Suppose $(M,g)$ is a two dimensional Riemannian manifold. Let $\gamma:(-\delta,\delta)\to M$ be a geodesic segment and suppose that $\gamma(0)$ is not conjugate to any other point in $(-\delta,\delta)$. Is it true that there always exists a solution to the Jacobi equation along $\gamma$ that is non-vanishing anywhere along $\gamma$?


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    $\begingroup$ I guess you mean orthogonlal Jacobi field to $\gamma$;otherwise $J=\gamma'$ solves your problem. For orthogonal Jacobi field the answer is no, in the unit sphere you can take $\delta$ arbitrary close to $\pi$ so, you geodesic has length almost $2\cdot\pi$ --- there is no Jacobi equation orthogonal to γ that is non-vanishing anywhere. $\endgroup$ – Anton Petrunin Mar 30 '19 at 1:42

The example in Anton's comment above shows that the result fails if one assumes only that the conjugate radius of each point along $\gamma$ is at least $\delta$. I'll add only that the result is true if the focal radius of each point along $\gamma$ is at least $\delta$ (see here for the definition of the focal radius). In that case, every Jacobi field $J$ along $\gamma$ satisfying $J(0) \neq 0$ and $\nabla_{\gamma'} J(0) = 0$ is nowhere vanishing.

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