I've become interested in finding geodesic on the intersection of riemann manifolds however it turns out much of the literature is way above my head so I'm looking into simpler cases.
Suppose we have two distinct surfaces $U$ and $V$ in $R^3$ that intersect. Since the surfaces are distinct and intersect the intersection must be a line.
If $A$ is a point on $U$ and $B$ is a point on $V$ how do you find the shortest path(s) between the points?
My inclination is that it boils down to a boundary value problem in the calculus of variations and finding the geodesic subject to the constraint that the end point is on the line of intersection but I cannot find a way to set the problem in general.
I would love help setting up the general problem or links to papers or articles that talk about this. I have a feeling that there isn't any nice solution but I'm hoping that there are some simplifications.
EDIT:
I realized that the geodesic on a linear surface is just a straight line so this is just a matter of minimizing the distance between $A$ and the line of intersection plus the distance between $B$ and the line of intersection.