Problem understanding Milnor Proof: (Theorem 1, Vector Fields chapter)
$begingroup$
(THIS QUESTION HAS BEEN EDITED)
I am reading Milnor's ''Topology From a Differentiable Viewpoint'' and in the chapter about vector fields, page 38, there is a theorem that states that:
Given any vector field $v$ on $M subset mathbb{R^n}$, ($M$ an m-dimensional, compact boundaryless manifold) with only nondegenerate zeros, then the index sum of $v$ is equal to the degree of the Gauss mapping.
So, I am going to avoid writing the proof (if anyone needs it to give a better answer I will write it, just prefer to avoid this if not necessary).
You have:
$N_{epsilon}$ a closed $epsilon$-neighborhood of $M$
$r$: $N_{epsilon} to M$, a differentiable map that maps $x$ to the point in $M$, closest to $x$. (Making $epsilon $ small enough this works well)
$w$ a vector field in $N_{epsilon}$ that extends $v$, given by: $w(x)= (x-r(x))+v(r(x))$
So.. What I think Milnor is trying to do, is to use Hopf's lemma (Lemma 3 in the book),you can see that $w$ points outwards along the boundary and if it has a zero, it must be a zero of $v$ (since $x-r(x)$ and $v(r(x))$ are orthogonal), so all its zeros are isolated, then you are in condition to apply Hopf's lemma.
Now he states that: For any $z in M$
$d_zw(h)= d_zv(h) $ in $T_zM$
and
$d_zw(h)=h$ in $T_zM$'s orthogonal complement.
So... maybe it is easy to see, but I cannot see why this is. Trying to calculate $frac{partial w_i}{partial x_j}$ for arbitrary $i,j$ seems usless, since I do not know how to calculate the derivative of $r(x)$. This is my first doubt.
My second doubt.. Supposing that the previous statement is true, do you have that $d_zw$ and $d_zv$ have the same determinant at any $z$, zero of $w$, hence the same index. Now Milnor uses Hopf's Lemma, to prove that the index sum of $w$ is equal the Gauss mapping. To finish this off I need to see that $v$ has the same zeros than $w$, why is this true? (You know that a zero of $w$ is a zero of $v$, why is the reciprocal true?)
I am really stuck with this, any help would be appreciated, thanks in advance.
(FROM HERE ON I EDITED IT)
So... I am going to answer my second doubt: If $z$ is a zero of $v$, then $z in M$ so $r(z)=z$ and thus $w(z)=v(z)=0$
And for my first doubt, I was thinking that I never used that $v$ has non-degenerate zeros (Or so I think), so maybe that is a hint to anwering my first doubt, cannot seem to see it though. If I had used that $v$ has only non-degenerate zeros, please tell me where. Thanks in advance.
differential-topology
edited Dec 18 '18 at 20:17
Card_Trick
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
$endgroup$
add a comment |
$begingroup$
(THIS QUESTION HAS BEEN EDITED)
I am reading Milnor's ''Topology From a Differentiable Viewpoint'' and in the chapter about vector fields, page 38, there is a theorem that states that:
Given any vector field $v$ on $M subset mathbb{R^n}$, ($M$ an m-dimensional, compact boundaryless manifold) with only nondegenerate zeros, then the index sum of $v$ is equal to the degree of the Gauss mapping.
So, I am going to avoid writing the proof (if anyone needs it to give a better answer I will write it, just prefer to avoid this if not necessary).
You have:
$N_{epsilon}$ a closed $epsilon$-neighborhood of $M$
$r$: $N_{epsilon} to M$, a differentiable map that maps $x$ to the point in $M$, closest to $x$. (Making $epsilon $ small enough this works well)
$w$ a vector field in $N_{epsilon}$ that extends $v$, given by: $w(x)= (x-r(x))+v(r(x))$
So.. What I think Milnor is trying to do, is to use Hopf's lemma (Lemma 3 in the book),you can see that $w$ points outwards along the boundary and if it has a zero, it must be a zero of $v$ (since $x-r(x)$ and $v(r(x))$ are orthogonal), so all its zeros are isolated, then you are in condition to apply Hopf's lemma.
Now he states that: For any $z in M$
$d_zw(h)= d_zv(h) $ in $T_zM$
and
$d_zw(h)=h$ in $T_zM$'s orthogonal complement.
So... maybe it is easy to see, but I cannot see why this is. Trying to calculate $frac{partial w_i}{partial x_j}$ for arbitrary $i,j$ seems usless, since I do not know how to calculate the derivative of $r(x)$. This is my first doubt.
My second doubt.. Supposing that the previous statement is true, do you have that $d_zw$ and $d_zv$ have the same determinant at any $z$, zero of $w$, hence the same index. Now Milnor uses Hopf's Lemma, to prove that the index sum of $w$ is equal the Gauss mapping. To finish this off I need to see that $v$ has the same zeros than $w$, why is this true? (You know that a zero of $w$ is a zero of $v$, why is the reciprocal true?)
I am really stuck with this, any help would be appreciated, thanks in advance.
(FROM HERE ON I EDITED IT)
So... I am going to answer my second doubt: If $z$ is a zero of $v$, then $z in M$ so $r(z)=z$ and thus $w(z)=v(z)=0$
And for my first doubt, I was thinking that I never used that $v$ has non-degenerate zeros (Or so I think), so maybe that is a hint to anwering my first doubt, cannot seem to see it though. If I had used that $v$ has only non-degenerate zeros, please tell me where. Thanks in advance.
differential-topology
edited Dec 18 '18 at 20:17
Card_Trick
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
$endgroup$
add a comment |
$begingroup$
(THIS QUESTION HAS BEEN EDITED)
I am reading Milnor's ''Topology From a Differentiable Viewpoint'' and in the chapter about vector fields, page 38, there is a theorem that states that:
Given any vector field $v$ on $M subset mathbb{R^n}$, ($M$ an m-dimensional, compact boundaryless manifold) with only nondegenerate zeros, then the index sum of $v$ is equal to the degree of the Gauss mapping.
So, I am going to avoid writing the proof (if anyone needs it to give a better answer I will write it, just prefer to avoid this if not necessary).
You have:
$N_{epsilon}$ a closed $epsilon$-neighborhood of $M$
$r$: $N_{epsilon} to M$, a differentiable map that maps $x$ to the point in $M$, closest to $x$. (Making $epsilon $ small enough this works well)
$w$ a vector field in $N_{epsilon}$ that extends $v$, given by: $w(x)= (x-r(x))+v(r(x))$
So.. What I think Milnor is trying to do, is to use Hopf's lemma (Lemma 3 in the book),you can see that $w$ points outwards along the boundary and if it has a zero, it must be a zero of $v$ (since $x-r(x)$ and $v(r(x))$ are orthogonal), so all its zeros are isolated, then you are in condition to apply Hopf's lemma.
Now he states that: For any $z in M$
$d_zw(h)= d_zv(h) $ in $T_zM$
and
$d_zw(h)=h$ in $T_zM$'s orthogonal complement.
So... maybe it is easy to see, but I cannot see why this is. Trying to calculate $frac{partial w_i}{partial x_j}$ for arbitrary $i,j$ seems usless, since I do not know how to calculate the derivative of $r(x)$. This is my first doubt.
My second doubt.. Supposing that the previous statement is true, do you have that $d_zw$ and $d_zv$ have the same determinant at any $z$, zero of $w$, hence the same index. Now Milnor uses Hopf's Lemma, to prove that the index sum of $w$ is equal the Gauss mapping. To finish this off I need to see that $v$ has the same zeros than $w$, why is this true? (You know that a zero of $w$ is a zero of $v$, why is the reciprocal true?)
I am really stuck with this, any help would be appreciated, thanks in advance.
(FROM HERE ON I EDITED IT)
So... I am going to answer my second doubt: If $z$ is a zero of $v$, then $z in M$ so $r(z)=z$ and thus $w(z)=v(z)=0$
And for my first doubt, I was thinking that I never used that $v$ has non-degenerate zeros (Or so I think), so maybe that is a hint to anwering my first doubt, cannot seem to see it though. If I had used that $v$ has only non-degenerate zeros, please tell me where. Thanks in advance.
differential-topology
edited Dec 18 '18 at 20:17
Card_Trick
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
$endgroup$
(THIS QUESTION HAS BEEN EDITED)
I am reading Milnor's ''Topology From a Differentiable Viewpoint'' and in the chapter about vector fields, page 38, there is a theorem that states that:
Given any vector field $v$ on $M subset mathbb{R^n}$, ($M$ an m-dimensional, compact boundaryless manifold) with only nondegenerate zeros, then the index sum of $v$ is equal to the degree of the Gauss mapping.
So, I am going to avoid writing the proof (if anyone needs it to give a better answer I will write it, just prefer to avoid this if not necessary).
You have:
$N_{epsilon}$ a closed $epsilon$-neighborhood of $M$
$r$: $N_{epsilon} to M$, a differentiable map that maps $x$ to the point in $M$, closest to $x$. (Making $epsilon $ small enough this works well)
$w$ a vector field in $N_{epsilon}$ that extends $v$, given by: $w(x)= (x-r(x))+v(r(x))$
So.. What I think Milnor is trying to do, is to use Hopf's lemma (Lemma 3 in the book),you can see that $w$ points outwards along the boundary and if it has a zero, it must be a zero of $v$ (since $x-r(x)$ and $v(r(x))$ are orthogonal), so all its zeros are isolated, then you are in condition to apply Hopf's lemma.
Now he states that: For any $z in M$
$d_zw(h)= d_zv(h) $ in $T_zM$
and
$d_zw(h)=h$ in $T_zM$'s orthogonal complement.
So... maybe it is easy to see, but I cannot see why this is. Trying to calculate $frac{partial w_i}{partial x_j}$ for arbitrary $i,j$ seems usless, since I do not know how to calculate the derivative of $r(x)$. This is my first doubt.
My second doubt.. Supposing that the previous statement is true, do you have that $d_zw$ and $d_zv$ have the same determinant at any $z$, zero of $w$, hence the same index. Now Milnor uses Hopf's Lemma, to prove that the index sum of $w$ is equal the Gauss mapping. To finish this off I need to see that $v$ has the same zeros than $w$, why is this true? (You know that a zero of $w$ is a zero of $v$, why is the reciprocal true?)
I am really stuck with this, any help would be appreciated, thanks in advance.
(FROM HERE ON I EDITED IT)
So... I am going to answer my second doubt: If $z$ is a zero of $v$, then $z in M$ so $r(z)=z$ and thus $w(z)=v(z)=0$
And for my first doubt, I was thinking that I never used that $v$ has non-degenerate zeros (Or so I think), so maybe that is a hint to anwering my first doubt, cannot seem to see it though. If I had used that $v$ has only non-degenerate zeros, please tell me where. Thanks in advance.
differential-topology
differential-topology
edited Dec 18 '18 at 20:17
Card_Trick
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
edited Dec 18 '18 at 20:17
Card_Trick
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
edited Dec 18 '18 at 20:17
Card_Trick
935
edited Dec 18 '18 at 20:17
Card_Trick
935
edited Dec 18 '18 at 20:17
Card_Trick
935
935
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
asked Nov 25 '18 at 21:43
Bajo FondoBajo Fondo
405215
405215
add a comment |
add a comment |
0
active
oldest
votes
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%2f3013454%2fproblem-understanding-milnor-proof-theorem-1-vector-fields-chapter%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3013454%2fproblem-understanding-milnor-proof-theorem-1-vector-fields-chapter%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
