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15

2

$begingroup$

Let $Msubset mathbb{R}^n$ be a compact smooth manifold embedded in $mathbb{R}^n$, we define $$mathfrak{X}(M) := {X: M to mathbb{R}^n; Xmbox{ is smooth and } X(p) in T_p M subset mathbb{R}^n, forall p in M }.$$

Choosing an atlas ${(varphi_i,U_i)}_{i=1}^{n}$, and compacts $K_i subset U_i$, such that $$bigcup_{i=1}^n K_i = M,$$ we define the $|cdot |_r$ norm as

begin{align*}|cdot|_r : mathfrak{X}(M)&to mathbb{R}\
X &to max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_i)}left| text{d}^{j}left( Xcircvarphi_i right) right|right},
end{align*}

then we named $mathfrak{X}^r(M)$ as the complete Banach space $(mathfrak{X}(M),|cdot|_r)$ (it is possible to prove that the topology of $mathfrak{X}^r(M)$ does not depend on the selected atlas).

My Question: Let $X in mathfrak{X}(M)$ and $Y$ be a smooth vector field on $M$ defined just in a compact $K subset M$ such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap K)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Ycircvarphi_i right) right|right}<varepsilon,$$
is it possible extend $Y$ to a vector field $tilde{Y}$ such that

1) $left.tilde{Y}right|_{K} = Y$,

2) $|X-tilde Y|_r < Acdotvarepsilon$ , where $A $ is a constant that depends only on the manifold $K$ ?

The compact $K$ is a connected submanifold with boundary of $M$, such that $dim K = dim M$.

Edit: I changed $|X-tilde Y|_r < varepsilon$ to $|X-tilde Y|_r < Acdotvarepsilon$ after Moishe Kohan's comment.

---

My ideas

First, I extend $Y$ by a smooth vector field $Z$ $in mathfrak{X}(M)$, by the continuity of $Z$, so there exists a neighborhood $U$ of $K$, such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap U)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Zcircvarphi_i right) right|right}<varepsilon,$$

and then choosing a partition of unity $ {phi_1, phi_2}$ subordinate to the cover ${U,Msetminus K}$ we can define
$$tilde{Y} = phi_1 Z + phi_2 X, $$
however I could not guarantee that $|X - tilde{Y}|_r < varepsilon$, because I can not control de derivatives of $phi_1$ and $phi_2$. Does anyone know how should I proceed?

real-analysis analysis differential-geometry manifolds differential-topology

share|cite|improve this question

edited Apr 1 at 15:24

Matheus Manzatto

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is a form of Whitney extension theorem/problem. Take a look here: annals.math.princeton.edu/wp-content/uploads/… I suspect you cannot keep the same $epsilon$: Fefferman's result (and some follow up papers) show that you can find an extension with an error $Aepsilon$ where $A$ is some constant depending only on dimension.
  $endgroup$
  – Moishe Kohan
  Mar 28 at 3:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx for the reference. Once the same $A$ holds for every function. It would be enough to solve my problem. The only complication I am seeing right know it is the fact that this result is for $mathbb{R}^n $ and not manifolds. Do you know how to generalize to manifolds this result?
  $endgroup$
  – Matheus Manzatto
  Mar 28 at 9:13
  

  
  
  
  

add a comment | 

15

2

$begingroup$

Let $Msubset mathbb{R}^n$ be a compact smooth manifold embedded in $mathbb{R}^n$, we define $$mathfrak{X}(M) := {X: M to mathbb{R}^n; Xmbox{ is smooth and } X(p) in T_p M subset mathbb{R}^n, forall p in M }.$$

Choosing an atlas ${(varphi_i,U_i)}_{i=1}^{n}$, and compacts $K_i subset U_i$, such that $$bigcup_{i=1}^n K_i = M,$$ we define the $|cdot |_r$ norm as

begin{align*}|cdot|_r : mathfrak{X}(M)&to mathbb{R}\
X &to max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_i)}left| text{d}^{j}left( Xcircvarphi_i right) right|right},
end{align*}

then we named $mathfrak{X}^r(M)$ as the complete Banach space $(mathfrak{X}(M),|cdot|_r)$ (it is possible to prove that the topology of $mathfrak{X}^r(M)$ does not depend on the selected atlas).

My Question: Let $X in mathfrak{X}(M)$ and $Y$ be a smooth vector field on $M$ defined just in a compact $K subset M$ such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap K)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Ycircvarphi_i right) right|right}<varepsilon,$$
is it possible extend $Y$ to a vector field $tilde{Y}$ such that

1) $left.tilde{Y}right|_{K} = Y$,

2) $|X-tilde Y|_r < Acdotvarepsilon$ , where $A $ is a constant that depends only on the manifold $K$ ?

The compact $K$ is a connected submanifold with boundary of $M$, such that $dim K = dim M$.

Edit: I changed $|X-tilde Y|_r < varepsilon$ to $|X-tilde Y|_r < Acdotvarepsilon$ after Moishe Kohan's comment.

---

My ideas

First, I extend $Y$ by a smooth vector field $Z$ $in mathfrak{X}(M)$, by the continuity of $Z$, so there exists a neighborhood $U$ of $K$, such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap U)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Zcircvarphi_i right) right|right}<varepsilon,$$

and then choosing a partition of unity $ {phi_1, phi_2}$ subordinate to the cover ${U,Msetminus K}$ we can define
$$tilde{Y} = phi_1 Z + phi_2 X, $$
however I could not guarantee that $|X - tilde{Y}|_r < varepsilon$, because I can not control de derivatives of $phi_1$ and $phi_2$. Does anyone know how should I proceed?

real-analysis analysis differential-geometry manifolds differential-topology

share|cite|improve this question

edited Apr 1 at 15:24

Matheus Manzatto

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This is a form of Whitney extension theorem/problem. Take a look here: annals.math.princeton.edu/wp-content/uploads/… I suspect you cannot keep the same $epsilon$: Fefferman's result (and some follow up papers) show that you can find an extension with an error $Aepsilon$ where $A$ is some constant depending only on dimension.
  $endgroup$
  – Moishe Kohan
  Mar 28 at 3:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thx for the reference. Once the same $A$ holds for every function. It would be enough to solve my problem. The only complication I am seeing right know it is the fact that this result is for $mathbb{R}^n $ and not manifolds. Do you know how to generalize to manifolds this result?
  $endgroup$
  – Matheus Manzatto
  Mar 28 at 9:13
  

  
  
  
  

add a comment | 

$begingroup$

Let $Msubset mathbb{R}^n$ be a compact smooth manifold embedded in $mathbb{R}^n$, we define $$mathfrak{X}(M) := {X: M to mathbb{R}^n; Xmbox{ is smooth and } X(p) in T_p M subset mathbb{R}^n, forall p in M }.$$

Choosing an atlas ${(varphi_i,U_i)}_{i=1}^{n}$, and compacts $K_i subset U_i$, such that $$bigcup_{i=1}^n K_i = M,$$ we define the $|cdot |_r$ norm as

begin{align*}|cdot|_r : mathfrak{X}(M)&to mathbb{R}\
X &to max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_i)}left| text{d}^{j}left( Xcircvarphi_i right) right|right},
end{align*}

then we named $mathfrak{X}^r(M)$ as the complete Banach space $(mathfrak{X}(M),|cdot|_r)$ (it is possible to prove that the topology of $mathfrak{X}^r(M)$ does not depend on the selected atlas).

My Question: Let $X in mathfrak{X}(M)$ and $Y$ be a smooth vector field on $M$ defined just in a compact $K subset M$ such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap K)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Ycircvarphi_i right) right|right}<varepsilon,$$
is it possible extend $Y$ to a vector field $tilde{Y}$ such that

1) $left.tilde{Y}right|_{K} = Y$,

2) $|X-tilde Y|_r < Acdotvarepsilon$ , where $A $ is a constant that depends only on the manifold $K$ ?

The compact $K$ is a connected submanifold with boundary of $M$, such that $dim K = dim M$.

Edit: I changed $|X-tilde Y|_r < varepsilon$ to $|X-tilde Y|_r < Acdotvarepsilon$ after Moishe Kohan's comment.

---

My ideas

First, I extend $Y$ by a smooth vector field $Z$ $in mathfrak{X}(M)$, by the continuity of $Z$, so there exists a neighborhood $U$ of $K$, such that
$$max_{substack{iin{1,...,n} \ jin {0,...,r}}}left{sup_{x in varphi^{-1}_i(K_icap U)}left| text{d}^{j}left( Xcircvarphi_i right) - text{d}^{j}left( Zcircvarphi_i right) right|right}<varepsilon,$$

and then choosing a partition of unity $ {phi_1, phi_2}$ subordinate to the cover ${U,Msetminus K}$ we can define
$$tilde{Y} = phi_1 Z + phi_2 X, $$
however I could not guarantee that $|X - tilde{Y}|_r < varepsilon$, because I can not control de derivatives of $phi_1$ and $phi_2$. Does anyone know how should I proceed?

real-analysis analysis differential-geometry manifolds differential-topology

share|cite|improve this question

edited Apr 1 at 15:24

Matheus Manzatto

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

$endgroup$

share|cite|improve this question

edited Apr 1 at 15:24

Matheus Manzatto

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

share|cite|improve this question

edited Apr 1 at 15:24

Matheus Manzatto

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626

asked Mar 23 at 23:45

Matheus ManzattoMatheus Manzatto

1,3091626