian manifold with sectional positive curvature and without conjugate points. I've tried the projective space, but the identfication used to construct it does not take out the conjugated points: instead of it, this identificates them. Does anyone know some simple example?

Answer 1

I don't think there is any example of what you seek.

More specifically, this paper has a proof of the following claim:

Assume $M$ is complete and has no conjugate points. If there is a point at which the Ricci curvature in all directions is non-negative, then $M$ is flat. That is, all sectional curvatures are $0$.

The contrapositive gives that there is no $M$ with positive sectional curvature and no conjugate points.