Vtyjkyuk

$M$ a complete Riemannian manifold with nonpositive sectional curvature. Show that $|d(exp_p)_v(w)| geq |w|$ for all $v in T_p M$, and all $w in T_v(T_p M)$.

Update:

Based on the hints I've gotten, here is my attempted solution.

We will compare $M$ with $tildeM=T_p M approx T_v T_p M$ via the Rauch comparison theorem. Let $gamma:[0,1] rightarrow M$ be a geodesic and let $tildegamma: [0,1] rightarrow tildeM$ be a comparison geodesic (which means that $gamma$ and $tildegamma$ have the same speed). Note that $tildegamma$ also has no conjugate points because curvature zero. Write $gamma(0)=p$ and $gamma'(0)=v$.

Let $J(t)$ be the geodesic along $gamma$ with $J(0)=0$ and $J'(0)=w$; then $J(t)$ can be written

$J(t)=d(exp_p)_tgamma'(0)(tJ'(0))$

so we see that $J(1)=d(exp_p)tv(tw)$.

The Rauch comparison theorem gives us that, for a Jacobi field $tildeJ$ along $tildegamma$ with

1. $tildeJ(0)=J(0)=0$



2. $|tildeJ'(0)|=|J'(0)|$




3. $langle tildeJ'(0),gamma'(0) rangle = langle J'(0), gamma'(0) rangle$

we have $|tildeJ| leq |J|$, so if we had a Jacobi field $tildeJ$ along $tildegamma$ with $tildeJ(1)=w$ we'd be done.

There is a theorem that says if there is no conjugate points along $tildegamma$ then there is a unique Jacobi field $tildeJ$ along $tildegamma$ with $tildeJ(0)=0$ and $tildeJ(1)=w$ but I am unsure how how to get that the highlighted conditions (2) and (3) above hold.