I will try to explain my question via a trivial analogy.
Let $a$,$b$ be two points in 2D Euclidean space. I would like to make point $a$ closer and close to $b$. Let $L=||a-b||^2=(a_x-b_x)^2+(a_y-b_y)^2$, where $a_x$ is the x-coordinate of $a$ and so on. Then, minimizing w.r.t $a$ we have
$$L_x=\partial_x ||a-b||^2 = 2(a_x-b_x) $$ $$L_y=\partial_y ||a-b||^2 = 2(a_y-b_y) $$ and overall we can use gradient descent to obtain $$a_x := a_x - \alpha L_x $$ $$a_y := a_y - \alpha L_y $$ Eventually we will obtain $a=b$.
Now on to my question. Assume instead that $a,b$ lie on a manifold that's a a cylinder (without its interior, a cylinder with a fixed radius), i.e. $a=(a_\theta, a_h)$ where $a_\theta$ is the angle and $a_h$ is the height. How can I 'walk' or advance using a gradient-descend-type flow from point $a$ to point $b$? I think the curve should be the geodesic connecting the two points, but I'm not sure how to formulate the scheme correctly.
Thanks.
A curve $c(t)$, parametrised by arc length, on an oriented surface is a geodesic if and only if its acceleration vector $\ddot{c}(t)$ is normal to the surface. See differential geometry of surfaces - Wikipedia.
From there, you'll be able to prove that a curve joining the two points with minimal length is the smallest helix joining the two points.