Dgdrxrt

Approximately how much travel time was saved by the opening of the Suez Canal in 1869?

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

What defenses are there against being summoned by the Gate spell?

Why doesn't a class having private constructor prevent inheriting from this class? How to control which classes can inherit from a certain base?

How much RAM could one put in a typical 80386 setup?

What's that red-plus icon near a text?

Is it unprofessional to ask if a job posting on GlassDoor is real?

How does one intimidate enemies without having the capacity for violence?

Two films in a tank, only one comes out with a development error – why?

Has there ever been an airliner design involving reducing generator load by installing solar panels?

Rock identification in KY

Convert two switches to a dual stack, and add outlet - possible here?

Roll the carpet

Why do I get two different answers for this counting problem?

Today is the Center

Alternative to sending password over mail?

how to check a propriety using r studio

What typically incentivizes a professor to change jobs to a lower ranking university?

How to move a thin line with the black arrow in Illustrator?

Could an aircraft fly or hover using only jets of compressed air?

Are the number of citations and number of published articles the most important criteria for a tenure promotion?

Does object always see its latest internal state irrespective of thread?

Was any UN Security Council vote triple-vetoed?

Which country benefited the most from UN Security Council vetoes?

About prove of converse of Frobenius theorem in manifolds

Visualizing Frobenius TheoremIntegrable ManifoldsFrobenius Condition for Singular Integrable DistributionsHolomorphic Frobenius TheoremAre all smooth manifolds the zero locus of a smooth function?existence of integral of one form.Definition of a distribution and integral manifoldsImmersion of the manifold in the Flowout TheoremSmooth distribution: first examples.trouble understanding Moser theorem

1

$begingroup$

Theorem

$M$:a smooth manifold
$D$:$c$-dim, smooth distribution on M

If for all $m$ in $M$, there exists a integral manifold of $D$ which includes $m$, $D$ is involutive.

Proof(?)
Let $X$ and $Y$ are smooth vector field on $M$ lying in $D$. Fix $m$ in $M$ and $(N,Ψ)$:Integral manifold of $D$ at $m$. There exist vector field $Z$ and $W$ on $N$ such that $Z$ is $Ψ$-related with $X$ and $W$ is so on with $Y$. If $Z$ and $W$ is smooth, $[X,Y]$ and $[Z,W]$ are $Ψ$-related
. Because $(N,Ψ)$ is integral manifold, $[X,Y]_m$ in $D(m)$.

But, I don’t understand why $Z$ and $W$ are smooth. Please tell me the reason.

differential-geometry

share|cite|improve this question

edited Mar 29 at 14:13

稲垣真郷

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

$endgroup$

add a comment | 

1

$begingroup$

Theorem

$M$:a smooth manifold
$D$:$c$-dim, smooth distribution on M

If for all $m$ in $M$, there exists a integral manifold of $D$ which includes $m$, $D$ is involutive.

Proof(?)
Let $X$ and $Y$ are smooth vector field on $M$ lying in $D$. Fix $m$ in $M$ and $(N,Ψ)$:Integral manifold of $D$ at $m$. There exist vector field $Z$ and $W$ on $N$ such that $Z$ is $Ψ$-related with $X$ and $W$ is so on with $Y$. If $Z$ and $W$ is smooth, $[X,Y]$ and $[Z,W]$ are $Ψ$-related
. Because $(N,Ψ)$ is integral manifold, $[X,Y]_m$ in $D(m)$.

But, I don’t understand why $Z$ and $W$ are smooth. Please tell me the reason.

differential-geometry

share|cite|improve this question

edited Mar 29 at 14:13

稲垣真郷

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

$endgroup$

add a comment | 

$begingroup$

Theorem

$M$:a smooth manifold
$D$:$c$-dim, smooth distribution on M

If for all $m$ in $M$, there exists a integral manifold of $D$ which includes $m$, $D$ is involutive.

Proof(?)
Let $X$ and $Y$ are smooth vector field on $M$ lying in $D$. Fix $m$ in $M$ and $(N,Ψ)$:Integral manifold of $D$ at $m$. There exist vector field $Z$ and $W$ on $N$ such that $Z$ is $Ψ$-related with $X$ and $W$ is so on with $Y$. If $Z$ and $W$ is smooth, $[X,Y]$ and $[Z,W]$ are $Ψ$-related
. Because $(N,Ψ)$ is integral manifold, $[X,Y]_m$ in $D(m)$.

But, I don’t understand why $Z$ and $W$ are smooth. Please tell me the reason.

differential-geometry

share|cite|improve this question

edited Mar 29 at 14:13

稲垣真郷

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

$endgroup$

share|cite|improve this question

edited Mar 29 at 14:13

稲垣真郷

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

share|cite|improve this question

edited Mar 29 at 14:13

稲垣真郷

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62

asked Mar 29 at 10:03

稲垣真郷稲垣真郷

62