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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.

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  • $\begingroup$ Are you considering a Riemannian manifold $(M,g)$ with metric? 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 $\endgroup$
    – Jaap Eldering
    Commented Nov 11, 2021 at 11:11
  • $\begingroup$ In a more general sense I don't really care about the metric on $M$, I was saying that if we do have a metric $g$ on $M$ then the Sasaki metric has totally geodesic fibers but does not have positive injectivity radius. But I was asking if there exists a metric in $TM$ with these two properties .@JaapEldering $\endgroup$
    – Someone
    Commented Nov 11, 2021 at 11:16
  • $\begingroup$ The question was answered in the MSE answer here in greater generality. As a general rule, if you post a question at MSE and then, the same question, at MO, you should link to the cross-post to avoid duplication of efforts. $\endgroup$
    – Moishe Kohan
    Commented Dec 29, 2021 at 23:13
  • $\begingroup$ The question was not answered since we don't know how to control the injectivity radius of the metric on $N$ @MoisheKohan. We only know how to to do it in the tubular neighborhood. Hence I tried to see if anyone here an idea to solve the problem in this particular case. $\endgroup$
    – Someone
    Commented Dec 29, 2021 at 23:17

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