Mixing
vector field of real projective space
$begingroup$
Let $RP^n$ the real projective space. This is a manifold and i have take the usual charts in order to prove it. The problem is that i don't know how to define a vector field on that.
Since $RP^n$ is a set of lines that passes through the point (0,...,0) could I claim that the vector space is a map that takes every point of $RP^n$ - line and sends it to a vector $overline {OP}$ ,where O is the origin and P is a point of the line ?
Is there a concrete example on $PR^2$?
manifolds
asked Jan 27 at 15:48
janejane
264
$endgroup$
add a comment |
$begingroup$
Let $RP^n$ the real projective space. This is a manifold and i have take the usual charts in order to prove it. The problem is that i don't know how to define a vector field on that.
Since $RP^n$ is a set of lines that passes through the point (0,...,0) could I claim that the vector space is a map that takes every point of $RP^n$ - line and sends it to a vector $overline {OP}$ ,where O is the origin and P is a point of the line ?
Is there a concrete example on $PR^2$?
manifolds
asked Jan 27 at 15:48
janejane
264
$endgroup$
add a comment |
$begingroup$
Let $RP^n$ the real projective space. This is a manifold and i have take the usual charts in order to prove it. The problem is that i don't know how to define a vector field on that.
Since $RP^n$ is a set of lines that passes through the point (0,...,0) could I claim that the vector space is a map that takes every point of $RP^n$ - line and sends it to a vector $overline {OP}$ ,where O is the origin and P is a point of the line ?
Is there a concrete example on $PR^2$?
manifolds
asked Jan 27 at 15:48
janejane
264
$endgroup$
Let $RP^n$ the real projective space. This is a manifold and i have take the usual charts in order to prove it. The problem is that i don't know how to define a vector field on that.
Since $RP^n$ is a set of lines that passes through the point (0,...,0) could I claim that the vector space is a map that takes every point of $RP^n$ - line and sends it to a vector $overline {OP}$ ,where O is the origin and P is a point of the line ?
Is there a concrete example on $PR^2$?
manifolds
manifolds
asked Jan 27 at 15:48
janejane
264
asked Jan 27 at 15:48
janejane
264
asked Jan 27 at 15:48
janejane
264
asked Jan 27 at 15:48
janejane
264
asked Jan 27 at 15:48
janejane
264
264
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We can visualize $RP^2$ as the plane in $mathbb R^3$ at $z=1$.
We can write a point as $(x:y:1)$. Or more generally as $(x:y:z)$, which we consider equivalent to $(frac xz:frac yz:1)$ if $zne 0$. Consequently all points on a line through the origin are equivalent. And a point $(x:y:0)$ is a point at infinity, which is also an element of $RP^2$.
An obvious choice for a local chart is $phi: (x:y:1)mapsto (x,y)$.
Taking the partial derivatives gives us the representations $fracpartial{partial x}=(1:0:0)$ and $fracpartial{partial y}=(0:1:0)$. Note that these correspond exactly with the standard unit vectors in the plane $z=1$.
A vector field $X: RP^2to TRP^2$ can now locally be written as:
$$X(mathbf x) = X(x:y:1) = xi(mathbf x)fracpartial{partial x} + eta(mathbf x)fracpartial{partial y} = (xi(mathbf x):eta(mathbf x):0)$$
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
$endgroup$
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
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
});
}
});
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%2f3089729%2fvector-field-of-real-projective-space%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
$begingroup$
We can visualize $RP^2$ as the plane in $mathbb R^3$ at $z=1$.
We can write a point as $(x:y:1)$. Or more generally as $(x:y:z)$, which we consider equivalent to $(frac xz:frac yz:1)$ if $zne 0$. Consequently all points on a line through the origin are equivalent. And a point $(x:y:0)$ is a point at infinity, which is also an element of $RP^2$.
An obvious choice for a local chart is $phi: (x:y:1)mapsto (x,y)$.
Taking the partial derivatives gives us the representations $fracpartial{partial x}=(1:0:0)$ and $fracpartial{partial y}=(0:1:0)$. Note that these correspond exactly with the standard unit vectors in the plane $z=1$.
A vector field $X: RP^2to TRP^2$ can now locally be written as:
$$X(mathbf x) = X(x:y:1) = xi(mathbf x)fracpartial{partial x} + eta(mathbf x)fracpartial{partial y} = (xi(mathbf x):eta(mathbf x):0)$$
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
$endgroup$
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
add a comment |
$begingroup$
We can visualize $RP^2$ as the plane in $mathbb R^3$ at $z=1$.
We can write a point as $(x:y:1)$. Or more generally as $(x:y:z)$, which we consider equivalent to $(frac xz:frac yz:1)$ if $zne 0$. Consequently all points on a line through the origin are equivalent. And a point $(x:y:0)$ is a point at infinity, which is also an element of $RP^2$.
An obvious choice for a local chart is $phi: (x:y:1)mapsto (x,y)$.
Taking the partial derivatives gives us the representations $fracpartial{partial x}=(1:0:0)$ and $fracpartial{partial y}=(0:1:0)$. Note that these correspond exactly with the standard unit vectors in the plane $z=1$.
A vector field $X: RP^2to TRP^2$ can now locally be written as:
$$X(mathbf x) = X(x:y:1) = xi(mathbf x)fracpartial{partial x} + eta(mathbf x)fracpartial{partial y} = (xi(mathbf x):eta(mathbf x):0)$$
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
$endgroup$
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
add a comment |
$begingroup$
We can visualize $RP^2$ as the plane in $mathbb R^3$ at $z=1$.
We can write a point as $(x:y:1)$. Or more generally as $(x:y:z)$, which we consider equivalent to $(frac xz:frac yz:1)$ if $zne 0$. Consequently all points on a line through the origin are equivalent. And a point $(x:y:0)$ is a point at infinity, which is also an element of $RP^2$.
An obvious choice for a local chart is $phi: (x:y:1)mapsto (x,y)$.
Taking the partial derivatives gives us the representations $fracpartial{partial x}=(1:0:0)$ and $fracpartial{partial y}=(0:1:0)$. Note that these correspond exactly with the standard unit vectors in the plane $z=1$.
A vector field $X: RP^2to TRP^2$ can now locally be written as:
$$X(mathbf x) = X(x:y:1) = xi(mathbf x)fracpartial{partial x} + eta(mathbf x)fracpartial{partial y} = (xi(mathbf x):eta(mathbf x):0)$$
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
$endgroup$
We can visualize $RP^2$ as the plane in $mathbb R^3$ at $z=1$.
We can write a point as $(x:y:1)$. Or more generally as $(x:y:z)$, which we consider equivalent to $(frac xz:frac yz:1)$ if $zne 0$. Consequently all points on a line through the origin are equivalent. And a point $(x:y:0)$ is a point at infinity, which is also an element of $RP^2$.
An obvious choice for a local chart is $phi: (x:y:1)mapsto (x,y)$.
Taking the partial derivatives gives us the representations $fracpartial{partial x}=(1:0:0)$ and $fracpartial{partial y}=(0:1:0)$. Note that these correspond exactly with the standard unit vectors in the plane $z=1$.
A vector field $X: RP^2to TRP^2$ can now locally be written as:
$$X(mathbf x) = X(x:y:1) = xi(mathbf x)fracpartial{partial x} + eta(mathbf x)fracpartial{partial y} = (xi(mathbf x):eta(mathbf x):0)$$
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
answered Feb 2 at 20:18
Klaas van AarsenKlaas van Aarsen
4,3421822
4,3421822
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
add a comment |
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
1
1
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
$begingroup$
oh...i get it ..thanks...
$endgroup$
– jane
Feb 3 at 17:33
add a comment |
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%2f3089729%2fvector-field-of-real-projective-space%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
