Bounds on the norm $|J'|$, where $J$ is a Jacobi field

Question

I am trying to understand the well known formula for the norm of Jacobi fields $J$ along a geodesic $\gamma(t)$ with $J(0)=0$, i.e. $$ J''(t)+ R(J,\gamma'),\gamma'=0. $$ The formula $$ (\ast) \quad g(J(t),J(t)) = O(t^2) $$ holds on a Riemannian manifold $(M,g)$ (see e.g. here). In this link the Jacobi equations are used to compute the first two derivatives in $t=0$ and then use a Taylor expansion to argue that the remainder is of higher order. But what I do not understand in this proof is how to make sense of the remainder because it involves $J(t)$ again: $$ (g(J(t),J(t)))''= 2(g(J''(t),J(t))+ g(J'(t),J'(t)). $$ So a Taylor expansion yields for some $s_t \in [0,t]$ $$ g(J(t),J(t))= 0+ t\cdot 0 + t^2 \cdot (g( R(J,\gamma'(s_t)),\gamma'(s_t),J(s_t)))+ g(J'(s_t),J'(s_t)). $$ How do we estimate the term $g(J'(s_t),J'(s_t))$? For example, how do rule out that $$ g(J'(s_t),J'(s_t)) \geq \frac{1}{t}? $$

Answer 1

We use the following trick for second order ODE: Let $f(t) = |J(t)|^2 + |J'(t)|^2$. Then

\begin{align*} f' &= 2g(J, J') + 2 g(J' , J'') \\ &= 2g(J, J') + 2 g(J' , R(J, \gamma')\gamma')) \\ &\le 2|J| |J'| + 2C |J| |J'| \\ &\le (C+1)( |J|^2 + |J'|^2) = (C+1)f. \end{align*}

Here $C$ is the bound on $|R|$. Thus we have the estimates

$$ f(t) \le f(0) e^{(C+1)t}$$

and this implies

$$ |J'(t) |^2 \le f(t) \le f(0) e^{(C+1)t} = |J'(0)|^2 e^{(C+1)t}$$ since $|J(0)| = 0$.