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 $S^1$ beginning at p also end at p? If $M$ is the Möbius band, there exist such points.