Let $M$ be a complete Riemannian manifold, $x \in M$ a point, and $\gamma_s(t)$ be a family of geodesics starting at $x$ at time $t=0$, $s \in [-\epsilon,\epsilon]$. The Jacobi field $J(s,t)$ of this family satisfies $J(s,0) = 0$. Moreover, the Taylor expansion of the length in $t$ looks like
$$ J(0,t) = 0 + t^2 |J'(t)|^2 + t^4 R + O(t^6) $$
where $R$ denotes the sectional curvature of the plan spanned by the family at the initial point. Intuitively, this suggests in particular that negative curvature implies that the geodesics' spreading out. However, this is just a local result. Plus, $J$ isn't analytic in general, so the Taylor estimate might not hold on the nose even in any small neighborhood.
Thus my question: is there a compact Riemannian manifold with non-positive curvature everywhere such that a pair of geodesics starting from the same point with different initial direction meets up somewhere else?
EDIT Yes there are, given in the comment below: a surface of genus $g \geq 2$ has many closed geodesics, which serve as examples I want.
A new question: what about those for $M$ which is diffeomorphic to $\mathbb{R}^n$?