Differentiable structure on product of manifolds to yield inclusion maps as imbeddings
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I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows:
"Let $M$ and $N$ have class $C^r$; let $M$ be non-bounded. Construct a $C^r$ differentiable structure on $Mtimes N$ such that the natural inclusions of $M$ and $N$ into $Mtimes N$ are imbeddings. Why do we require M to be non-bounded?"
My initial effort was to define the differentiable structure of $Mtimes N$, denoted $mathcal{D}_{Mtimes N}$, to be the crossings of the neighborhoods of $M$ and $N$ and then the concatenation of their homeomorphisms i.e. ${(Utimes V, (h,g)) | (U,h)in mathcal{D}_M, (V,g)in mathcal{D}_N}.$ This I believe gets you that the inclusion maps are imbeddings. But I do not understand where I need that M is non-bounded. Any help with this is greatly appreciated.
differential-topology product-space
asked Jan 10 at 20:58
mnewmanmnewman
265
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add a comment |
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I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows:
"Let $M$ and $N$ have class $C^r$; let $M$ be non-bounded. Construct a $C^r$ differentiable structure on $Mtimes N$ such that the natural inclusions of $M$ and $N$ into $Mtimes N$ are imbeddings. Why do we require M to be non-bounded?"
My initial effort was to define the differentiable structure of $Mtimes N$, denoted $mathcal{D}_{Mtimes N}$, to be the crossings of the neighborhoods of $M$ and $N$ and then the concatenation of their homeomorphisms i.e. ${(Utimes V, (h,g)) | (U,h)in mathcal{D}_M, (V,g)in mathcal{D}_N}.$ This I believe gets you that the inclusion maps are imbeddings. But I do not understand where I need that M is non-bounded. Any help with this is greatly appreciated.
differential-topology product-space
asked Jan 10 at 20:58
mnewmanmnewman
265
$endgroup$
$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14
add a comment |
$begingroup$
I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows:
"Let $M$ and $N$ have class $C^r$; let $M$ be non-bounded. Construct a $C^r$ differentiable structure on $Mtimes N$ such that the natural inclusions of $M$ and $N$ into $Mtimes N$ are imbeddings. Why do we require M to be non-bounded?"
My initial effort was to define the differentiable structure of $Mtimes N$, denoted $mathcal{D}_{Mtimes N}$, to be the crossings of the neighborhoods of $M$ and $N$ and then the concatenation of their homeomorphisms i.e. ${(Utimes V, (h,g)) | (U,h)in mathcal{D}_M, (V,g)in mathcal{D}_N}.$ This I believe gets you that the inclusion maps are imbeddings. But I do not understand where I need that M is non-bounded. Any help with this is greatly appreciated.
differential-topology product-space
asked Jan 10 at 20:58
mnewmanmnewman
265
$endgroup$
I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows:
"Let $M$ and $N$ have class $C^r$; let $M$ be non-bounded. Construct a $C^r$ differentiable structure on $Mtimes N$ such that the natural inclusions of $M$ and $N$ into $Mtimes N$ are imbeddings. Why do we require M to be non-bounded?"
My initial effort was to define the differentiable structure of $Mtimes N$, denoted $mathcal{D}_{Mtimes N}$, to be the crossings of the neighborhoods of $M$ and $N$ and then the concatenation of their homeomorphisms i.e. ${(Utimes V, (h,g)) | (U,h)in mathcal{D}_M, (V,g)in mathcal{D}_N}.$ This I believe gets you that the inclusion maps are imbeddings. But I do not understand where I need that M is non-bounded. Any help with this is greatly appreciated.
differential-topology product-space
differential-topology product-space
asked Jan 10 at 20:58
mnewmanmnewman
265
asked Jan 10 at 20:58
mnewmanmnewman
265
asked Jan 10 at 20:58
mnewmanmnewman
265
asked Jan 10 at 20:58
mnewmanmnewman
265
asked Jan 10 at 20:58
mnewmanmnewman
265
265
$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14
add a comment |
$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14
$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14
$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14
add a comment |
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$begingroup$
To avoid corners.
$endgroup$
– Lord Shark the Unknown
Jan 10 at 21:14