Manifolds with nonpositive radial curvature

Question

How can one construct examples of Riemannian manifolds which have nonpositive radial curvature about some point, but are not nonpositively curved everywhere? (I presume that they exist, but do not know for sure.) The question may be formulated precisely as follows.

A (complete) Riemannian manifold $M$ has a pole at a point $p$, provided that the exponential map $exp_p \colon T_p M\to M$ is a diffeomorphism. The radial curvatures of $M$ (with respect to $p$) are the sectional curvatures with respect to the planes tangent to geodesics emanating from $p$. If all the radial curvatures are nonpositive, then how can $M$ have any positive curvature?

It is easy to show that such examples cannot be constructed by means of a warped product metric on $S^{n-1}\times[0,\infty)$. So $M$ cannot be too symmetric with respect to $p$. I would be particularly interested in $3$-dimensional examples.

Answer 1

Assume that $M$ is three-dimenional and $\exp_p\colon\mathbb{R}^3\to M$ is a diffeomorphism. Let $g_r$ be the Riemannian metric induced on $\mathbb{S}^{2}$ by the map $x\mapsto \exp_p(r\cdot x)$.

Note that $M$ has negative radial curvature if $g_r$ grows fast, say if $\tfrac{\partial^2}{\partial r^2}g_r>g_r$. But it gives absolutely no conditions on the shape of $(\mathbb{S}^{2},g_r)$ for large $r$. So you can choose $(\mathbb{S}^{2},g_r)$ to be very positively curved at some point; say bigger than $(\tfrac{\partial}{\partial r}g_r(V,V)/g_r(V,V))^2$ for any tangent vector $V$. Applying Gauss formula, you get that the sectional curvature in the direction tangent to the sphere is positive.