Mixing
Smoothness is local
$begingroup$
Let us consider a map $f:Mto R$ where $M$ is a smooth manifold. If every point $pin M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function.
My idea is to prove that any two coordinate charts from any two atlases are smoothly compatible (if $f|_U$ is smoothly than $fcirc varphi^{-1}$ is smoothly for any $varphi$ from the atlas that defines the smooth structure on $U$). Is that ok? If yes, how can I prove that?
Thank you!
smooth-manifolds smooth-functions
asked Jan 22 at 17:18
mipmip
425
$endgroup$
add a comment |
$begingroup$
Let us consider a map $f:Mto R$ where $M$ is a smooth manifold. If every point $pin M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function.
My idea is to prove that any two coordinate charts from any two atlases are smoothly compatible (if $f|_U$ is smoothly than $fcirc varphi^{-1}$ is smoothly for any $varphi$ from the atlas that defines the smooth structure on $U$). Is that ok? If yes, how can I prove that?
Thank you!
smooth-manifolds smooth-functions
asked Jan 22 at 17:18
mipmip
425
$endgroup$
add a comment |
$begingroup$
Let us consider a map $f:Mto R$ where $M$ is a smooth manifold. If every point $pin M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function.
My idea is to prove that any two coordinate charts from any two atlases are smoothly compatible (if $f|_U$ is smoothly than $fcirc varphi^{-1}$ is smoothly for any $varphi$ from the atlas that defines the smooth structure on $U$). Is that ok? If yes, how can I prove that?
Thank you!
smooth-manifolds smooth-functions
asked Jan 22 at 17:18
mipmip
425
$endgroup$
Let us consider a map $f:Mto R$ where $M$ is a smooth manifold. If every point $pin M$ has a neighborhood $U$ such that $f|_U$ is smooth, prove that $f$ is a smooth function.
My idea is to prove that any two coordinate charts from any two atlases are smoothly compatible (if $f|_U$ is smoothly than $fcirc varphi^{-1}$ is smoothly for any $varphi$ from the atlas that defines the smooth structure on $U$). Is that ok? If yes, how can I prove that?
Thank you!
smooth-manifolds smooth-functions
smooth-manifolds smooth-functions
asked Jan 22 at 17:18
mipmip
425
asked Jan 22 at 17:18
mipmip
425
asked Jan 22 at 17:18
mipmip
425
asked Jan 22 at 17:18
mipmip
425
asked Jan 22 at 17:18
mipmip
425
425
add a comment |
add a comment |
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