# Conjugate points

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 $$\gamma$$?

Thanks,

• 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. – 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.