Consider two people living on two different smooth 1-manifolds $S$ and $T$ as shown in figure 1. The manifold $S$ is a bump function joining the points $A$ and $B$ and the manifold $T$ is formed by joining two identical bump functions at the points $P$ and $Q$. Let the points marked as $H$ denote the homes of the two guys $X$ and $Y$ living on the manifolds $S$ and $T$ respectively with distances (along the curve) from $H$ and $B$ (and $H$ and $Q$) being 40 km.
Let $X$ and $Y$ are equipped with cars which do not have a reverse gear. Every time $X$ starts on a drive from his home $H$ it is certain that he reaches back home after a 80Km drive. But same is not the case with $Y$. It is not sure how much long a drive does he need to return back to his home or will he even ever return back to his home.
My question is what kind of a manifolds are $S$ and $T$ and how are they related. Is there any paradox here which needs to be resolved. Life of $Y$ can be made more complicated by joining the points $P$ and $Q$ with some more bump functions each lying on different planes.