Let $f: M\to S^1$ be a Riemannian submersion, and also a fiber bundle. $M$ may be non-compact, possibly with boundary. For $x\in S^1$, consider the fiber $f^{-1}(x)\subset M$. Is there a point $p\in f^{-1}(x)$, such that the Horizontal lift of $S^1$ beiginning at p also end at p? If $M$ is the Mobius band, there exist such points.