the smoothness of an induced orthonomal vector field in normal neighborhood
$begingroup$
Let $M$ be a Riemannian manifold and p be a point on M. Let U be a normal neighborhood about p. Fix a vector $v_p∈T_pM$.And use parallel transport,we can transport is to $T_qM$ for any $qin U$,because there is a geodesic line from p to q.Then we get a vector field on U.
How to find it is smooth?
In this question
enter link description here
Dear Jack Lee solved this using the smoothness of solution to ODE,the smoothness of solution to smooth ODE is dependent on the initial condition.But as i know, it is true locally, how to use the theorem when the"t" is large, in this problem ,we need t=1.
riemannian-geometry geodesic
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
$endgroup$
add a comment |
$begingroup$
Let $M$ be a Riemannian manifold and p be a point on M. Let U be a normal neighborhood about p. Fix a vector $v_p∈T_pM$.And use parallel transport,we can transport is to $T_qM$ for any $qin U$,because there is a geodesic line from p to q.Then we get a vector field on U.
How to find it is smooth?
In this question
enter link description here
Dear Jack Lee solved this using the smoothness of solution to ODE,the smoothness of solution to smooth ODE is dependent on the initial condition.But as i know, it is true locally, how to use the theorem when the"t" is large, in this problem ,we need t=1.
riemannian-geometry geodesic
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
$endgroup$
add a comment |
$begingroup$
Let $M$ be a Riemannian manifold and p be a point on M. Let U be a normal neighborhood about p. Fix a vector $v_p∈T_pM$.And use parallel transport,we can transport is to $T_qM$ for any $qin U$,because there is a geodesic line from p to q.Then we get a vector field on U.
How to find it is smooth?
In this question
enter link description here
Dear Jack Lee solved this using the smoothness of solution to ODE,the smoothness of solution to smooth ODE is dependent on the initial condition.But as i know, it is true locally, how to use the theorem when the"t" is large, in this problem ,we need t=1.
riemannian-geometry geodesic
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
$endgroup$
Let $M$ be a Riemannian manifold and p be a point on M. Let U be a normal neighborhood about p. Fix a vector $v_p∈T_pM$.And use parallel transport,we can transport is to $T_qM$ for any $qin U$,because there is a geodesic line from p to q.Then we get a vector field on U.
How to find it is smooth?
In this question
enter link description here
Dear Jack Lee solved this using the smoothness of solution to ODE,the smoothness of solution to smooth ODE is dependent on the initial condition.But as i know, it is true locally, how to use the theorem when the"t" is large, in this problem ,we need t=1.
riemannian-geometry geodesic
riemannian-geometry geodesic
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
asked Dec 12 '18 at 9:31
gongbabaihedigongbabaihedi
13
13
add a comment |
add a comment |
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