The differential structure of infinite Mobius strip
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As the title,how to solve the following problem:
Problem: Consider the cylinder
$$
C={(x,y,z)inmathbb{R}^3|x^2+y^2=1}
$$
and identify the point $(x,y,z)$ with $(-x,-y,-z)$.Show that the quotient space of $C$ by this equivalence relation can be given a differentiable structure (infinite Mobius band).
Please give me more details,thanks!
differential-geometry differential-topology riemannian-geometry
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
$endgroup$
add a comment |
$begingroup$
As the title,how to solve the following problem:
Problem: Consider the cylinder
$$
C={(x,y,z)inmathbb{R}^3|x^2+y^2=1}
$$
and identify the point $(x,y,z)$ with $(-x,-y,-z)$.Show that the quotient space of $C$ by this equivalence relation can be given a differentiable structure (infinite Mobius band).
Please give me more details,thanks!
differential-geometry differential-topology riemannian-geometry
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
$endgroup$
$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
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Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
1
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37
add a comment |
$begingroup$
As the title,how to solve the following problem:
Problem: Consider the cylinder
$$
C={(x,y,z)inmathbb{R}^3|x^2+y^2=1}
$$
and identify the point $(x,y,z)$ with $(-x,-y,-z)$.Show that the quotient space of $C$ by this equivalence relation can be given a differentiable structure (infinite Mobius band).
Please give me more details,thanks!
differential-geometry differential-topology riemannian-geometry
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
$endgroup$
As the title,how to solve the following problem:
Problem: Consider the cylinder
$$
C={(x,y,z)inmathbb{R}^3|x^2+y^2=1}
$$
and identify the point $(x,y,z)$ with $(-x,-y,-z)$.Show that the quotient space of $C$ by this equivalence relation can be given a differentiable structure (infinite Mobius band).
Please give me more details,thanks!
differential-geometry differential-topology riemannian-geometry
differential-geometry differential-topology riemannian-geometry
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
edited Dec 16 '18 at 13:00
daoqiangliu
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
asked Dec 16 '18 at 8:55
daoqiangliudaoqiangliu
163
163
$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
$begingroup$
Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
1
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37
add a comment |
$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
$begingroup$
Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
1
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37
$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
$begingroup$
Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
$begingroup$
Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
1
1
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37
add a comment |
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$begingroup$
Hello, do you know what they mean when they ask for a differentiable structure?
$endgroup$
– Prototank
Dec 16 '18 at 13:01
$begingroup$
Yes, of course !
$endgroup$
– daoqiangliu
Dec 16 '18 at 13:04
1
$begingroup$
Why don't you add to your question the meaning of "can be given a differentiable structure" (there are multiple (equivalent) definitions, and it'd be nice for us to know which one you're using). Then you can say how far you've gotten in making this meaning match your situation. And then maybe we can help.
$endgroup$
– John Hughes
Dec 16 '18 at 13:37