Mixing
Is it implicitly assumed that a product of manifolds have the product smooth manifold structure?
$begingroup$
If $M$ and $N$ are smooth manifolds and $(U,varphi)$ is a smooth chart of $M times N$, is it standard for a text to assume that $M times N$ has the product smooth manifold structure without implicitly stating so? I am following John Lee's introduction to smooth manifolds, and wonder if I can assume that $(pi_1(U), pi_1 circ varphi)$ is a smooth chart of $M$.
manifolds smooth-manifolds smooth-functions
edited Jan 21 at 20:55
Bernard
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
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add a comment |
$begingroup$
If $M$ and $N$ are smooth manifolds and $(U,varphi)$ is a smooth chart of $M times N$, is it standard for a text to assume that $M times N$ has the product smooth manifold structure without implicitly stating so? I am following John Lee's introduction to smooth manifolds, and wonder if I can assume that $(pi_1(U), pi_1 circ varphi)$ is a smooth chart of $M$.
manifolds smooth-manifolds smooth-functions
edited Jan 21 at 20:55
Bernard
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
$endgroup$
1
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01
add a comment |
$begingroup$
If $M$ and $N$ are smooth manifolds and $(U,varphi)$ is a smooth chart of $M times N$, is it standard for a text to assume that $M times N$ has the product smooth manifold structure without implicitly stating so? I am following John Lee's introduction to smooth manifolds, and wonder if I can assume that $(pi_1(U), pi_1 circ varphi)$ is a smooth chart of $M$.
manifolds smooth-manifolds smooth-functions
edited Jan 21 at 20:55
Bernard
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
$endgroup$
If $M$ and $N$ are smooth manifolds and $(U,varphi)$ is a smooth chart of $M times N$, is it standard for a text to assume that $M times N$ has the product smooth manifold structure without implicitly stating so? I am following John Lee's introduction to smooth manifolds, and wonder if I can assume that $(pi_1(U), pi_1 circ varphi)$ is a smooth chart of $M$.
manifolds smooth-manifolds smooth-functions
manifolds smooth-manifolds smooth-functions
edited Jan 21 at 20:55
Bernard
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
edited Jan 21 at 20:55
Bernard
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
edited Jan 21 at 20:55
Bernard
122k741116
edited Jan 21 at 20:55
Bernard
122k741116
edited Jan 21 at 20:55
Bernard
122k741116
122k741116
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
asked Jan 21 at 20:23
Marius JonssonMarius Jonsson
705414
705414
1
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01
add a comment |
1
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01
1
1
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01
add a comment |
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1
$begingroup$
$pi_1 circ varphi$ cannot be a chart for $M$.
$endgroup$
– Paul Frost
Jan 21 at 22:01