I am studying Jacobi fields on Riemannian manifolds, and I am trying to construct a simple example of a point with a "non-trivial" conjugate point (one with multiplicity $m \in \{ 1, \ldots , n-2 \}$, where multiplicity measures the maximum number of linearly independent Jacobi fields satisfying $J(0)=0=J(t_0)$) to get a feel for what it would mean for a conjugate point to have non-trivial multiplicity.
It seems trivial to me to see that you can't construct such an example for a $2$-manifold, which implies that all conjugate points on $2$-manifolds have maximum multiplicity, meaning that conjugate points for $2$-manifolds are where nearby geodesics bump into one another. Please correct me if my reasoning here is faulty.
I'm trying to imagine/reason what might happen for a $3$-manifold. If the multiplicity of a particular conjugate point is $1$, then does this mean that geodesics will converge to a particular submanifold? I am finding it hard to get a geometric interpretation so any input on this would be helpful.