Let $M$ a Riemannan manifold with non-positive seccional curvature. Prove that
$|(d \exp_{p})_{v} w|\geq |w|$, for all $p \in M$, all $v \in T_{p}M$ and all $w \in T_{v}(T_{p}M)$.
I imagine that this problem is a direct application of the Rauch comparison theorem. But, I can not to realize all assumptions. Let $\gamma:[0,a]\to \mathbb{R}^{n}$ a unit geodesic $\gamma(t)=tv$, but the sectional curvature of Euclidean space is $0$, thus a Jacobi field along $\gamma $ is $J(t)=tw(t)$, where $<w(t),\gamma(t)>=0$ and $|w(t)|=1$, observe that $w$ is a parallel vector field along $\gamma$, which not depend on $t$, since is a parallel transport on $\mathbb{R}^{n}$. Now, my question is: it is possible create a unit geodesic $\bar{\gamma}$ on $M$ with a Jacobi field $\bar{J}$ s.t, $\bar{J}'(0)=w$.? If is possible, the Jacobi field on $M$,$\bar{J}=(d\exp_p)_{tv}(tw)$ is greater than (length)whic the Jacobi Field on $\mathbb{R}^{n}$, $J(t)=tw$. If t=1, the problem is over.
Any tips? Thanks