$X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?commuting vector fields, commuting flowsThe set of complete vector fieldsDoes local flow of left-invariant vector field commute with the left-translation operator?What is the importance of conformal vector fields on Riemannian manifolds?How do I flow in a circle on a set of vector fields?How does the Torsion of two vector fields act on their corresponding flows?Flow of sum of commuting vector fieldsComputing Flow of vectors fields with partial derivativesCan we characterise concircular vector fields by their flow?Flow in the direction of a complex vectorShowing that these vector fields commute on the image
Should I tell management that I intend to leave due to bad software development practices?
Diode datasheet reading
Can one be a co-translator of a book, if he does not know the language that the book is translated into?
Is it unprofessional to ask if a job posting on GlassDoor is real?
Why doesn't H₄O²⁺ exist?
Why is consensus so controversial in Britain?
How to draw the figure with four pentagons?
Forgetting the musical notes while performing in concert
Has there ever been an airliner design involving reducing generator load by installing solar panels?
I would say: "You are another teacher", but she is a woman and I am a man
Were any external disk drives stacked vertically?
How much of data wrangling is a data scientist's job?
What killed these X2 caps?
How can saying a song's name be a copyright violation?
Doing something right before you need it - expression for this?
How badly should I try to prevent a user from XSSing themselves?
Why is the ratio of two extensive quantities always intensive?
Is there a way to gain immortality short of becoming a Lich or Vampire?
Latex document compiles but tikzpicture is not showing up
Anagram holiday
What is going on with Captain Marvel's blood colour?
Facing a paradox: Earnshaw's theorem in one dimension
Can a virus destroy the BIOS of a modern computer?
Can I make "comment-region" comment empty lines?
$X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?
commuting vector fields, commuting flowsThe set of complete vector fieldsDoes local flow of left-invariant vector field commute with the left-translation operator?What is the importance of conformal vector fields on Riemannian manifolds?How do I flow in a circle on a set of vector fields?How does the Torsion of two vector fields act on their corresponding flows?Flow of sum of commuting vector fieldsComputing Flow of vectors fields with partial derivativesCan we characterise concircular vector fields by their flow?Flow in the direction of a complex vectorShowing that these vector fields commute on the image
$begingroup$
If $X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?
I'm kind of confused on what the flow is.
I know that the respective flows for $Phi_t^X$ and $Phi_t^Y$ commute, but what is the flow for the addition of two vector fields?
Thank you.
differential-geometry vector-fields
edited 2 days ago
Daniele Tampieri
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
If $X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?
I'm kind of confused on what the flow is.
I know that the respective flows for $Phi_t^X$ and $Phi_t^Y$ commute, but what is the flow for the addition of two vector fields?
Thank you.
differential-geometry vector-fields
edited 2 days ago
Daniele Tampieri
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
If $X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?
I'm kind of confused on what the flow is.
I know that the respective flows for $Phi_t^X$ and $Phi_t^Y$ commute, but what is the flow for the addition of two vector fields?
Thank you.
differential-geometry vector-fields
edited 2 days ago
Daniele Tampieri
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
If $X, Y$ are two complete vector fields with$[X, Y] = 0$, what is the resulting flow of $X+Y$?
I'm kind of confused on what the flow is.
I know that the respective flows for $Phi_t^X$ and $Phi_t^Y$ commute, but what is the flow for the addition of two vector fields?
Thank you.
differential-geometry vector-fields
differential-geometry vector-fields
edited 2 days ago
Daniele Tampieri
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
Daniele Tampieri
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
Daniele Tampieri
2,65221022
edited 2 days ago
Daniele Tampieri
2,65221022
edited 2 days ago
Daniele Tampieri
2,65221022
2,65221022
asked Mar 29 at 1:22
fuwba2fuwba2
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Mar 29 at 1:22
fuwba2fuwba2
182
asked Mar 29 at 1:22
fuwba2fuwba2
182
182
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
fuwba2 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
First recall the following useful fact about commuting vector fields: if $X$ and $Y$ are complete vector fields such that $[X,Y]=0$, then $left(Phi_t^Xright)^*Y=Y$ for all $tinmathbbR$. Here $Phi_t^X$ is the time $t$ flow of $X$, and the pullback of vector fields is defined by
$$
left[left(Phi_t^Xright)^*Yright](p)=left(dPhi_-t^Xright)_Phi_t^X(p)(Y(Phi_t^X(p))).
$$
For a proof of this fact, see for instance commuting vector fields, commuting flows.
Now, we can prove that the flow of $X+Y$ is given by
$$
Phi_t^X+Y=Phi_t^XcircPhi_t^Y.
$$
Indeed, using the chain rule we have
beginalign
fracddtleft(Phi_t^XcircPhi_t^Yright)(p)&=left.fracddsright|_s=tleft(Phi_s^XcircPhi_t^Yright)(p)+left.fracddsright|_s=tleft(Phi_t^XcircPhi_s^Yright)(p)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)left(left.fracddsright|_s=tPhi_s^Y(p)right)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)big(Y(Phi_t^Y(p)big)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left[left(Phi_-t^Xright)^*(Y)right](Phi_t^XcircPhi_t^Y(p))\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+Ybig((Phi_t^XcircPhi_t^Y)(p)big),
endalign
using in the last equality that $left(Phi_-t^Xright)^*(Y)=Y$, as mentioned above. This proves the statement.
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
fuwba2 is a new contributor. Be nice, and check out our Code of Conduct.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166621%2fx-y-are-two-complete-vector-fields-withx-y-0-what-is-the-resulting-fl%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
First recall the following useful fact about commuting vector fields: if $X$ and $Y$ are complete vector fields such that $[X,Y]=0$, then $left(Phi_t^Xright)^*Y=Y$ for all $tinmathbbR$. Here $Phi_t^X$ is the time $t$ flow of $X$, and the pullback of vector fields is defined by
$$
left[left(Phi_t^Xright)^*Yright](p)=left(dPhi_-t^Xright)_Phi_t^X(p)(Y(Phi_t^X(p))).
$$
For a proof of this fact, see for instance commuting vector fields, commuting flows.
Now, we can prove that the flow of $X+Y$ is given by
$$
Phi_t^X+Y=Phi_t^XcircPhi_t^Y.
$$
Indeed, using the chain rule we have
beginalign
fracddtleft(Phi_t^XcircPhi_t^Yright)(p)&=left.fracddsright|_s=tleft(Phi_s^XcircPhi_t^Yright)(p)+left.fracddsright|_s=tleft(Phi_t^XcircPhi_s^Yright)(p)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)left(left.fracddsright|_s=tPhi_s^Y(p)right)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)big(Y(Phi_t^Y(p)big)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left[left(Phi_-t^Xright)^*(Y)right](Phi_t^XcircPhi_t^Y(p))\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+Ybig((Phi_t^XcircPhi_t^Y)(p)big),
endalign
using in the last equality that $left(Phi_-t^Xright)^*(Y)=Y$, as mentioned above. This proves the statement.
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
$endgroup$
add a comment |
$begingroup$
First recall the following useful fact about commuting vector fields: if $X$ and $Y$ are complete vector fields such that $[X,Y]=0$, then $left(Phi_t^Xright)^*Y=Y$ for all $tinmathbbR$. Here $Phi_t^X$ is the time $t$ flow of $X$, and the pullback of vector fields is defined by
$$
left[left(Phi_t^Xright)^*Yright](p)=left(dPhi_-t^Xright)_Phi_t^X(p)(Y(Phi_t^X(p))).
$$
For a proof of this fact, see for instance commuting vector fields, commuting flows.
Now, we can prove that the flow of $X+Y$ is given by
$$
Phi_t^X+Y=Phi_t^XcircPhi_t^Y.
$$
Indeed, using the chain rule we have
beginalign
fracddtleft(Phi_t^XcircPhi_t^Yright)(p)&=left.fracddsright|_s=tleft(Phi_s^XcircPhi_t^Yright)(p)+left.fracddsright|_s=tleft(Phi_t^XcircPhi_s^Yright)(p)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)left(left.fracddsright|_s=tPhi_s^Y(p)right)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)big(Y(Phi_t^Y(p)big)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left[left(Phi_-t^Xright)^*(Y)right](Phi_t^XcircPhi_t^Y(p))\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+Ybig((Phi_t^XcircPhi_t^Y)(p)big),
endalign
using in the last equality that $left(Phi_-t^Xright)^*(Y)=Y$, as mentioned above. This proves the statement.
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
$endgroup$
add a comment |
$begingroup$
First recall the following useful fact about commuting vector fields: if $X$ and $Y$ are complete vector fields such that $[X,Y]=0$, then $left(Phi_t^Xright)^*Y=Y$ for all $tinmathbbR$. Here $Phi_t^X$ is the time $t$ flow of $X$, and the pullback of vector fields is defined by
$$
left[left(Phi_t^Xright)^*Yright](p)=left(dPhi_-t^Xright)_Phi_t^X(p)(Y(Phi_t^X(p))).
$$
For a proof of this fact, see for instance commuting vector fields, commuting flows.
Now, we can prove that the flow of $X+Y$ is given by
$$
Phi_t^X+Y=Phi_t^XcircPhi_t^Y.
$$
Indeed, using the chain rule we have
beginalign
fracddtleft(Phi_t^XcircPhi_t^Yright)(p)&=left.fracddsright|_s=tleft(Phi_s^XcircPhi_t^Yright)(p)+left.fracddsright|_s=tleft(Phi_t^XcircPhi_s^Yright)(p)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)left(left.fracddsright|_s=tPhi_s^Y(p)right)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)big(Y(Phi_t^Y(p)big)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left[left(Phi_-t^Xright)^*(Y)right](Phi_t^XcircPhi_t^Y(p))\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+Ybig((Phi_t^XcircPhi_t^Y)(p)big),
endalign
using in the last equality that $left(Phi_-t^Xright)^*(Y)=Y$, as mentioned above. This proves the statement.
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
$endgroup$
First recall the following useful fact about commuting vector fields: if $X$ and $Y$ are complete vector fields such that $[X,Y]=0$, then $left(Phi_t^Xright)^*Y=Y$ for all $tinmathbbR$. Here $Phi_t^X$ is the time $t$ flow of $X$, and the pullback of vector fields is defined by
$$
left[left(Phi_t^Xright)^*Yright](p)=left(dPhi_-t^Xright)_Phi_t^X(p)(Y(Phi_t^X(p))).
$$
For a proof of this fact, see for instance commuting vector fields, commuting flows.
Now, we can prove that the flow of $X+Y$ is given by
$$
Phi_t^X+Y=Phi_t^XcircPhi_t^Y.
$$
Indeed, using the chain rule we have
beginalign
fracddtleft(Phi_t^XcircPhi_t^Yright)(p)&=left.fracddsright|_s=tleft(Phi_s^XcircPhi_t^Yright)(p)+left.fracddsright|_s=tleft(Phi_t^XcircPhi_s^Yright)(p)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)left(left.fracddsright|_s=tPhi_s^Y(p)right)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left(dPhi_t^Xright)_Phi_t^Y(p)big(Y(Phi_t^Y(p)big)\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+left[left(Phi_-t^Xright)^*(Y)right](Phi_t^XcircPhi_t^Y(p))\
&=Xbig((Phi_t^XcircPhi_t^Y)(p)big)+Ybig((Phi_t^XcircPhi_t^Y)(p)big),
endalign
using in the last equality that $left(Phi_-t^Xright)^*(Y)=Y$, as mentioned above. This proves the statement.
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
answered Apr 1 at 10:55
studiosusstudiosus
2,259815
2,259815
add a comment |
add a comment |
fuwba2 is a new contributor. Be nice, and check out our Code of Conduct.
fuwba2 is a new contributor. Be nice, and check out our Code of Conduct.
fuwba2 is a new contributor. Be nice, and check out our Code of Conduct.
fuwba2 is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3166621%2fx-y-are-two-complete-vector-fields-withx-y-0-what-is-the-resulting-fl%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
