Let $(Q,g)$ be a (compact) Riemannian manifold with injectivity radius $\rho>0$. There is a natural metric $\tilde g$ on the tangent bundle $TQ$ which is known as the Sasaki metric and which makes $\pi:TQ\rightarrow Q$ a Riemannian submersion. Denote its injectivity radius by $\tilde\rho$. Obviously $\tilde\rho\leq\rho$ holds, since the zero section is totally geodesic in $TQ$. But is something known about lower bounds? For example, is it true that $\tilde\rho>0$ or even $\tilde\rho=\rho$?

5$\begingroup$ Maybe you know this already: if $(Q, g)$ is not flat, then the sectional curvatures of the Sasaki metric on $TQ$ are both unbounded below and unbounded above. (See eg Propositions 7.67.8 in GudmundssonKappos "On the geometry of tangent bundles" ams.org/mathscinetgetitem?mr=1888866 .) To me, the latter makes it seem unlikely that $\tilde\rho>0$. $\endgroup$ – macbeth Apr 17 '12 at 22:30

$\begingroup$ Yes I knew this, but I was not sure about consequences on the injectivity radius. Thanks anyway! $\endgroup$ – Dawidow Apr 18 '12 at 7:02
If the manifold is not flat then $\bar \rho=0$.
It is sufficient to show that given $\epsilon>0$ there are two tangent vectors $v,w\in T_pQ$ such that $vw=\epsilon$, but the minimizing geodesic does not lie in $T_pQ$.
We assume that curvature at $p$ does not vanish. Consider a loop $\gamma$ based at $p$ with length $\delta<\epsilon$ and nontrivial integral curvature $R$. Choose generic $v$, so $w=R v\ne v$. We can assume that $vw=\epsilon$. A horizontal lift of $\gamma$ connects $v$ to $w$ and has length $\delta$.