riemannian geometry - Jacobi fields on a "bump surface"

ORIGINAL ANSWER DELETED

EDIT: I neglected to account for the need to parameterize by arclength. And I think I also misunderstood and thought that you wanted only the Jacobi field that fixes the center. You want to solve for an Jacobi field, given a point (away from the center) and a vector at that point, right?

So that's definitely not as easy as I thought. Here are my thoughts:

1) I think the already proposed surface given by a spherical cap glued to a pseudosphere is already a good enough question. In my experience you never really need a $C^2$ surface, and something with piecewise continuous curvature is almost always enough. I encourage you to try it.

2) As for the more general approach, I no longer have any easy answer, but here are some thoughts:

Let the surface be given by $(r,theta) mapsto X(r,theta) = (rcostheta, rsintheta, f(r))$. If $s$ be the arclength parameter along a radial geodesic, then $s'(r) = sqrt{1 + f'(r)^2}$. One Jacobi field $J_1(r,theta)$ is given simply by

$J_1(r,theta) = partial X/partialtheta = re_theta$, where $e_theta = (-sintheta, costheta, 0)$ is a unit vector field that is orthogonal to and parallel along any radial geodesic.

If we view $r$ as a function of $s$, then the Jacobi equation says that $r'' + Kr = 0$, where $K$ is the Gauss curvature. It suffices to solve for one more Jacobi field $J_2 = h(s)e_theta$ independent of $J_1$. The Jacobi equation for $J_2$ is given by $h'' + Kh = 0$. Since $r$ is already a solution, we can try to solve for $h$ using variation of parameters.

So the goal is to find an even function $f$ with an inflection point such that the function

$s(r) = int_0^r sqrt{1 + f'(t)^2} dt$

can be explicitly integrated and inverted. I suggest trying something like $f(r) = 1/(1+r^2)$.