There is a classical formula based on Taylor series for the square norm of a Jacobi field $J$ with $J(0) = 0$ along a geodesic on a Riemannian manifold $M$. I am interested on a possible estimate for the square norm of a Jacobi field $J$ such that $J(0) = X\neq 0$. Or simply, it would be nice to understand the relation between $J(0)$ and $J'(0)$. Can we guarantee under any assumption that $\langle J(0),J'(0)\rangle \neq 0$?
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No you can‘t guarantee that, since vor any $v,w \in T_pM$ there exists a Jacobi field $J$ with $J(0)=v$ and $J‘(0)=w$. – Frieder Jäckel Feb 26 '19 at 18:54
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@FriederJäckel, in fact I meant "some assumption" in the sense that "is this true in any case?", not "any" in the sense "every". But now I realized that the Jacobi field is determined by the initial condition, so we can choose $J$ satisfying the desired requirement. – L.F. Cavenaghi Feb 26 '19 at 20:09