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 $\tilde\rho=\rho$?
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 $|v-w|=\epsilon$, 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 $|v-w|=\epsilon$. A horizontal lift of $\gamma$ connects $v$ to $w$ and has length $\delta$.