Intersection of two plaques is an open subset of both plaques.
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
4217
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add a comment |
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
4217
$endgroup$
add a comment |
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
4217
$endgroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
4217
asked Dec 22 '18 at 6:43
VicVic
4217
edited Dec 23 '18 at 5:58
Vic
asked Dec 22 '18 at 6:43
VicVic
4217
asked Dec 22 '18 at 6:43
VicVic
4217
asked Dec 22 '18 at 6:43
VicVic
4217
4217
add a comment |
add a comment |
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