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Suppose we have a compact riemannian manifold $(M,g)$ and we endow $TM$ with the Sasaki metric $\tilde g$. Now I am interested in understanding the injectivity radius of $(TM,\tilde g)$ but I am confused. I have seen answers that claim that this injectivity radius will be $0$, for example here Injectivity radius of the Sasaki metric, they claim that if the manifold $M$ is not flat then the injectivity radius of $\tilde g$ is zero.

However I have seen papers and books claim that we have an upper bound for the sectional curvatures of the sasaki metric and that it's injectivity radius is bounded away from zero, however no proof is given . For example see page $3$ of https://arxiv.org/pdf/math/0210468.pdf below Proposition $2.2$.

For that reason I am confused, what is it ? Is the injectivity radius positive or not ?

Any enlightment is appreciated, thanks in advance.

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    $\begingroup$ It seems that there is two different notions of Sasaki metrics here. One of them is the one on the tangent bundle mentioned in the referenced MO post, the other is a class of compact manifolds related to Kahler geometry, i.e. a Sasaki manifold is to a Kahler manifold what a contact manifold is to a symplectic manifold. I think this is the reason for the apparent contradiction. $\endgroup$
    – Nick L
    Commented Nov 9, 2021 at 16:50
  • $\begingroup$ And so , just so that I get this straight, the injectivity radius of the natural metric, the sasaki metric, in $TM$ induced from a metric in $M$ will be $0$? @NickL $\endgroup$
    – Someone
    Commented Nov 10, 2021 at 16:24

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