Let $M$ be (path) connected two manifold. Then is it true that for two points $p\neq q$, there are at least two disjoint (except at endpoints) paths between $p$ and $q$?
My attempt: There should be at least one path, and if we can cover the path with regular coordinate balls. Then we "wiggle" the path to make another one. Is this a plausible idea?
Also, if the statement is true, is it true that the higher dimensional manifolds share the same property?
Basically, the existence of a piecewise-linear path between $p$ and $q$ is deduced from the connectivity of the manifold (the set of points connected with $p$ by a path is nonempty, open and closed).
At each point $p$ on that path, you can pick a chart $\phi: U \rightarrow M$ such that a point $P$ of $U$ is sent to $p$ and a ball $B_r (P)$ is sent to a neighborhood of $p$, and a piecewise-linear path in $U$ is sent to the original path. Now you take finitely many that covers the entire path (invoking compactness of the path), and work inside each such Euclidean neighborhood. Here the restriction is that the dimension shall be greater than 1. That case, A ball minus a piecewise linear segment is still connected and path-connected.