Unextendable vector fields on $mathbb{S}^2$












0












$begingroup$



Let $ M = mathbb{S}^2 $ be a manifold with the coordinate patch $U$parameterized using spherical coordinates with $(phi, theta) in (0,2pi)times (0,pi)$. Let $Y = frac{partial}{partial phi}$ be vector fields over $U$. Show $Y$ cannot be extended continously to a vector field in $chi(M)$.




I am aware of the idea that I have to show that if I consider a patch that covers the points $ M -{U}$ then the vector field $Y$ won't be defined at least on the intersection of the patches. However, I am unsure how to achieve this. Moreover, is it correct to suggest that the information about the spherical coordinate parametrization is redundant since they have already provided me the $Y$ vector field?










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$endgroup$












  • $begingroup$
    What is $chi(M)$ supposed to be?
    $endgroup$
    – Randall
    Dec 3 '18 at 21:05












  • $begingroup$
    The set of all vector fields in M
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 21:05










  • $begingroup$
    Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 21:56










  • $begingroup$
    I believe no, because it doesnt depend on it.
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 22:00










  • $begingroup$
    I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 23:17
















0












$begingroup$



Let $ M = mathbb{S}^2 $ be a manifold with the coordinate patch $U$parameterized using spherical coordinates with $(phi, theta) in (0,2pi)times (0,pi)$. Let $Y = frac{partial}{partial phi}$ be vector fields over $U$. Show $Y$ cannot be extended continously to a vector field in $chi(M)$.




I am aware of the idea that I have to show that if I consider a patch that covers the points $ M -{U}$ then the vector field $Y$ won't be defined at least on the intersection of the patches. However, I am unsure how to achieve this. Moreover, is it correct to suggest that the information about the spherical coordinate parametrization is redundant since they have already provided me the $Y$ vector field?










share|cite|improve this question











$endgroup$












  • $begingroup$
    What is $chi(M)$ supposed to be?
    $endgroup$
    – Randall
    Dec 3 '18 at 21:05












  • $begingroup$
    The set of all vector fields in M
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 21:05










  • $begingroup$
    Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 21:56










  • $begingroup$
    I believe no, because it doesnt depend on it.
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 22:00










  • $begingroup$
    I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 23:17














0












0








0





$begingroup$



Let $ M = mathbb{S}^2 $ be a manifold with the coordinate patch $U$parameterized using spherical coordinates with $(phi, theta) in (0,2pi)times (0,pi)$. Let $Y = frac{partial}{partial phi}$ be vector fields over $U$. Show $Y$ cannot be extended continously to a vector field in $chi(M)$.




I am aware of the idea that I have to show that if I consider a patch that covers the points $ M -{U}$ then the vector field $Y$ won't be defined at least on the intersection of the patches. However, I am unsure how to achieve this. Moreover, is it correct to suggest that the information about the spherical coordinate parametrization is redundant since they have already provided me the $Y$ vector field?










share|cite|improve this question











$endgroup$





Let $ M = mathbb{S}^2 $ be a manifold with the coordinate patch $U$parameterized using spherical coordinates with $(phi, theta) in (0,2pi)times (0,pi)$. Let $Y = frac{partial}{partial phi}$ be vector fields over $U$. Show $Y$ cannot be extended continously to a vector field in $chi(M)$.




I am aware of the idea that I have to show that if I consider a patch that covers the points $ M -{U}$ then the vector field $Y$ won't be defined at least on the intersection of the patches. However, I am unsure how to achieve this. Moreover, is it correct to suggest that the information about the spherical coordinate parametrization is redundant since they have already provided me the $Y$ vector field?







differential-geometry






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 4 '18 at 9:54







mathnoob123

















asked Dec 3 '18 at 20:57









mathnoob123mathnoob123

693417




693417












  • $begingroup$
    What is $chi(M)$ supposed to be?
    $endgroup$
    – Randall
    Dec 3 '18 at 21:05












  • $begingroup$
    The set of all vector fields in M
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 21:05










  • $begingroup$
    Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 21:56










  • $begingroup$
    I believe no, because it doesnt depend on it.
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 22:00










  • $begingroup$
    I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 23:17


















  • $begingroup$
    What is $chi(M)$ supposed to be?
    $endgroup$
    – Randall
    Dec 3 '18 at 21:05












  • $begingroup$
    The set of all vector fields in M
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 21:05










  • $begingroup$
    Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 21:56










  • $begingroup$
    I believe no, because it doesnt depend on it.
    $endgroup$
    – mathnoob123
    Dec 3 '18 at 22:00










  • $begingroup$
    I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
    $endgroup$
    – Ted Shifrin
    Dec 3 '18 at 23:17
















$begingroup$
What is $chi(M)$ supposed to be?
$endgroup$
– Randall
Dec 3 '18 at 21:05






$begingroup$
What is $chi(M)$ supposed to be?
$endgroup$
– Randall
Dec 3 '18 at 21:05














$begingroup$
The set of all vector fields in M
$endgroup$
– mathnoob123
Dec 3 '18 at 21:05




$begingroup$
The set of all vector fields in M
$endgroup$
– mathnoob123
Dec 3 '18 at 21:05












$begingroup$
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
$endgroup$
– Ted Shifrin
Dec 3 '18 at 21:56




$begingroup$
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
$endgroup$
– Ted Shifrin
Dec 3 '18 at 21:56












$begingroup$
I believe no, because it doesnt depend on it.
$endgroup$
– mathnoob123
Dec 3 '18 at 22:00




$begingroup$
I believe no, because it doesnt depend on it.
$endgroup$
– mathnoob123
Dec 3 '18 at 22:00












$begingroup$
I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:17




$begingroup$
I don't understand what you just said. What does the vector field $Y$ look like in a neighborhood of the north or south pole on the sphere?
$endgroup$
– Ted Shifrin
Dec 3 '18 at 23:17










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