Horizontal lifts and integral curves
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Let $[0,b]$be an interval and let $t$ be the standard coordinate on this interval. Suppose that $pi:E rightarrow [0,b]$ is a vector bundle with a given Ehresmann connection. Recall that a Ehresmann connection in this situation is a smooth distribution $mathcal{H}$ on the total space $E$ such that:
$mathcal{H}$ is complementary to the vertical bundle $$TE = mathcal{H} oplus mathcal{V}mathcal{E}$$
$mathcal{H}$ is homogeneous, that is, $$T_y mu_r(mathcal{H}_y) = mathcal{H}_{ry}$$ for all $y in E, r in mathbb{R},$ where $mu_r:E rightarrow E$ is the multiplication map
Let $tilde{partial}$ denote the horizontal lift of $partial/partial t$ and let $0 leq t_0 < b.$ I want to show that there is a fixed $epsilon >0$ that depends only on $t_0$ such that all maximal integral curves of $tilde{partial}$ originating in the fixed fiber $E_{t_0}$ are defined at least on $[t_0,e).$
The hint I have been given is that I should endow $E$ with a bundle metric and consider all integral curves originating in the unit sphere in $E_{t_0}$ and then use the second property of an Ehresmann connection, but I do not see how to proceed with this hint. Any further tips, or solutions are welcome.
differential-geometry
asked Nov 21 at 20:40
Dedalus
1,99711936
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up vote
0
down vote
favorite
Let $[0,b]$be an interval and let $t$ be the standard coordinate on this interval. Suppose that $pi:E rightarrow [0,b]$ is a vector bundle with a given Ehresmann connection. Recall that a Ehresmann connection in this situation is a smooth distribution $mathcal{H}$ on the total space $E$ such that:
$mathcal{H}$ is complementary to the vertical bundle $$TE = mathcal{H} oplus mathcal{V}mathcal{E}$$
$mathcal{H}$ is homogeneous, that is, $$T_y mu_r(mathcal{H}_y) = mathcal{H}_{ry}$$ for all $y in E, r in mathbb{R},$ where $mu_r:E rightarrow E$ is the multiplication map
Let $tilde{partial}$ denote the horizontal lift of $partial/partial t$ and let $0 leq t_0 < b.$ I want to show that there is a fixed $epsilon >0$ that depends only on $t_0$ such that all maximal integral curves of $tilde{partial}$ originating in the fixed fiber $E_{t_0}$ are defined at least on $[t_0,e).$
The hint I have been given is that I should endow $E$ with a bundle metric and consider all integral curves originating in the unit sphere in $E_{t_0}$ and then use the second property of an Ehresmann connection, but I do not see how to proceed with this hint. Any further tips, or solutions are welcome.
differential-geometry
asked Nov 21 at 20:40
Dedalus
1,99711936
My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59
add a comment |
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0
down vote
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up vote
0
down vote
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Let $[0,b]$be an interval and let $t$ be the standard coordinate on this interval. Suppose that $pi:E rightarrow [0,b]$ is a vector bundle with a given Ehresmann connection. Recall that a Ehresmann connection in this situation is a smooth distribution $mathcal{H}$ on the total space $E$ such that:
$mathcal{H}$ is complementary to the vertical bundle $$TE = mathcal{H} oplus mathcal{V}mathcal{E}$$
$mathcal{H}$ is homogeneous, that is, $$T_y mu_r(mathcal{H}_y) = mathcal{H}_{ry}$$ for all $y in E, r in mathbb{R},$ where $mu_r:E rightarrow E$ is the multiplication map
Let $tilde{partial}$ denote the horizontal lift of $partial/partial t$ and let $0 leq t_0 < b.$ I want to show that there is a fixed $epsilon >0$ that depends only on $t_0$ such that all maximal integral curves of $tilde{partial}$ originating in the fixed fiber $E_{t_0}$ are defined at least on $[t_0,e).$
The hint I have been given is that I should endow $E$ with a bundle metric and consider all integral curves originating in the unit sphere in $E_{t_0}$ and then use the second property of an Ehresmann connection, but I do not see how to proceed with this hint. Any further tips, or solutions are welcome.
differential-geometry
asked Nov 21 at 20:40
Dedalus
1,99711936
Let $[0,b]$be an interval and let $t$ be the standard coordinate on this interval. Suppose that $pi:E rightarrow [0,b]$ is a vector bundle with a given Ehresmann connection. Recall that a Ehresmann connection in this situation is a smooth distribution $mathcal{H}$ on the total space $E$ such that:
$mathcal{H}$ is complementary to the vertical bundle $$TE = mathcal{H} oplus mathcal{V}mathcal{E}$$
$mathcal{H}$ is homogeneous, that is, $$T_y mu_r(mathcal{H}_y) = mathcal{H}_{ry}$$ for all $y in E, r in mathbb{R},$ where $mu_r:E rightarrow E$ is the multiplication map
Let $tilde{partial}$ denote the horizontal lift of $partial/partial t$ and let $0 leq t_0 < b.$ I want to show that there is a fixed $epsilon >0$ that depends only on $t_0$ such that all maximal integral curves of $tilde{partial}$ originating in the fixed fiber $E_{t_0}$ are defined at least on $[t_0,e).$
The hint I have been given is that I should endow $E$ with a bundle metric and consider all integral curves originating in the unit sphere in $E_{t_0}$ and then use the second property of an Ehresmann connection, but I do not see how to proceed with this hint. Any further tips, or solutions are welcome.
differential-geometry
differential-geometry
asked Nov 21 at 20:40
Dedalus
1,99711936
asked Nov 21 at 20:40
Dedalus
1,99711936
edited Nov 22 at 17:56
asked Nov 21 at 20:40
Dedalus
1,99711936
asked Nov 21 at 20:40
Dedalus
1,99711936
asked Nov 21 at 20:40
Dedalus
1,99711936
1,99711936
My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59
add a comment |
My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59
My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59
add a comment |
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My first guess would be: use a local trivialization around $t_0$, so that $Esupset U=Itimes mathbb{R}^m$, where $I$ is an interval containing $t_0$. Since the $t$ component of the lift is non-zero at $t_0$, by continuity it shouldn't vanish in a neighborhood. What is $partial delta$? I don't know about the subject, so I apologize for any mistake.
– user90189
Nov 22 at 2:15
@user90189 I managed to solve this exercise using another idea. If you have any thoughts on my recently posted question related to this, please tell me!
– Dedalus
Nov 24 at 0:28
Please tag this as exercise and, if you solved it, please post the answer, so we can learn from it and upvote it
– magma
Nov 24 at 8:59