I am working out some math involving integral curves of a gradient field on a smooth, connected manifold with boundary. Let's assume the function that provides the gradient field is smooth or even analytic and has no singularities or critical points. So far I assumed that such integral curves would always start and end somehow on the boundary or be circular.
A colleague pointed out that these curves are not guaranteed to exists unless the manifold is complete (which cannot be assumed as it has a boundary). The issue he points out is that integral curves are only defined locally and it might not be possible to connect the local solution domains if their radius is not bounded from below. However, he didn't know more details on this issue, so I am now left with the task to find out if there is an actual issue.
I think what he describes is the situation that the integral curves would be connected but not be path-connected. This can maybe happen if the manifold was something like the graph of sin(1/x). However, that's not a manifold: In connected manifolds are path connected I found that every connected manifold is also path connected and in https://en.wikipedia.org/wiki/Connected_space#Path_connectedness it is stated that the graph of sin(1/x) (on (0, 1]) is connected but not path-connected.
My question: Can it happen on a connected manifold that integral curves of a vector field are not path-connected? If so, what regularity requirements from the vector field and the manifold are needed to justify the assumption that integral curves are path-connected? Does it make a difference if the vector field is a gradient field of a function? In that case, what regularity requirements are needed on the function? (obviously it must be differentiable to yield a gradient field, but what else?)
I read through these lecture notes: http://math.stanford.edu/~conrad/diffgeomPage/handouts/intcurve.pdf On page 18 it is stated: “we can “glue” all such integral curves; on the union of their open interval domains we obviously get the unique maximal integral curve of the desired sort”. I am not sure if I can conclude an answer to one of my questions from that. On the other hand I think if there was a potential pitfall with path-connectedness that would have been discussed in such notes.