If $p: widetildeM to M$ is a covering map and $X$ a vector field on $M$, then exists a vector field satisfying $dp widetildeX = X circ h $

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I'm reading the book "Dynamical Systems on Surfaces - C.Godbillon", and on page 2, he claims the following result:




Let $widetildeM$ and $M$ be smooth manifolds without boundary, $p: widetildeMto M$ a smooth covering map and $X$ a smooth vector field on $M$. Then there exists a uniquely defined smooth vector field $widetildeX$ on $widetildeM$ such that $$textdp(x) widetildeX(x) = X circ p(x) $$




I would like to demonstrate this result (I only need the existence part), but I'm not getting much progress.



Can anyone help me (just a reference is enough)?




differential-topology dynamical-systems smooth-manifolds homotopy-theory covering-spaces




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    Have you thought of using the fact that a covering map is a local diffeomorphism?
    – Tyrone
    19 hours ago














up vote
2
down vote

favorite
1












I'm reading the book "Dynamical Systems on Surfaces - C.Godbillon", and on page 2, he claims the following result:




Let $widetildeM$ and $M$ be smooth manifolds without boundary, $p: widetildeMto M$ a smooth covering map and $X$ a smooth vector field on $M$. Then there exists a uniquely defined smooth vector field $widetildeX$ on $widetildeM$ such that $$textdp(x) widetildeX(x) = X circ p(x) $$




I would like to demonstrate this result (I only need the existence part), but I'm not getting much progress.



Can anyone help me (just a reference is enough)?




differential-topology dynamical-systems smooth-manifolds homotopy-theory covering-spaces




share|cite|improve this question















  • 1




    Have you thought of using the fact that a covering map is a local diffeomorphism?
    – Tyrone
    19 hours ago












up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I'm reading the book "Dynamical Systems on Surfaces - C.Godbillon", and on page 2, he claims the following result:




Let $widetildeM$ and $M$ be smooth manifolds without boundary, $p: widetildeMto M$ a smooth covering map and $X$ a smooth vector field on $M$. Then there exists a uniquely defined smooth vector field $widetildeX$ on $widetildeM$ such that $$textdp(x) widetildeX(x) = X circ p(x) $$




I would like to demonstrate this result (I only need the existence part), but I'm not getting much progress.



Can anyone help me (just a reference is enough)?




differential-topology dynamical-systems smooth-manifolds homotopy-theory covering-spaces




share|cite|improve this question











I'm reading the book "Dynamical Systems on Surfaces - C.Godbillon", and on page 2, he claims the following result:




Let $widetildeM$ and $M$ be smooth manifolds without boundary, $p: widetildeMto M$ a smooth covering map and $X$ a smooth vector field on $M$. Then there exists a uniquely defined smooth vector field $widetildeX$ on $widetildeM$ such that $$textdp(x) widetildeX(x) = X circ p(x) $$




I would like to demonstrate this result (I only need the existence part), but I'm not getting much progress.



Can anyone help me (just a reference is enough)?





differential-topology dynamical-systems smooth-manifolds homotopy-theory covering-spaces





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









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Matheus Manzatto

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  • 1




    Have you thought of using the fact that a covering map is a local diffeomorphism?
    – Tyrone
    19 hours ago












  • 1




    Have you thought of using the fact that a covering map is a local diffeomorphism?
    – Tyrone
    19 hours ago







1




1




Have you thought of using the fact that a covering map is a local diffeomorphism?
– Tyrone
19 hours ago




Have you thought of using the fact that a covering map is a local diffeomorphism?
– Tyrone
19 hours ago















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