Problem at proof of Cartan's theorem about the relation between metric and curvature in do Carmo's book

Question

I'm reading DoCarmo's book, Riemannian Geometry and i don't understand something. At page 157, Cartan's theorem.

My question is, why can we take a Jacobi field $J$ in such a way that $J(0)=0$ and $J(l)=v?$

Answer 1

A Jacobi field $J$ with initial conditions $J(0)=0$ and $J‘(0)=w$ along the geodesic $\gamma(t)=exp_p(tv)$ can explicitly be written as \begin{equation} J(t)=(d exp_p)_{tv}(tw). \end{equation} For small enough $t$ the differential of the exponential map at $tv$ is bijective. So for any $\nu \in T_{\gamma(t)}M$ there is some $u \in T_pM$ with \begin{equation} (d exp_p)_{tv}(u)=\nu. \end{equation} Then the Jacobi field with initial condition $J(0)=0$ and $J‘(0)=\frac{u}{t}$ is the one you‘re looking for.