You're going about this backwards
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 fields are useful field that fixes the center. You want to solve for calculating an Jacobi field, given a point (away from the curvature center) and a vector at that point, right?
So that's definitely not vice versaas easy as I thought. Especially for radial geodesics of Here are my thoughts:
1) I think the already proposed surface given by a radially symmetric Riemannian metricspherical cap glued to a pseudosphere is already a good enough question.
You can do what In my experience you want with any radially symmetric never really need a $C^2$ surface, and something with piecewise continuous curvature is almost always enough. There's no need I encourage you to go searching for a "nice" onetry it.
To find the formula
2) As for the Jacobi field of more general approach, I no longer have any easy answer, but here are some thoughts:
Let the surface be given by $(r,\theta) \mapsto X(r,\theta) = (r\cos\theta, r\sin\theta, f(r))$. If $s$ be the arclength parameter along a radial geodesic, just use the definition of a then $s'(r) = \sqrt{1 + f'(r)^2}$. One Jacobi field $J_1(r,\theta)$ is given simply by
$J_1(r,\theta) = \partial X/\partial\theta = re_\theta$, where $e_\theta = (-\sin\theta, \cos\theta, 0)$ is a unit vector field that is orthogonal to and not parallel along any radial geodesic.
If we view $r$ as a function of $s$, then the differential Jacobi equation it happens to satisfy.
In particularsays that $r'' + Kr = 0$, you don't need to find where $K$ is the Gauss curvaturefirst. In fact, if you look at the formula It suffices to solve for the Gauss curvature 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 radially symmetric surface written in polar co-ordinatessolution, you will see that the best way we can try to compute solve for $h$ using variation of parameters.
So the Gauss curvature goal is via to find an even function $f$ with an inflection point such that the Jacobi field of radial geodesicsfunction
$s(r) = \int_0^r \sqrt{1 + f'(t)^2} dt$
can be explicitly integrated and inverted. I suggest trying something like $f(r) = 1/(1+r^2)$.
