Jacobi field and the metric I'm reading about Jacobi fields lately, and have noticed some features of it (and it's derivative) with respect to the metric. Thinking about that, I had an non-based, purely intuitive thought that the following feature may holds, but I have a really weak proving skills (I'm just reading related book in my spare time, my last formal education was calculus and some linear algebra 15 years ago). I was hoping to get this clarified. 
Let $(M,g)$ be a Riemannian manifold, and let $\gamma_s(t)=\exp_p(t(v+su))$ where $u,v\in T_pM$ for some $p\in M$. Assume M have a non-positive sectional curvature, show that $g(J(t),J'(t))>0$ , where $g,J$ are the metirc tensor, and the Jacobi field of $\gamma_s$ respectively.
Thanks.

Let $T(t)=\dot{\gamma_0}(t)$ we have 
$$D_Tg(J,J')=g(J',J')+g(J,J'')$$
$$=|J'|^2-g(J,R(J,T)T))$$
$$=|J'|^2-K(J,T)(|J|^2|T|^2-g^2(T,J))\ge 0$$
So $g(J,J')$ is increasing and $g(J(0),J'(0))=g(0,J'(0))=0$.
