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$begingroup$

Be M a smooth manifold, $N subseteq M$ a submanifold and $ι : N rightarrow M$ the embedding, and be P another smooth manifold. I want to show the following:
(a) If $f:Mrightarrow P$ is smooth ,the restriction $f|_N :Nrightarrow P$ is smooth.
(b) A map $f : P rightarrow N$ is smooth if $ι circ f : P rightarrow M$ is smooth.

For a: If $f:M rightarrow P$ is smooth, there is an open subset $U subseteq M$ with $f_U$ smooth. $Nsubseteq M$ is an embedded manifold, therefore one can choose a neightborhood $U$ such that $U cap N subseteq U$. Hence, there is for every point in $N$ a function $f_U cap N: U cap N rightarrow P$ and because one can chooose an arbitrary large ambient $U$, the function $f_N$ is smooth.

Is this argument correct? And how would I extend it to the second question?

differential-geometry

share|cite|improve this question

edited 14 hours ago

John Smith

asked 15 hours ago

John SmithJohn Smith

919

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here is a more straightforward argument for (a). The embedding $i$ is smooth by definition, and the restriction is just $fcirc i$, which is smooth as the composition of smooth maps.
  $endgroup$
  – Amitai Yuval
  11 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I thought of that argument. But for an exam or smth. would this be enough?
  $endgroup$
  – John Smith
  11 hours ago
  

  
  
  
  

add a comment | 

1

1

$begingroup$

Be M a smooth manifold, $N subseteq M$ a submanifold and $ι : N rightarrow M$ the embedding, and be P another smooth manifold. I want to show the following:
(a) If $f:Mrightarrow P$ is smooth ,the restriction $f|_N :Nrightarrow P$ is smooth.
(b) A map $f : P rightarrow N$ is smooth if $ι circ f : P rightarrow M$ is smooth.

For a: If $f:M rightarrow P$ is smooth, there is an open subset $U subseteq M$ with $f_U$ smooth. $Nsubseteq M$ is an embedded manifold, therefore one can choose a neightborhood $U$ such that $U cap N subseteq U$. Hence, there is for every point in $N$ a function $f_U cap N: U cap N rightarrow P$ and because one can chooose an arbitrary large ambient $U$, the function $f_N$ is smooth.

Is this argument correct? And how would I extend it to the second question?

differential-geometry

share|cite|improve this question

edited 14 hours ago

John Smith

asked 15 hours ago

John SmithJohn Smith

919

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Here is a more straightforward argument for (a). The embedding $i$ is smooth by definition, and the restriction is just $fcirc i$, which is smooth as the composition of smooth maps.
  $endgroup$
  – Amitai Yuval
  11 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, I thought of that argument. But for an exam or smth. would this be enough?
  $endgroup$
  – John Smith
  11 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Be M a smooth manifold, $N subseteq M$ a submanifold and $ι : N rightarrow M$ the embedding, and be P another smooth manifold. I want to show the following:
(a) If $f:Mrightarrow P$ is smooth ,the restriction $f|_N :Nrightarrow P$ is smooth.
(b) A map $f : P rightarrow N$ is smooth if $ι circ f : P rightarrow M$ is smooth.

For a: If $f:M rightarrow P$ is smooth, there is an open subset $U subseteq M$ with $f_U$ smooth. $Nsubseteq M$ is an embedded manifold, therefore one can choose a neightborhood $U$ such that $U cap N subseteq U$. Hence, there is for every point in $N$ a function $f_U cap N: U cap N rightarrow P$ and because one can chooose an arbitrary large ambient $U$, the function $f_N$ is smooth.

Is this argument correct? And how would I extend it to the second question?

differential-geometry

share|cite|improve this question

edited 14 hours ago

John Smith

asked 15 hours ago

John SmithJohn Smith

919

$endgroup$

share|cite|improve this question

edited 14 hours ago

John Smith

asked 15 hours ago

John SmithJohn Smith

919

share|cite|improve this question

edited 14 hours ago

John Smith

asked 15 hours ago

John SmithJohn Smith

919

asked 15 hours ago

John SmithJohn Smith

919

asked 15 hours ago

John SmithJohn Smith

919