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Multi tool use

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Service Entrance Breakers Rain Shield

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Transversality to a family of submanifolds

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1

$begingroup$

Let $E$ be an $n$-dimensional smooth manifold obtained as the total space of a fiber bundle with codimension-$m$ fibers, and let $X subset E$ be a codimension-$k$ submanifold. (More generally, one could take $E$ to be any $n$-manifold with a codimension-$m$ foliation.)

Let $A subset X$ be the set of points $x in X$ where $X$ is tangent to the fiber containing $x$. Is there an appropriate notion of transversality for families of submanifolds that lets us understand $A$? After a small isotopy of $X$, can we assume that $Asubset X$ is a smooth submanifold, and can its (co)dimension be expressed in terms of $k$, $m$, and $n$?

differential-topology smooth-manifolds

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asked Mar 22 at 2:25

KyleKyle

4,04111126

$endgroup$

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1

$begingroup$

Let $E$ be an $n$-dimensional smooth manifold obtained as the total space of a fiber bundle with codimension-$m$ fibers, and let $X subset E$ be a codimension-$k$ submanifold. (More generally, one could take $E$ to be any $n$-manifold with a codimension-$m$ foliation.)

Let $A subset X$ be the set of points $x in X$ where $X$ is tangent to the fiber containing $x$. Is there an appropriate notion of transversality for families of submanifolds that lets us understand $A$? After a small isotopy of $X$, can we assume that $Asubset X$ is a smooth submanifold, and can its (co)dimension be expressed in terms of $k$, $m$, and $n$?

differential-topology smooth-manifolds

share|cite|improve this question

asked Mar 22 at 2:25

KyleKyle

4,04111126

$endgroup$

add a comment | 

$begingroup$

Let $E$ be an $n$-dimensional smooth manifold obtained as the total space of a fiber bundle with codimension-$m$ fibers, and let $X subset E$ be a codimension-$k$ submanifold. (More generally, one could take $E$ to be any $n$-manifold with a codimension-$m$ foliation.)

Let $A subset X$ be the set of points $x in X$ where $X$ is tangent to the fiber containing $x$. Is there an appropriate notion of transversality for families of submanifolds that lets us understand $A$? After a small isotopy of $X$, can we assume that $Asubset X$ is a smooth submanifold, and can its (co)dimension be expressed in terms of $k$, $m$, and $n$?

differential-topology smooth-manifolds

share|cite|improve this question

asked Mar 22 at 2:25

KyleKyle

4,04111126

$endgroup$

share|cite|improve this question

asked Mar 22 at 2:25

KyleKyle

4,04111126

share|cite|improve this question

asked Mar 22 at 2:25

KyleKyle

4,04111126

asked Mar 22 at 2:25

KyleKyle

4,04111126

asked Mar 22 at 2:25

KyleKyle

4,04111126