Suppose we have a cotangent bundle $T^*M$ of a compact manifold $M$. If we would consider the Sasaki-metric on $T^*M$ we would be able to find a metric which has totally geodesic fibers.

Now I also know that , by this question Do cotangent bundles have "bounded geometry"?, that $T^*M$ has bounded geometry, and hence there will exist a metric on it such that it has positive injectivity radius.

It has been mentioned to me that the Sasaki metric does not have positive injectivity radius. Therefore, my question is that if we can find a metric on $T^*M$ with positive injectivity radius and totally geodesic fibers ?

Any insight is appreciated, thanks in advance.

Edit: An idea would be to consider something in it's conformal class. That is maybe it's possible to find a smooth function $f:T^*M\rightarrow \mathbb{R}$ such that $f.g_{S}$, where $g_S$ is the sasaki metric, has positive injectivity radius and such that $\text{grad}(f)$ has only vertical components. If this is possible then the result follows by looking at the second fundamental forms and using the fact that the fibers are totally geodesic for the sasaki metric. However I am not sure we can find such a function.

Maybe it is not possible to find such a metric however I was not able to come up with a contradiction.

withmetric? From your question talking about the Sasaki metric it seems you are and in that case your presumption that $T^* M$ has bounded geometry, I think is wrong, see mathoverflow.net/questions/94322/… or also my answer and the discussions under it at mathoverflow.net/a/212723/3928 $\endgroup$