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YCor
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For a fiber bundle $f:M\to S^1$, is there a point whose Horizontalhorizontal lift is a circle?

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 Horizontalhorizontal lift of $S^1$ beiginningbeginning at p also end at p? If $M$ is the MobiusMöbius band, there exist such points.

For a fiber bundle $f:M\to S^1$, is there a point whose Horizontal lift is a circle?

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.

For a fiber bundle $f:M\to S^1$, is there a point whose horizontal lift is a circle?

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.

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For a fiber bundle $f:M\to S^1$, is there a point whose Horizontal lift is a circle?

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.