Unextendable vector fields on $mathbb{S}^2$
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
asked Dec 3 '18 at 20:57
mathnoob123
693417
|
show 3 more comments
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
asked Dec 3 '18 at 20:57
mathnoob123
693417
What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
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?
– Ted Shifrin
Dec 3 '18 at 23:17
|
show 3 more comments
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
asked Dec 3 '18 at 20:57
mathnoob123
693417
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
differential-geometry
asked Dec 3 '18 at 20:57
mathnoob123
693417
asked Dec 3 '18 at 20:57
mathnoob123
693417
edited Dec 4 '18 at 9:54
asked Dec 3 '18 at 20:57
mathnoob123
693417
asked Dec 3 '18 at 20:57
mathnoob123
693417
asked Dec 3 '18 at 20:57
mathnoob123
693417
693417
What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
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?
– Ted Shifrin
Dec 3 '18 at 23:17
|
show 3 more comments
What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
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?
– Ted Shifrin
Dec 3 '18 at 23:17
What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
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?
– Ted Shifrin
Dec 3 '18 at 23:17
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?
– Ted Shifrin
Dec 3 '18 at 23:17
|
show 3 more comments
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What is $chi(M)$ supposed to be?
– Randall
Dec 3 '18 at 21:05
The set of all vector fields in M
– mathnoob123
Dec 3 '18 at 21:05
Look at the vector field $Y$ in cartesian coordinates. As you approach $theta=0$ or $theta=pi$, does it have a limiting value?
– Ted Shifrin
Dec 3 '18 at 21:56
I believe no, because it doesnt depend on it.
– mathnoob123
Dec 3 '18 at 22:00
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?
– Ted Shifrin
Dec 3 '18 at 23:17