Parallel Transport on a Hyperbola

Question

Disclaimer: I'm very new to fancy TeX pictures..as you can see by my reputation 1 on this exchange.

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I recently came across this post, describing how to the draw parallel transport of a vector on a sphere. For a paper I'm writing, I'd like to do the same on the upper-sheet of a hyperboloid, but I have absolutely no idea how to do this beyond this post showing how to draw a hyperboloid in LaTeX.

In short, how can I code up this picture

where:

• The blue thing is supposed to be a hyperboloid of one sheet
• The red squares are supposed to be parts of tangent spaces at two points in the hyperboloid
• The green arrows are supposed to be orthogonal vectors
• The brown line is supposed to be a curve travelling along the hyperboloid

So far I have the following (terrible) attempt...

\begin{asy}
import graph3;
unitsize(1cm);
% Setup
settings.render=0;
settings.prc=false;
size3(25);
% Hyperboloid
triple f(pair t){return (sinh(t.y)cos(t.x)/2,sinh(t.y)sin(t.x)/2, cosh(t.y));}
surface s=surface(f,(0,-2) ,(2*pi,2),nu=40,nv=40,Spline);
draw(s,green+opacity(0.9),render(merge=true));
% Path
path3 circX=arc(sinh(t.y),sinh(t.y),4,normal=x);
\end{asy}