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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2$\begingroup$ Every such bundle is a mapping torus of a self-diffeomorphisms of $f^{-1}(x)$. The horizontal lift you ask for exists if and only if this diffeomorphism has a fixed point. For $f^{-1}(x)=(-1,1)$ the Mobius bundle is induced by $t\mapsto -t$ which indeed has a fixed point. $\endgroup$– archipelagoCommented Sep 23, 2020 at 12:34
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$\begingroup$ @archipelago: For Mobius band, this surely holds. I ask the general case. $\endgroup$– mathmetricgeometryCommented Sep 23, 2020 at 12:42
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1$\begingroup$ @mathmetricgeometry: archipelago has explained the general case: a mapping torus. $\endgroup$– Ben McKayCommented Sep 23, 2020 at 12:47
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1$\begingroup$ @archipelago: Thank you! Just consider the cyliner $f:S^1\times (-\infty,+\infty)\to S^1$, the diffeomorphisms of $f^{-1}(x)$ is $(0,t)\to (1,t+1)$. Then the Horizontal lifting lines are spiral rising lines. $\endgroup$– mathmetricgeometryCommented Sep 23, 2020 at 13:07
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$\begingroup$ @archipelago: I think the above cylinder is not a counter example. Their Horizontal lifting are circles. Klein bottle to $S^1$ is a counter example, the fibers are $S^1$, the diff is reflexion, no fixed point. Need Horizontal lifting of base $S^1$ twice, then it's a closed curve. $\endgroup$– mathmetricgeometryCommented Sep 23, 2020 at 15:56
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1 Answer
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The 2-sheeted cover of the circle by itself is a counterexample. By "the" horizontal lift, do you mean "a" horiontal lift? If so, and if the fibres are connected, then the answer is positive.
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$\begingroup$ The second assertion doesn't seem to be true: the Klein bottle (viewed as the non-orientable $S^1$-bundle over $S^1$) does not have a section. $\endgroup$ Commented Sep 25, 2020 at 7:11
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1$\begingroup$ I cannot agree, Marco. It does have a section (which is weaker than being a product). $\endgroup$ Commented Sep 26, 2020 at 12:31
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$\begingroup$ Oops. Non-orientable things are confusing. You're right. $\endgroup$ Commented Sep 28, 2020 at 7:27