Let's work on a Riemannian manifold $M$ of nonpositive sectional curvature.

Fix a unit-speed geodesic $\beta$, and a Jacobi field $\eta$ over it. Assume that $\eta(0)$ is nonzero and orthogonal to $\beta'(0)$, and that $\eta'(0)$ (i.e. $\nabla_{\beta'} \eta (0)$) equals $0$.

Under these conditions, it's known that $\eta(t) \neq 0$ for every $t$ (indeed, the minimum of $\|\eta(t)\|$ is attained at $t=0$). However, I'd like to show that $\eta(t)$ stays nonzero in a constant direction''; more precisely:

QUESTION: Let $\zeta(t)$ be the parallel transport of $\eta(0)$ along $\beta$. Is it true that $\langle \eta(t), \zeta(t) \rangle > 0$ for every $t$?

If the answer is no, then it looks like the Jacobi field ``turns around'' the geodesic. Is it possible? In that case, I ask:

QUESTION 2: What if we additionally assume that $M$ is a symmetric space?