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$M$ compact, $0$ is a regular value of $f:Mtomathbb R,$ show that $f^-1(0)$ is diffeomorphic to $f^-1(varepsilon)$ for small $varepsilon.$

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$begingroup$

The exact statement of the problem is:

If $M$ is compact and $0$ is a regular value of $f:Mtomathbb R,$ then there is a neighborhood $U$ of $0inmathbb R$ such that $f^-1(U)$ is diffeomorphic to $f^-1(0)times U$ by a diffeomorphism $phi:f^-1(0)times Uto f^-1(U)$ with $f(phi(p,t))=t.$

I want to find a vector field $X$ on a neighborhood of $f^-1(0)$ which can be pushed forward to $d/dt$ on $mathbb R.$ From there it seems like I could then find a coordinate neighborhood $x$ about each point in $f^-1(0)$ such that in this neighborhood $X$ looks like $partial/partial x^1,$ use a compactness argument, and somehow get the proof out of this. However, I don't know how to go about defining such a vector field $X.$ I'm also wondering whether this is true when $M$ is not compact, but I don't exactly know how I would go about proving this.

Even just a hint would be helpful.

differential-topology smooth-manifolds vector-fields smooth-functions

share|cite|improve this question

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Choose a Riemannian metric on $M$ and use the gradient field.
  $endgroup$
  – Neal
  Mar 12 at 19:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Neal I don't have any results about Riemannian metrics. A solution using vector fields is what I think I'm supposed to be doing.
  $endgroup$
  – D. Brogan
  Mar 12 at 22:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can always get a Riemannian metric on $M$ with a partition of unity. If you're not comfortable with that, take the gradient of $f$ on each open set in an oriented submanifold atlas of $f^-1(0)$ and use compactness + a partition of unity.
  $endgroup$
  – Neal
  Mar 12 at 22:45
  
  

  
  
  
  

add a comment | 

0

$begingroup$

The exact statement of the problem is:

If $M$ is compact and $0$ is a regular value of $f:Mtomathbb R,$ then there is a neighborhood $U$ of $0inmathbb R$ such that $f^-1(U)$ is diffeomorphic to $f^-1(0)times U$ by a diffeomorphism $phi:f^-1(0)times Uto f^-1(U)$ with $f(phi(p,t))=t.$

I want to find a vector field $X$ on a neighborhood of $f^-1(0)$ which can be pushed forward to $d/dt$ on $mathbb R.$ From there it seems like I could then find a coordinate neighborhood $x$ about each point in $f^-1(0)$ such that in this neighborhood $X$ looks like $partial/partial x^1,$ use a compactness argument, and somehow get the proof out of this. However, I don't know how to go about defining such a vector field $X.$ I'm also wondering whether this is true when $M$ is not compact, but I don't exactly know how I would go about proving this.

Even just a hint would be helpful.

differential-topology smooth-manifolds vector-fields smooth-functions

share|cite|improve this question

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

$endgroup$

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  Choose a Riemannian metric on $M$ and use the gradient field.
  $endgroup$
  – Neal
  Mar 12 at 19:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Neal I don't have any results about Riemannian metrics. A solution using vector fields is what I think I'm supposed to be doing.
  $endgroup$
  – D. Brogan
  Mar 12 at 22:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can always get a Riemannian metric on $M$ with a partition of unity. If you're not comfortable with that, take the gradient of $f$ on each open set in an oriented submanifold atlas of $f^-1(0)$ and use compactness + a partition of unity.
  $endgroup$
  – Neal
  Mar 12 at 22:45
  
  

  
  
  
  

add a comment | 

$begingroup$

The exact statement of the problem is:

If $M$ is compact and $0$ is a regular value of $f:Mtomathbb R,$ then there is a neighborhood $U$ of $0inmathbb R$ such that $f^-1(U)$ is diffeomorphic to $f^-1(0)times U$ by a diffeomorphism $phi:f^-1(0)times Uto f^-1(U)$ with $f(phi(p,t))=t.$

I want to find a vector field $X$ on a neighborhood of $f^-1(0)$ which can be pushed forward to $d/dt$ on $mathbb R.$ From there it seems like I could then find a coordinate neighborhood $x$ about each point in $f^-1(0)$ such that in this neighborhood $X$ looks like $partial/partial x^1,$ use a compactness argument, and somehow get the proof out of this. However, I don't know how to go about defining such a vector field $X.$ I'm also wondering whether this is true when $M$ is not compact, but I don't exactly know how I would go about proving this.

Even just a hint would be helpful.

differential-topology smooth-manifolds vector-fields smooth-functions

share|cite|improve this question

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

$endgroup$

share|cite|improve this question

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

share|cite|improve this question

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

asked Mar 12 at 19:42

D. BroganD. Brogan

643513

asked Mar 12 at 19:42

D. BroganD. Brogan

643513