Bound of $D\exp$ in Riemannian manifold with $|\text{sec} M| \leq K$ and $\text{inj}M \geq i_0$

Solution 1:

This seems wrong. On the unit $n$-sphere, we have $K=1$ and $i_0=\pi$. Whatever $f(n,1,\pi)$ is, $\exp(f(n,1,\pi))$ is positive. But $D\exp^{−1}_p$ becomes singular as you approach the point antipodal to $p$, so there's no positive upper bound on $\|D\exp^{−1}_p\|$.

One way to see why that's true is by using Jacobi fields. Choose two orthonormal tangent vectors $v,w$ at $p$. There's a Jacobi field $J$ along the geodesic $\gamma_v$ that satisfies $J(0)=0$ and $D_tJ(0)=w$ (e.g., see my Intro to Riemannian Manifolds, Prop. 10.10). Then $D(\exp_p)|_{\pi v}(w) = J(\pi)=0$ (see the proof of Prop. 10.20 in IRM). If we let $x_t = D(\exp_p)|_{tv}(w)$, this means that as $t\nearrow\pi$, $\|x_t\|\to 0$, and therefore $\|D(\exp_p^{-1})|_{\exp_p(tv)}(x_t)\|/\|x_t\|= \|w\|/\|x_t\| \to \infty$, and thus the operator norm of $D(\exp_p^{-1})$ increases without bound. Any other norm is equivalent to this one.