9
$\begingroup$

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 $\tilde\rho=\rho$?

$\endgroup$
2
  • 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 7.6-7.8 in Gudmundsson-Kappos "On the geometry of tangent bundles" ams.org/mathscinet-getitem?mr=1888866 .) To me, the latter makes it seem unlikely that $\tilde\rho>0$. $\endgroup$
    – macbeth
    Apr 17, 2012 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, 2012 at 7:02

1 Answer 1

6
$\begingroup$

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 $\delta$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.